Page 201 - Fundamentals of Reservoir Engineering

STABILIZED INFLOW EQUATIONS 139

Combining these two results in equ. (6.11), and including the mechanical skin factor,
results in the modified inflow equation

qµ r 3
pp wf = ln e − + S (6.12)
−
2kh r w 4
π
6.3 STEADY STATE SOLUTION

The steady state solution of the diffusitivy equation can be derived using precisely the
same mathematical steps as for the semi-steady state solution only, in this case, since
∂p/∂t = 0, the diffusivity equation is reduced to
1 ∂ r ∂ p = 0 (6.13)
r
r ∂ r ∂

which is the radial form of the Laplace equation. Because of the simple form of
equ. (6.13) the mathematics involved in obtaining inflow equations expressed in terms
of p e and p is somewhat easier than in the previous section. The derivation of these
equations will therefore be left as an exercise for the reader. The solutions of the radial
diffusivity equation for both steady state and semi-steady state flow conditions are
summarised in table 6.1.

STEADY STATE SEMI-STEADY STATE

2
General relationship pp = qµ In r pp qµ ln r r
−
2
between p and r wf 2kh w r − wf = 2kh r w − 2r e
π
π

Inflow equations p − p = qµ In r e qµ r e 1
expressed e wf 2kh r w p − p wf = 2kh ln r w − 2
e
π
π
in terms of p = p e at r = r e
Inflow equations qµ r e 1 qµ r e 3
−
−
expressed in terms of pp wf = 2kh ln r w − 2 pp wf = 2kh ln r w − 4
π
π

the average pressure
TABLE 6.1
Radial inflow equations for stabilized flow conditions
qµ
N.B. To express in field units (stb/d, psi, mD, ft.) the term should be replaced by
2kh
π
141.2q B o , in each of the equations in table 6.1 In addition the mechanical skin factor
µ
kh
can be included in the equations as shown in equs. (4.27) and (6.7).
As an alternative, the skin factor can be accounted for in the inflow equations by
artificially changing the wellbore radius. For example, including the skin factor,
equ. (6.12) can be expressed as