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1D, transient, homogeneous, isotropic, confined, no sink/source term

Governing Eqn. for Reservoir Problem. 1D, transient, homogeneous, isotropic, confined, no sink/source term. Explicit solution Implicit solution. Explicit Approximation. Explicit Solution. Eqn. 4.11 (W&A). Everything on the RHS of the equation is known.

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1D, transient, homogeneous, isotropic, confined, no sink/source term

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

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

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

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

  9. Water Balance Storage = V(t2)- V(t1) 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 out Flow in Storage Storage

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

  11. h1 h2 datum x 0 L = 100 m At t = tss the system reaches a new steady state: h(x) = ((h2 –h1)/ L) x + h1 (Eqn. 4.12 W&A) spreadsheet

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