Page 211 - Fundamentals of Reservoir Engineering

OILWELL TESTING 149

The constant terminal rate solution is therefore the equation of p wf versus t for constant
rate production for any value of the flowing time. The pressure decline, fig. 7.1 (b), can
normally be divided into three sections depending on the value of the flowing time and
the geometry of the reservoir or part of the reservoir being drained by the well.

Initially, the pressure response can be described using a transient solution of the
diffusivity equation. It is assumed during this period that the pressure response at the
wellbore is not affected by the drainage boundary of the well and vice versa. This is
frequently referred to as the infinite reservoir case since, during the transient flow
period, the reservoir appears to be infinite in extent.

The transient phase is followed by the so-called late transient period during which the
influence of the drainage boundary begins to be felt. For a well producing from within a
no-flow boundary both the shape of the area drained and position of the well with
respect to the boundary are of major importance in determining the appropriate late
transient constant terminal rate solution.

Eventually, stabilised flow conditions will prevail which means that for the no-flow
boundary case the rate of change of wellbore pressure with respect to time is constant.
This corresponds to the semi-steady state condition described in Chapter 5, sec 3(b).

The constant terminal rate solution, for all values of the flowing time, was first
presented to the industry by Hurst and Van Everdingen in 1949. In their classic paper
1
on the subject , the authors solved the radial diffusivity equation using the Laplace
transform for both the constant terminal rate and constant terminal pressure cases. The
latter, which is relevant to water influx calculations. will be described in Chapter 9.

The full Hurst and Van Everdingen solution, equ. 7.34, is a most intimidating
mathematical equation which contains as one of its components an infinite summation
of Bessel functions. The complexity is due to the wellbore pressure response during
the late transient period, since for transient and semi-steady state flow relatively simple
solutions can be obtained which will be described in sec. 7.3. The fact that the full
solution is so complex is rather unfortunate since the constant terminal rate solution of
the radial diffusivity equation can be regarded as the basic equation in wellbore
pressure analysis techniques. By superposition of such solutions, as will be shown in
sec. 7.5, the pressure response at the wellbore can be theoretically described for any
sequence of different rates acting for different periods of time, and this is the general
method employed in the analysis of any form of oil or gas well test.

7.3 THE CONSTANT TERMINAL RATE SOLUTION FOR TRANSIENT AND SEMI-
STEADY STATE FLOW CONDITIONS

During the initial transient flow period, it has been found that the constant terminal rate
solution of the radial diffusivity equation, determined using the Laplace transform, can
be approximated by the so-called line source solution which assumes that in
comparison to the apparently infinite reservoir the wellbore radius is negligible and the
wellbore itself can be treated as a line. This leads to a considerable simplification in the