Page 202 - Fundamentals of Reservoir Engineering

STABILIZED INFLOW EQUATIONS 140

qµ r 3
pp wf = ln e − (6.14)
−
2kh r′ w 4
π
in which

r′ = r e −s (6.15)
w
w
is the effective wellbore radius due to the presence of skin. If the formation is
damaged, so that the permeability close to the well is reduced, the skin factor is
positive. If, however, the well has been stimulated, for instance by acidising, then the
permeability close to the well can exceed the average formation permeability and the
skin factor is then negative. In either case the magnitude and sign of the skin factor can
be determined from pressure buildup analysis as will be described in Chapter 7, sec. 7.

6.4 EXAMPLE OF THE APPLICATION OF THE STABILIZED INFLOW EQUATIONS

The solution of the diffusivity equation under semi-steady state flow conditions has
been described in detail in section 6.2 since the mathematical approach is quite
general and can be applied to more complex radial flow problems. Consider, for
instance, the case of a well which has been stimulated by steam soaking, refer
Chapter 4, sec. 7. In this type of stimulation several thousand tons of steam are
injected into the well and, upon re-opening, the well will produce at a greatly increased
rate. As a first approximation it will be assumed that, due to the steam injection, the
1
temperature distribution can be described by a temperature step function so that, for
r w < r < r h, the temperature T s is uniform and initially equal to the condensing steam
temperature at the sandface. During production, T s will decrease due to heat losses by
conduction and convection. For r > r h, the temperature is the original reservoir
temperature T r. The situation at any time during the production cycle is shown in
fig. 6.2,

p e
Pressure T, µ oh
s
T, µ oc p h
r
p wf

r
r w r h e
Fig. 6.2 Pressure profile during the steam soak production phase

where µ oh and µ oc are the viscosities of the oil at temperatures T s and T r, respectively. If
the inflow equations are formulated under steady state flow conditions, the result will
be as follows