Flow Regimes - Cased-Hole Log Analysis and Reservoir Performance Monitoring - page 55

Fig. 4.5 Slip velocities. Courtesy Schlumberger (data from Zuber & Findlay 1965)

Figure 4.6 shows a section of pipe with internal cross-sectional area A, in which
a mixture of oil and water is flowing. The velocity of the water is V w . The velocity
of the oil is V o . Since the oil is traveling faster by an amount V s , the slip velocity, we
may write

=+.
The flow rates are q w and q o . If we define y w as the fraction of the cross-sectional
area taken up by the water, we may write:

VVV
o w s

qyAV
w

= ,

w w

= (

)

q 1- y AV ,

o w o

=+ (

) .

qqq
t

totalflow

o w

Combining all these equations it is possible to derive an expression for y w :

( ) ± ( ) +

2

AV q - q - AV 4 qAV

s t t s ws

.
That is, we find that y w , the fraction of the pipe area actually occupied by water, is a
quadratic function of the actual water cut. Figure 4.7 shows a plot of y w , the water
holdup, against total flow as a function of water cut for flow in a 6-in. ID pipe. Note
that, for example, with a 50 % water cut at a flow rate of 200 B/D some 90 % of the
pipe area is in fact occupied with water.

y =

w

2 AV

s