2. General 3D equation used in MODFLOW Block centered grid xixi i-1 i i+1  x i-1/2  x i+1/2

4. K values in the space between nodes is calculated using the harmonic mean

5. The default boundary condition is no flow. (active nodes)

6. No flow boundary K=0 Imaginary nodes (i,j,k) (i+1) = 0

8. MODFLOW Grid

9. Conductance Darcy’s law

10. In general, conductance = KA/  L Also: leakance = K v /thickness resistance = thickness/K v (resistance is an analytic element term)

11. Vertical Conductance Terms VCONT = K v /  z = Conductance/(  x  y) = “leakance” MODFLOW uses special terms (VCONT) for vertical conductance, which is required in 3D problems. Modflow.exe requires the user to supply values for the VCONT arrays. GW Vistas computes the values in the VCONT arrays for you.

12. Vertical conductance is defined using leakance, or VCONT in MODFLOW language. We also have horizontal conductance along rows and along columns.

13. See MODFLOW manual (p. 2-19) or 2 nd ed. of textbook by Domenico & Schwartz (Eqn. 7.15) for MODFLOW’s FD equation

14. Types of Layers (LAYCON array) Confined Unconfined Convertible (Useful to think in terms of a layer transmissivity.)

15. Conductance Confined layer transmissivity = (KR i,j-1/2,k )(  v k ) Unconfined layer transmissivity = (KR i,j-1/2,k )(h i,j,k -BOT i,j,k ) h datum bot layer 1 layer 2

16. MODFLOW asks for values of horizontal grid spacing (i.e.,  r and  c) but not vertical grid spacing (  v). Vertical grid spacing (  v) is calculated by MODFLOW from user supplied values of the bottom (BOT) of each layer. (GwVistas asks for both top and bottom of each layer.) TOP Input of Grid Spacing BOT

17. Note that a top unconfined layer actually has no top. The head value in the top layer could potentially rise to infinity. Land surface elevation is not used in MODFLOW, except in the ET package.

18. Examples of solution techniques that combine matrix solution with iteration: IADI (see chapter 5 of W&A) SSOR* SIP* PCG2* *Used in MODFLOW

19. Schematic of solution procedure tt Note: m is time level and n is iteration level. m-1 m m+1

20. The 3D eqn. with an implicit approximation generates a coefficient matrix with 7 diagonals

21. Coefficient Matrix for a model with 2 rows, 3 columns, 2 layers. Matrix is sparse and symmetric.

22. SSOR

23. The coefficient matrix has 5 diagonals Coefficient matrix is Symmetric and banded

24. From MODFLOW manual

25. & PCG2 required 1 is required head dependent BCs MODFLOW88/96 Packages

26. Input assembly for the BAS Package