Page 262 - Fundamentals of Reservoir Engineering

OILWELL TESTING 199

whether the latter time coincides with the end of a simulation time step or not. There
are then two ways of comparing the observed well pressure with the simulated grid
block pressures.

The first of these is to calculate the average pressure within the no-flow boundary at
the time of survey, using the MBH or Dietz method, and compare this with the volume
averaged pressure over all the grid blocks and partial blocks within the natural no-flow
boundary. This is a rather tedious business. A simpler, approximate method has been
16
15
introduced by van Poollen and further described by Earlougher . This consists of
using the Horner buildup plot in conjunction with the Dietz method to calculate the so-
called "dynamic grid block pressure" p d which is simply the average pressure in the grid
block containing the well at the time of survey. The analysis seeks to determine at what
t +∆ t
value of log d should the Horner plot be entered so that the pressure read from
t ∆ d
the hypothetical linear buildup has risen to be equal to the dynamic pressure, i.e.
p ws(LIN) = p d. Again, equ. (7.63) can be applied but in this case t DA must be evaluated
using the grid block area rather than that of the no-flow boundary and C A takes on the
fixed value of 19.1. The reasoning behind the latter choice is that the grid block
boundary is not a no-flow boundary. Instead the boundary condition corresponds more
closely to that of steady state flow and for such Dietz has only presented one case
corresponding to a well producing from the centre of a circle for which C A = 19.1,
fig 6.4. Thus the rectangular grid block shape is approximated as circular with area
equal to that of the grid block. Therefore, the Horner plot is entered for a value of

t +∆ t
log d = log (19.1 t DA ) (7.64)
t ∆ d

and p d read from the linear buildup as shown in fig. 7.21. Again, use of equ. (7.64)
depends on the fact that the well is flowing under stabilised conditions at the time of
survey. Normally, in this case t >>∆t d and van Poollen, using this assumption, has
presented an expression for explicitly calculating the closed in time at which
p WS(LIN) = p d. This can readily be obtained from equ. (7.64), as

φµ cA φµ cr 2 e
π
t ∆= =
d
0.000264x19.1k 0.005042k
or

φµ cr 2 e
t ∆= 623
d
k
where r e is the radius of the circle with area equivalent to that of the grid block. This
approximate but speedy method for comparing observed with simulated pressures is
very useful in performing a history match on well pressures.

The following two exercises will illustrate the application of the pressure buildup
techniques, described in this section, to an under saturated oil reservoir.