q K grad h grad is a vector

q = - K grad h grad is a vector Darcy’s law

q is a vector Transient mass balance equation:

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

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

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

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

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

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

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

Iteration indices Solution by iteration Gauss-Seidel Iteration

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

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

1 1 2 3 10 9 8 10 2 4 7 6 9. 75 3 10 5 4 10 5 10 6 j 9 i 8 7 Example with specified head boundary conditions

1 1 2 3 10 9 8 10 2 9. 75 4 7 6 8. 44 3 10 5 4 10 5 10 6 j 9 i 8 7 Example with specified head boundary conditions

1 1 2 3 10 9 8 9. 75 8. 44 10 2 9. 94 4 7 6 3 10 4 10 5 10 6 j 9 5 8 i 7 Example with specified head boundary conditions

1 1 2 3 10 9 8 10 2 4 7 6 9. 94 3 10 4 10 5 10 6 j i 9 5 8 7 Example with specified head boundary conditions

i j i, j Gauss-Seidel Iteration

C 4=(C 3+C 5+B 4+D 4)/4

Example of spreadsheet formula Toth Problem