Page 477 - Petrophysics

RADIAL FLOW SYSTEMS 445

For drainage areas with mixed outer boundaries and Ap = pe - pw, the
expression for p~ during pseudosteady state is [22]:

(7.83)

As noted earlier, after an extended time of production at a constant rate,
the bottom-hole flowing pressure, pw, becomes a linear function of time.
A Cartesian plot of pw vs. time should yield a straight line with slope m*.
For bounded drainage areas:

(7.84)

where m* is expressed in psi/hr. The pore volume of the drainage area.
Vp(ft3), is:

Vp = OhA = nr:h@ (7.85)

Equation 7.84 is commonly used to calculate the pore volume, Vp, of a
bounded reservoir. For mixed-boundary systems, such as in reservoirs
under the influence of a partial water drive or in unbalanced injection
patterns, the pseudosteady-state flow regime occurs only for small values
of the drainage boundary index f. In this case, the slope of the straight
line portion that corresponds to pseudosteady state, m*, can be obtai-
ned from the derivative of Equation 7.83 with respect to dimensionless
time tDA:

apD = pb = 2n(l - f) (7.86)
at DA
Substituting for p~ and tDA, and solving explicitly for m* = dp/dt gives:

(7.87)

Equation 7.87 can be used to calculate Vp if f is known from pressure
transient testing [22]. If the pore volume is known from other sources,
then Equation 7.87 provides a way to calculate the index f. Note that for
f = 0, i.e., the drainage boundary is closed, Equations 7.84 and 7.87 are
identical. For the rare case where f = 1, the rate of change of pressure
with time, dp/dt, is zero, and steady-state flow becomes the dominant
regime.