1. 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.

3. 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)

4. Mathematical Model of the Toth Problem
h = c x + zo
Laplace Equation
2D, steady state

5. 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)

6. Toth Problem z x Analytical Solution Numerical Solution
h = c x + zo
Mathematical
model x
Analytical Solution
Numerical Solution
continuous solution
discrete solution

7. Toth Problem z x z x Analytical Solution Numerical Solution
h = c x + zo
Mathematical
model x
Analytical Solution
Numerical Solution
h(x,z) = zo + cs/2 – 4cs/2 …
(eqn. 2.1 in W&A) z x
continuous solution
discrete solution

8. Toth Problem z x z x Analytical Solution Numerical Solution
h = c x + zo
Mathematical
model x
Analytical Solution
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

9. Hinge line
Add a water balance
& compute water
balance error
Example of spreadsheet formula

10. OUT IN
Q= KIA
Hinge line
OUT – IN = 0

11. Hinge line
Add a water balance
& compute water
balance error

12. Q = KIA=K(h/z)(x)(1) A
x=z
 Q = K h z x z x
1 m

13. Mesh centered grid: area needed in water balance
(x/2) x
No Flow
Boundary x
(x/2)
water
table
nodes

14. x=z
 Q = K h

15. Block centered grid: area needed in water balance
No flow boundary x x
water
table
nodes

16. K as a Tensor

17. div q = 0 q = - K grad h Steady state mass balance eqn. Darcy’s law z
equipotential line
grad h q
grad h x
Isotropic
Anisotropic
Kx = Kz
Kx  Kz

18. div q = 0 q = - K grad h steady state mass balance eqn. Darcy’s 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)

19. (homogeneous and isotropic conditions)
div q = 0
steady state mass balance eqn.
q = - K grad h
Darcy’s law
Assume K = a constant
(homogeneous and isotropic conditions)
Laplace Equation

20. Governing Eqn. for TopoDrive
2D, steady-state, heterogeneous, anisotropic

21. global local z z’ x’ x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0
bedding planes x’  x
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K’x
0 K’y 0
K’z

22. q = - K grad h
Kxx
0 Kyy 0
Kzz qx qy qz
= -

23. 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

24. q = - K grad h
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz qx qy qz
= -

26. This is the form of the governing equation used in MODFLOW.

27. global local z z’ x’ x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0
bedding planes x’  x
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K’x
0 K’y 0
K’z

28. z local global z’ grad h q q’ x’ Kz’=0 x 
Assume that there is no flow across impermeable bedding planes z
local
global z’
grad h q q’ x’
Kz’=0  x

30. [K] = [R]-1 [K’] [R] global local z z’ x’ q q’ x Kxx Kxy Kxz
bedding planes x’ q q’  x
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K’x
0 K’y 0
K’z
[K] = [R]-1 [K’] [R]