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Where, T sea is seawater temperature, °C; k m is thermal conductivity of drilling fluid,
W/(m•°C). Subscript: r stands for riser.
(3) Control equation for drill string section below static mud
π
4
D
2
pi ρ m C m
∂T p
∂t
+ ρ m C m Q m
∂T p
∂z
−
T a − T p
1
π D po h po
+
1
π D pi h pi
= Q m P p
(3)
(4) The governing equation of the annular segment below the seabed mud line
π
4
(D 2
ai − D 2
po )ρ m C m
∂T a
∂t
+ ρ m C m Q m
∂T a
∂z
−
T f − T a
1
π D po h po
+ 1
2π k f
ln
D h
D ao
+
1
π D ai h ai
= Q m P a (4)
Where, T f represents formation temperature, °C; k f represents the thermal conductivity of the formation, W/(m•°C); D h represents the outer diameter of the cement ring,
m.
(5) The governing equation of the riser annulus below the static mud
π
4
(D 2
ri − D 2
po )ρ m C m
∂T a
∂t
+ ρ m C m Q m
∂T a
∂z
−
T sea − T a
1
π D po h po
+ 1
2π k r
ln
D ro
D ri
+
1
π D ri h ri
= Q m P a (5)
Where, k r represents the heat conductivity of riser, W/(m•°C)
(6) Governing equation of annular section at static mud
π
4
(D
2
ri − D
2
po )ρ m C m
∂T a
∂t
+ ρ m C m Q m
∂T a
∂z
−
T sea − T a
1
2π k r
ln
D ro
D ri
= Q m P a
(6)
(7) Mud return line control equation
π
4
D
2
bi ρ m C m
∂T b
∂t
+ ρ m C m Q m
∂T b
∂z
−
T sea − T b
1
π D bo h bo
+
1
2π k b
ln
D bo
D bi
+
1
π D bi h bi
= Q m P b
(7)
Where, T b represents the return pipeline temperature, °C; k b represents the thermal
conductivity of the return pipeline, W/(m•°C). Subscript: b represents the return line.
2.2.3 Auxiliary Equation
(1) Pressure loss calculation formula
P =
2f ρ m υ 2
m
D
(8)
Where, f is the friction factor, dimensionless; υ m is the flow velocity, m/s. D is
hydraulic diameter, m.
(2) The calculation formula of convective heat transfer coefficient
Laminar: h =
4.12k m
D
Turbulent: h =
0.027Re 0.8 Pr 0.033 k m
D
(9)