1. Finite Difference Method

2. Derivation of 3-D GW Flow Equation from Darcy’s Lawz y
Mass In - Mass Out = Change in Storage

3. Steady State Flow
Replace qx, qy, and qz with Darcy using Kx, Ky, and Kz
Divide out constant , and assume Kx= Ky= Kz = K

4. Transient Flow
h is related to q through the soil water characteristic curve

5. Linear Second-order PDEs
Linear second-order PDEs are of the form
where A - H are functions of x and y only
Elliptic PDEs: B2 - AC < 0
(steady state equations)
Parabolic PDEs: B2 - AC = 0
(transfer equations)
Hyperbolic PDEs: B2 - AC > 0
(wave equations)

6. Difference vs Differential

7. Formulas for 1st, 2nd Derivatives

8. y x Discretization of the solution domain Vertical (j index). . . 5 4
Vertical
(j index) 3 2 1 1 2 3 4 5 . . .
N-2
N-1 N x
Horizontal
(i index)

9. (i,j+1)
(i,j)
(i-1,j)
(i+1,j)
(i,j-1)

10. Type 1 (Dirichlet) Boundary ConditionM
M-1
M-2 . . . 5 4 3 2 1 1 2 3 4 5 . . .
N-2
N-1 N
Variable Specified on Boundary

11. Type 2 (Neumann) Boundary Condition. . . 5 4 3 2 1 1 2 3 4 5 . . .
N-2
N-1 N
Derivative of Variable Specified on Boundary (usually zero)

12. Variable Conductivity
Steady State Flow
Variable Conductivity K2 K1 Dx Dx Dx

13. K2 K1

14. Steady State Flow

15. Steady State Flow
Steady State Flow
Single Conductivity