Page 467 - Petrophysics

RADIAL FLOW SYSTEMS 43s

volumetric rate. The exact form of these flow equations depends on
the nature of external reservoir boundaries. Three basic outer boundary
conditions exist: infinite pressure, constant pressure, and no-flow.
STEADY-STATE FLOW

Strictly speaking, steady-state flow can occur only if the flow across the
drainage boundary, re, is equal to the flow across the wellbore wall at
well radius r,, and the fluid properties remain constant throughout the
reservoir. These conditions may never be met in a reservoir; however,
in petroleum reservoirs produced by a strong water drive, whereby the
water influx rate at re equals the well producing rate, the pressure change
with time is so slight that it is practically undetectable. In such cases, the
assumption of steady state is acceptable [ 181. Steady-state flow equations
are also useful in analyzing the reservoir conditions in the vicinity of the
wellbore for short periods of time, even in an unsteady-state system [ 191.
Mathematically true steady-state flow occurs when ap/llt = 0, which
reduces the diffusivity equation to:

(7.54)

Integrating this differential equation gives:

(7.5 5a)

where Ci is a constant of integration. For constant flow rate at the
wellbore, one can impose the following condition on the pressure
gradient at the well (from Darcy's law):

(7.55b)

Combining these two expressions and solving for Ci at the well:

qP
c. - - (7.55c)
- 2nkh
Substituting this term in Equation 7.55~; separating variables, and
integrating between rw and re, where the pressures are p, and pe,
respectively:

(7.5 5d)