Conceptual Model

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 h = c x + zo z Mathematical model x Analytical Solution Numerical Solution continuous solution discrete solution

7. 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 W&A) z x continuous solution discrete solution

8. 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 W&A) z hi,j =(hi+1,j +hi-1,j +hi,j+1 +hi,j-1)/4 x continuous solution discrete solution

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

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

11. Hingeline Add a water balance & compute water balance error

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

13. No Flow Boundary x (x/2) Mesh centered grid: area needed in water balance 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 Steady state mass balance eqn. q = - Kgrad h Darcy’s law z q equipotential line grad h q grad h x Isotropic Anisotropic Kx = Kz Kx Kz

18. div q = 0 steady state mass balance eqn. q = - Kgrad h Darcy’s law

19. div q = 0 steady state mass balance eqn. q = - Kgrad 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’ bedding planes x’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z

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

23. q = - Kgrad h Kxx Kxy Kxz KyxKyy Kyz Kzx KzyKzz K = K is a tensor with 9 components Kxx ,Kyy, Kzz are the principal components of K

24. q = - Kgrad 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’ bedding planes x’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z

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

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