where
( fm - 1)qg
M=
.
Xo L
The time to complete melt is
Xo L ln fm
tm =
.
(B10)
qg ( fm - 1)
This leads to the value of δ when melt is completed.
3L ln fm
Xo
δo =
+
- 1 .
(B11)
GCu fm - 1
2
We define a linear temperature distribution that will have the same sensible heat when thaw is completed.
Ti′ = Tf + G′x
(B12)
Xo
δo -
G′
3.
=
(B13)
δo + Xo
G
where G' is the equivalent geothermal gradient. Finally, the new temperature distribution at the beginning
of freeze is
Ti = Tf + bo x + co x 2
(B14)
where
Table B1. Melt relations for Prudhoe Bay.
G(δo - Xo )
GXo
G′
co =
bo =
,
.
ln fm
δο
(δo + Xo )
Melt time
-1
δo + Xo
2
~
fm - 1
f
(m)
(years)
fm
G
0.1
0.55
1.5584
7948.1
0.9064
108,926
This initial temperature distribution is shown as curve a
0.2
0.60
1.0118
5265.6
0.8636
85,654
in Figure 12. Table B1 shows some results for Prudhoe
0.3
0.65
0 .720
3833.5
0.8196
73,230
Bay. Note the long melt times even if f is as high as
0.4
0.70
0.5272
2887.3
0.7706
65,022
90%.
0.5
0.75
0.3863
2195.8
0.7139
59,023
0.6
0.80
0.2771
1659.9
0.646
54,373
0.8
0.90
0.1157
867.8
0.455
47,502
Freeze of cooled soil
Xo = 600 m, L = 30.21 cal/cm3, Cu = 0.6457 cal/cm3 C.
The freezing process is as discussed earlier except
qg = 1.35 106 cal/cm2 s, G = 0.0286C.
that the initial soil temperature is lowered as noted in
Figure 10. The basic equations used earlier are still valid
except that the coefficients of eq 6 and 7 change, owing to the new initial temperature given by eq B14.
The basic equation, replacing eq 9, is
1
1
dFσ
= k21 - 2 -
(B15)
dτ
σ
g
β2
β2 β 1 σSo
1 1 ρ21
(β - 1)(β + 1)2
- - -
- C21σ
F = -
++
- mo
(B16)
6g 3 ST
6 3 2
6
3
2ρ (g - 1)β
[
]
β
- k21σ -β + 2mo (β + 1) + 2σSo (β + 1) = 21
2
(B17)
ST
g
26
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