where

( fm - 1)qg

.

The time to complete melt is

.

(B10)

This leads to the value of δ when melt is completed.

3*L * ln *f*m

δo =

+

- 1 .

(B11)

2

We define a linear temperature distribution that will have the same sensible heat when thaw is completed.

(B12)

δo -

3.

=

(B13)

δo + *X*o

where *G' *is the equivalent geothermal gradient. Finally, the new temperature distribution at the beginning

of freeze is

(B14)

where

G′

,

.

δο

(δo + *X*o )

-1

δo + *X*o

2

~

fm - 1

f

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

The freezing process is as discussed earlier except

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

= *k*21 - 2 -

(B15)

σ

β2

β2 β 1 σ*S*o

1 1 ρ21

(β - 1)(β + 1)2

- - -

- *C*21σ

++

- *m*o

(B16)

6*g * 3 * S*T

6 3 2

6

3

2ρ (g - 1)β

[

]

β

- *k*21σ -β + 2*m*o (β + 1) + 2σ*S*o (β + 1) = 21

2

(B17)

26

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