Page 469 - Petrophysics

RADIAL FLOW SYSTEMS 437

Substituting the above expression into Equation 7.58, integrating, and
solving explicitly for the volumetric flow rate, one obtains (in oilfield
units):

(7.60)

assuming that the term </rz is negligible. It is important to emphasize
that Equations 7.56 and 7.60 are strictly valid for the case of a single
well in an infinite reservoir and strong water drive reservoir producing
at steady-state flow conditions. These equations also apply equally well
in an oil reservoir experiencing pressure maintenance by water injection
or gas injection.

PSEUDOSTEADY-STATE
FLOW

In bounded cylindrical reservoirs, the pseudosteady-state flow regime
is common at long producing times. In these reservoirs, also called
volumetric reservoirs, there can be no flow across the impermeable outer
boundary, such as a sealing fault, and fluid production must come from
the expansion and pressure decline of the reservoir. This condition of no
flow boundary is also encountered in a well that is offset on four sides.
If there is no flow across the external boundary, then after sufficiently
long producing time elapses the pressure decline throughout the
drainage volume becomes a linear function of time. Therefore, for a well
producing at a constant production rate, the rate of pressure decline is
constant:

(7.61)

where Vp is the drainage pore volume, which is equal to nrzh@, and
c is the compressibility of the fluid at the average reservoir pressure.
Substituting Equation 7.61 into the diffusivity equation (Equation 7.53,
integrating twice and solving for the flow rate (in oiLfield units) gives [4] :

(7.62)

If the external pressure, pe, is unknown, Equation 7.62 should be
derived in terms of the average reservoir pressure, p. The pressure p