Review

1. Review

2. 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.

3. Mathematical Model a set of equations that describes the physical and/or chemical processes occurring in a system. • Governing equation • Boundary conditions • Initial conditions for transient simulations

4. 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

5. div q = 0 Steady state mass balance eqn. q = - Kgrad h Darcy’s law z q equipotential line grad h q grad h x Isotropic Anisotropic Kx = Kz Kx Kz

6. global local z z’ bedding planes x’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z

7. q = - Kgrad h Kxx Kxy Kxz KyxKyy Kyz Kzx KzyKzz qx qy qz = -

8. General 3D equation Hetergeneous, anisotropic, transient, sink/source term 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 (including constant 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. Governing Eqn. for TopoDrive 2D, steady-state, heterogeneous, anisotropic

13. “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.

14. 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.

15. 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.

16. Reservoir Problem t = 0 confining bed 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.

17. Solution techniques… • Analytical solutions • Numerical solutions • finite difference (FD) methods • finite element (FE) methods • Analytic element methods (AEM)

18. Toth Problem h = c x + zo z Mathematical model x Analytical Solution Numerical Solution h(x,z) = zo + cs/2 – 4cs/2 … (eqn. 2.1 in W&A) z hi,j =(hi+1,j +hi-1,j +hi,j+1 +hi,j-1)/4 x continuous solution discrete solution

19. 200 ft Grid Design h = 100 110 90 ft Toth Problem mesh vs block centeredgridsanother view x = y = a = 20 ft

20. Three options for solving the set of algebraic equations • that result from applying the method of FD or FE: • Iteration • Direct solution by matrix inversion • A combination of iteration and matrix solution Note: The explicit solution for the transient flow equation is another solution technique, but in practice is never used.

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

22. Let x=y=a

23. 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

24. 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

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

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

27. Explicit Approximation Implicit Approximation

28. In general: where  = 1 for fully implicit  = 0.5 for Crank-Nicolson  = 0 for explicit

29. 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 Note that critical t value is directly dependent on grid spacing,  x.

30. Implicit solutions require iteration or direct solution by matrix inversion or a combination of iteration and matrix inversion.

31. n+1 t m+3 Iteration planes m+2 m+1 Solution of an Implicit transient FD equation by iteration n

32. Modeling rules/guidelines • Boundary conditions always affect • a steady state solution. • Initial conditions should be selected to represent a steady state configuration of heads. • In general, the accuracy of the numerical • solution improves with smaller grid spacing • and smaller time step • A water balance should always be included • in the simulation.

33. At steady state: h /t = 0 and V /t = 0

34. Implicit Solution t = 5 Early time Note relatively large change in storage

35. At late timestorage is small time inflow Inflow+storage