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R. Zhang et al.
1.1 Inside the Upper Part of Drill String
π
4
d
2
pi
∂(ρ u
ml c u
ml T p )
∂t
= π d pi h
u
po (T a − T pi ) − (Q m + Q hgs )
∂(ρ u
ml c u
ml T p )
∂z
+S
u
cp
(A-1)
If make A = π d 2
pi (ρc) u
ml /4, B = π d pi h u
pi , C = (ρcQ) u
ml , α ∗ = Q hgs /(Q hgs + Q m )
So, the above equation can be written as
A
T n
i,p − T
n−1
i,p
t
= B(T
n
i,a − T
n
i,p ) − C
T n
i,p − T n
i−1,p
z
+ (S cp )
n
i
(A-2)
Then, make a = A 1 /4t, b = B, c = C 1 //z
Thus, the above equation can be changed into:
(a + b + c)T
n
i,1 − bT
n
i,2 − cT
n
i−1,1 = aT
n−1
i,1 + (S cp )
n
i
(A-3)
1.2 Inside Lower Part of Drill String
π
4
d
2
pi
∂(ρ d
ml c d
ml T p )
∂t
= π d pi h
d
po (T a − T pi ) − (Q m + Q hgs (1−ε))
∂(ρ d
ml c d
ml T p )
∂z
+S
d
cp
(A-3)
If make A 1 = π d 2
pi (ρc) d
ml /4, B 1 = π d pi h d
pi , C 1 = (ρcQ) d
ml , α ∗ =Q hgs ∗
(1−ε)/(Q hgs ∗ (1−ε) + Q m )
So, the above equation can be written as
A 1
T n
i,p − T
n−1
i,p
t
= B 1 (T
n
i,a − T
n
i,p ) − C 1
T n
i,p − T n
i−1,p
z
+ (S cp )
n
i
(A-5)
To make a 1 = A 1 //t, b 1 =B 1 , c 1 = C 1 //z
Thus, the above equation finally changes to
(a 1 + b 1 + c 1 )T
n
i,1 − b 1 T
n
i,2 − c 1 T
n
i−1,1 = a 1 T
n−1
i,1 + (S cp )
n
i
(A-6)
2. In Annulus
2.2 In the Upper Part of Annulus
π
4
(d
2
w − d
2
po )
∂(ρ u
ml c u
ml T p )
∂t
= π d pi h
u
po (T a − T pi ) − (Q m + Q hgs ε))
∂(ρ u
ml c u
ml T p )
∂z
+S
u
cp
(A-7)
If make A = π(d 2
w − d 2
po )(ρc) u
ml /4, B = π d po h u
po , C = π d w h u
w , D = (ρcQ) u
ml ,
α ∗ = Q hgs ∗ ε/(Q hgs ∗ ε + Q m )
Then the above equation can be changed into
A
T n
i,2 − T
n−1
i,2
t
= B(T
n
i,1 − T
n
i,2 ) + C(T
n
i,3 − T
n
i,2 ) + D
T n
i,2 − T n
i−1,2
z
+ (S ca )
n
i
(A-8)
To make a = A 1 //t, b = B, c = C, d = D//z
(a + b + c + d )T
n
i,1 − bT
n
i,3 + dT
n
i−1,2 = aT
n−1
i,2 + (S ca )
n
i
(A-9)