Review

4. Review

5. 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. May include information on water chemistry.

6. Mathematical Model a set of equations that describes the physical and/or chemical processes occurring in a system.

7. Derivation of the Governing Equation Q R x y q z x y • Consider flux (q) through REV • OUT – IN = - Storage • Combine with: q = -Kgrad h

8. General 3D equation 2D confined: 2D unconfined w/ Dupuit assumptions: Storage coefficient (S) is either storativity or specific yield. S = Ss b & T = K b

9. Types of Boundary Conditions • Specified head • Specified flow (including no flow) • Head-dependent flow

10. From conceptual model to mathematical model…

11. Toth Problem Water table forms the upper boundary condition h = c x + zo Laplace Equation 2D, steady state Cross section through an unconfined aquifer.

12. “Confined” Island Recharge Problem We can treat this system as a “confined” aquifer if we assume that T= Kb. Areal view Water table is the solution. R h datum groundwater divide Poisson’s Eqn. ocean ocean b x = - L x = 0 x = L 2D horizontal flow through an unconfined aquifer where T=Kb.

13. Unconfined version of the Island Recharge Problem (Pumping can be accommodated by appropriate definition of the source/sink term.) Water table is the solution. R groundwater divide h ocean ocean b datum x = - L x = 0 x = L 2D horizontal flow through an unconfined aquifer under the Dupuit assumptions.

14. Vertical cross section through an unconfined aquifer with the water table as the upper boundary. 2D horizontal flow in a confined aquifer; solution is h(x,y), i.e., the potentiometric surface. 2D horizontal flow in an unconfined aquifer where v= h2. Solution is h(x,y), i.e., the water table. All three governing equations are the LaPlace Eqn.

15. Reservoir Problem t = 0 t > 0 datum x 0 L = 100 m BC: h (0, t) = 16 m; t > 0 h (L, t) = 11 m; t > 0 IC: h (x, 0) = 16 m; 0 < x < L (represents static steady state) 1D transient flow through a confined aquifer.

16. Solution techniques…

17. Three options: • Iteration • Direct solution by matrix inversion • A combination of iteration and matrix solution

18. Examples of Iteration methods include: Gauss-Seidel Iteration Successive Over-Relaxation (SOR)

19. Let x=y=a

20. Gauss-Seidel Formula for 2D Laplace Equation General SOR Formula Relaxation factor = 1 Gauss-Seidel < 1 under-relaxation >1 over-relaxation, typically between 1 and 2 (e.g., 1.8)

21. Gauss-Seidel Formula for 2D Poisson Equation (Eqn. 3.7 W&A) SOR Formula Relaxation factor = 1 Gauss-Seidel < 1 under-relaxation >1 over-relaxation

22. solution Iteration for a steady state problem. m+3 Iteration levels m+2 m+1 m (Initial guesses)

23. Transient Problems require time steps. Steady state n+3 t Time levels n+2 t n+1 t n Initial conditions (at steady state)

24. Explicit Approximation Implicit Approximation Or weighted average

25. Explicit solutions do not require iteration but are unstable with large time steps. • We can derive the stability criterion by writing • the explicit approx. in a form that looks like the SOR • iteration formula and setting the terms in the • position occupied by omega equal to 1. • For the 1D governing equation used in the reservoir • problem, the stability criterion is: < < or

26. Implicit solutions require iteration or direct solution by matrix inversion.

27. n+1 t m+3 Iteration planes m+2 m+1 n Solution by iteration

28. Modeling “Rules” • Boundary conditions always affect • a steady state solution. • Initial conditions should be selected to represent a steady state configuration of heads.