Page 243 - Introduction to Petroleum Engineering

230 TRANSIENT WELL TESTING
where p is the fluid pressure, r is the dimensionless radius, and t is the dimension-
D
D
less time. Dimensionless radius is defined as
r
r D (12.2)
r w
where r is the radial distance from the well (ft) and r is the well radius (ft).
w
Dimensionless time is defined in terms of a group of parameters, namely,
kt
.
t D 0 000264 2 (12.3)
r
c T i w
where k is the permeability (md), t is the time (hr), ϕ is the porosity (fraction), μ is
the viscosity of the mobile, single‐phase liquid (cp), and c is the total compress-
T
−1
ibility (psia ). The group of parameters k/ϕμc is called the diffusivity coefficient.
T
The subscript i is used to show that liquid viscosity μ and total compressibility c are
T
evaluated at initial pressure. Dimensionless radius increases as radial distance
increases, and dimensionless time increases as time increases.
The diffusivity equation is based on several assumptions: the equation applies to
single‐phase flow of a slightly compressible liquid; the liquid flows in the
horizontal, radial direction only; and changes in formation and liquid properties are
negligible. In practice, formation and liquid properties do depend on pressure and
can experience slight changes as pressure changes. As a matter of consistency,
initial values are used to analyze pressure transient tests. The horizontal flow
assumption implies that gravity effects are neglected. In addition, it is assumed
that flow rate is proportional to pressure gradient so that Darcy’s law applies.
A consequence of the assumptions is that the well is able to produce or inject liquid
throughout the thickness of the formation so that flow is only along the radial
direction. Total system compressibility is
c T c r c S o c S w c S g (12.4)
g
o
w
where c is the rock compressibility and fluid‐phase compressibility is the saturation‐
r
weighted average of oil‐, water‐, and gas‐phase compressibilities.
The following discussion applies to any well satisfying the previous assumptions.
In our case, we focus on PTT of oil wells, but similar techniques can be applied to
PTT of water wells. Gas wells are discussed in later sections.
The diffusivity equation is solved by treating the well as a line source with constant
flow rate. The line source assumption implies that the volume of a real well does not
have a significant impact on our model of the pressure transient test. In general, this
is a reasonable assumption because wells are usually only a few inches in diameter
and the pressure transient test can probe the reservoir to a radial distance of hundreds
of feet. The volume of the well does impact early time pressure response, as noted
previously, and must be considered when interpreting PTT results.
Solutions of the diffusivity equation depend on the specified boundary conditions.
During the infinite‐acting time, flow in a reservoir behaves as if it does not have an
outer boundary and the reservoir can be treated like an infinite reservoir. The solution