Page 175 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 113

q = constant

pressure
p = constant
e
q

p wf

r w r r e

Fig. 4.4 The radial flow of oil into a well under steady state flow conditions

In addition, for simplicity, the reservoir will be assumed to be completely homogeneous
in all reservoir parameters and the well perforated across the entire formation
thickness.

Under these circumstances, Darcy's law for the radial flow of single phase oil can be
expressed as

kA dp
q = (4.23)
µ dr

Since the flow rate is constant, it is the same across any radial area, A = 2πrh, situated
at distance r from the centre of the system. Therefore, equ. (4.23) can be expressed as
2rkh dp
π
q =
µ dr

and separating the variables and integrating

p qµ r dr
dp =
π
p wf 2kh w r r

where p wf is the conventional symbol for the bottom hole flowing pressure. The
integration results in

qµ r
pp = ln (4.24)
−
wf
2kh r w
π
which shows that the pressure increases logarithmically with respect to the radius, as
shown in fig. 4.4, the pressure drop being consequently much more severe close to the
well than towards the outer boundary. In particular, when r = r e then

qµ r
p − p wf = ln e (4.25)
e
2kh r w
π
When a well is being drilled it is always necessary to have a positive pressure
differential acting from the wellbore into the formation to prevent inflow of the reservoir
fluids. Because of this, some of the drilling mud will flow into the formation and the