1. 1D, transient, homogeneous, isotropic, confined, no sink/source term Explicit solution Implicit solution Governing Eqn. for Reservoir Problem

2. Explicit Approximation

3. Explicit Solution Eqn. 4.11 (W&A) Everything on the RHS of the equation is known. Solve explicitly for ; no iteration is needed.

4. Explicit approximations 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

5. Implicit Approx.

6. Solve for to produce the Gauss-Seidel iteration formula. Implicit Solution

7. Could also solve using SOR iteration. Gauss-Seidel value from previous slide.

8. tt Iteration planes n n+1 m+2 m+1 m+3

9. Water Balance  Storage = V(t 2 )- V(t 1 ) IN > OUT then  Storage is + OUT > IN then  Storage is – OUT - IN = -  Storage +- Convention: Water coming out of storage goes into the aquifer (+ column). Water going into storage comes out of the aquifer (- column). Flow in  Storage Flow out  Storage

10. Water Balance  V = S s  h (  x  y  z) tt tt  V = S  h (  x  y) tt tt In 1D Reservoir Problem,  y is taken to be equal to 1.

11. datum 0 L = 100 m x At t = t ss the system reaches a new steady state: h(x) = ((h 2 –h 1 )/ L) x + h 1 h2h2 h1h1 (Eqn. 4.12 W&A)