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Head controlled bottom boundary
∂Fn
∂hnj +1, p
=
Δz n
Δt j
−κ
C nj +1 + Δz n
∂S mj +,n1
+
∂hnj +1
K j +κ
K nj−+½κ
+ n+½
½ (Δz n−1 + Δz n ) ½ Δz n
j +κ
⎞
∂K nj−+½κ ⎛⎜ hnj−+11 − hnj +1
⎟ + κ ∂K n+½
1
+
∂hnj +1
∂hnj +1 ⎜⎝ ½ (Δz n−1 + Δz n ) ⎟⎠
⎛ hnj +1 − hbot
⎞
⎜
+ 1⎟
⎜ ½ Δz
⎟
n
⎝
⎠
Predefined groundwater levels
∂S mj +, n1*
∂K nj*+−κ½
K nj*+−κ½
∂Fi
Δzi j +1
=
+
−κ
C * + Δz n *
∂hnj*+1 Δt j n
∂hnj*+1 ½ ( Δz n * −1 + Δz n * )
∂hnj*+1
∂K nj*++κ½
K nj*++κ½
⎛ z * − z n * +1 ⎞
⎟
+⎜ n
+κ
⎜
⎟
∂hnj*+1
⎝ z n * − gwl ⎠ ½ (Δz n * + Δz n * +1 )
⎛ h j*+1 − h j*+1
⎞
⎜
n −1
n
+ 1⎟
⎜ ½ ( Δz * + Δz * ) ⎟
n −1
n
⎝
⎠
⎛⎛ z * − z * ⎞
⎞
hnj*+1
⎜⎜ n
n +1 ⎟
+ 1⎟
⎜ ⎜ z * − gwl ⎟ ½ (Δz * + Δz * ) ⎟
n
n +1
⎠
⎝⎝ n
⎠
Cauchy relation for the bottom boundary
∂Fn
∂hnj +1, p
=
Δz n
Δt j
−κ
Cnj +1 + Δz n
∂S mj +,n1
+
∂hnj +1
K nj−+½κ
K nj++½κ
+
½ (Δz n−1 + Δz n ) ½ Δz n + cK nj++½κ
j +κ
⎛ j +1
⎞
⎞
∂K nj−+½κ ⎛⎜ hnj−+11 − hnj +1
⎟ + κ ∂K n+½ ½ Δz n ⎜ hn − (ϕ − z n ) ⎟
1
+
⎜⎜
j +κ 2 ⎟
⎟
∂hnj +1
∂hnj +1 ⎜⎝ ½ (Δz n−1 + Δz n ) ⎟⎠
⎝ ½ Δz n + cK n+½ ⎠
(
)
Seepage face
hnj +1 + ½ Δz n < 0 →
∂F1
∂h1j +1
=
Δz1
Δt j
C1j +1 + Δz1
∂S mj +,11
∂h1j +1
+
∂K j +κ
K1j½+κ
+ κ 1j½+1
½ (Δz i + Δzi +1 )
∂h1
∂Fn
hnj +1 + ½ Δz n </ 0 →
∂hnj +1, p
=
Δz n
Δt j
−κ
C nj +1 + Δz n
⎛ h1j +1 − h2j +1
⎞
⎜
+ 1⎟
⎜ ½(Δz1 + Δz 2 ) ⎟
⎝
⎠
∂S mj +,n1
∂hnj +1
+
K nj−+½κ
K j +κ
+ n+½
½ (Δz n−1 + Δz n ) ½ Δz n
j +κ
⎞
∂K nj−+½κ ⎛⎜ hnj−+11 − hnj +1
⎟ + κ ∂K n+½
1
+
∂hnj +1
∂hnj +1 ⎜⎝ ½ (Δz n−1 + Δz n ) ⎟⎠
⎞
⎛ hnj +1
⎜
+ 1⎟
⎟
⎜ ½ Δz
n
⎠
⎝
Free drainage
∂Fn
∂hnj +1, p
=
Δz n
Δt j
−κ
Cnj +1 + Δz n
∂S mj +,n1
∂hnj +1
+
K nj−+½κ
K j +1
+ n+½
½ (Δz n−1 + Δz n ) ½ Δz n
j +1
⎞
∂K nj−+½κ ⎛⎜ hnj−+11 − hnj +1
⎟ + κ ∂K n+½
1
+
∂hnj +1
∂hnj +1 ⎜⎝ ½ (Δz n−1 + Δz n ) ⎟⎠
The internodal conductivity K nj++½1 is in the case of free drainage irrespective the value
of κ always treated implicitly
236
Alterra Report 1649 - 01