th

TS

xh

2

2

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

• Explicit solution • Implicit solution

Governing Eqn. for Reservoir Problem

thh

TS

xhhh n

ini

ni

ni

ni

1

211

)(2

Explicit Approximation

th

TS

xh

2

2

thh

TS

xhhh n

iniiii

1

211

)(2

thh

TS

xhhh n

ini

ni

ni

ni

1

211 2

Explicit Solution

))(

2( 2

111

xhhh

StThh

ni

ni

nin

ini

Eqn. 4.11(W&A)

Everything on the RHS of the equation is known.Solve explicitly for ; no iteration is needed.1n

ih

• Explicit approximations are unstable with large time steps.

• We can derive the stability criterion by writingthe explicit approx. in a form that looks like the SORiteration formula and setting the terms in theposition occupied by omega equal to 1.

• For the 1D governing equation used in the reservoirproblem, the stability criterion is:

1)(

22

xStT <

TxSt

2)(5.0 <or

thh

TS

xhhh n

ini

ni

ni

ni

1

2

11

111

)(2

Implicit Approx.

th

TS

xh

2

2

thh

TS

xhhh n

iniiii

1

211 2

thh

TS

xhhh n

ini

ni

ni

ni

1

2

11

111 2

Solve for 1nih to produce the Gauss-Seidel

iteration formula.

11)( mnih },)(,){( 1

111

1ni

mni

mni hhhfunction

Implicit Solution

Could also solve using SOR iteration.

])()[( )()( 1,

11,

1,

11,

mnji

mnji

mnji

mnji hhhh

Gauss-Seidel value fromprevious slide.

tIterationplanes

n

n+1

m+2

m+1

m+3

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 outof the aquifer (- column).

Flow in

Storage

Flow out

Storage

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.

datum

0 L = 100 mx

At t = tss the system reachesa new steady state:h(x) = ((h2 –h1)/ L) x + h1

h2

h1 02

2

xh

(Eqn. 4.12 W&A)