Mathematical Modeling of Steam Injection in Vertical Wells - Selected Topics of Computational and Experimental Fluid Mechanics - page 327

Fig. 1 Physical model and schematic representation of the well and porous medium discretization

Momentum balance equation

f m v m ˁ m
2 d ti g c

dP

dz =− ˁ m g cos

ʸ

+ ˁ m v m
g c

dv m
dz

− + (2)

g c

Energy balance equation

f p ˁ p u p +

v p
2 g c J c

Q loss

∂
∂ t

3600 xA ti +

p

h p +

(3)

v p
2 g c J c

ˁ p v sp g cos ʸ

=− ∂ z ˁ p v sp +

g c J c

p p

where the subscripts g, l and m represent gas, liquid and mixture, respectively, P is
pressure (which is the saturation pressure P sat )

, ˁ is density, v s is superficial velocity,

g is the gravity acceleration, is the local angle between the well and the vertical

direction, g c is the gravitational conversion constant, d ti is the inner diameter of the
tubing, f m is friction factor, v is velocity, Q loss is the fluid heat loss, A ti is the inner
tubing area, p is the phase index (liquid or gas), h is enthalpy, u is internal energy, f
is the gas volume fraction, J c is the mechanical equivalent of heat (788 ft-lbf/Btu),
and 3,600 converts hours to seconds.
The heat losses, from the water-steam mixture to the porous medium, are dimin-
ished by thermal resistances due to the wall well, insulation, annulus, casing and the
cementing wall (see Fig. 1 ), then we have that

ʸ