grad is a vector

1. q = - Kgrad h Darcy’s law grad is a vector

2. Transient mass balance equation: q is a vector

3. Steady State Mass Balance Equation with W = 0 gradq = 0   q = 0 (del notation) The dot product of grad and q is div q = 0

4. Anisotropic medium: K at a point varies with direction Kx Ky  Kz or Kx= Ky  Kz Heterogeneous medium: K varies in space

5. Characteristics of K in two dimensions A  Kx A B  B Kz Kx Kz Kx =Kz = K Homogeneous, isotropic Homogeneous, anisotropic B A A B Heterogeneous, anisotropic Heterogeneous, isotropic

6. Characteristics of K in two dimensions A  Kx A B  B Kz Kx =Kz = K Kx Kz Homogeneous, isotropic Homogeneous, anisotropic B A A B Heterogeneous, anisotropic Heterogeneous, isotropic

7. homogeneous, isotropic porous medium: Kx = Ky = Kz = K steady state conditions: W = 0 Laplace equation h = 0 2

8. Toth Problem (2D) Groundwater divide Groundwater divide Impermeable Rock 2D, steady state

9. i i, j-1 The 5 point star in a regular grid j i, j i-1, j i+1, j y i, j+1 x = y x

10. Iteration indices Solution by iteration Gauss-Seidel Iteration

11. Gauss-Seidel Iteration solution m+3 Iteration planes m+2 m+1 initial guesses m

12. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 2D example with specified head boundary conditions 10 5 10 5 (Assume an initial guess of h =10 for all interior nodes) 6 10 9 8 10 7 j

13. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 9.75 Example with specified head boundary conditions 10 5 10 5 6 10 9 8 10 7 j

14. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 9.75 10 6 Example with specified head boundary conditions 8.44 10 5 10 5 6 10 9 8 10 7 j

15. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 8.44 9.75 10 6 Example with specified head boundary conditions 9.94 10 5 10 5 6 10 9 8 10 7 j

16. 1 2 3 4 i 1 2 3 4 5 6 10 9 8 7 10 6 Example with specified head boundary conditions 9.94 10 5 10 5 6 10 9 8 10 7 j

17. i j i, j Gauss-Seidel Iteration

18. C4=(C3+C5+B4+D4)/4

19. Example of spreadsheet formula Toth Problem