Page 239 - Fundamentals of Reservoir Engineering

OILWELL TESTING 177

array being shown in fig. 7.10, each well producing at the same rate as the real well
within the boundary. The constant terminal rate solution for this complex system can
then be expressed as

2kh 4t D ∞ φµ ca 2 j
π
t
(p − p wf ) = p D ( ) = 1 2 ln + 1 2 ei (7.41)
i
D
qµ γ j2 4kt
=
in which the first term on the right hand side of the equation is the component of the
pressure drop due to the production of the well itself, within an infinite reservoir,
equ. (7.23), and the infinite summation is the contribution to the wellbore pressure drop
due to the presence of the infinite array of image wells which simulate the no-flow
boundary. The exponential integral function is the line source solution of the diffusivity
equation introduced in sec. 7.2, equ. (7.11), for the constant terminal rate case and is
necessitated by the fact that the distance a j between the producing well and the j th
image well is large so that the logarithmic expression of the line source solution,
equ. (7.10), is an unacceptable approximation and the full exponential integral solution
must be used. The infinite summation in equ. (7.41) is therefore an example of
superposition in space of the basic constant terminal rate solution of the diffusivity
equation. For further details of the mathematical technique the reader should consult
7
the appendices of the original MBH paper .
Using this method to determine p D (t D), MBH were able to evaluate equ. (7.40) for a
wide variety of boundary conditions and presented their results as plots of

4kh ( p − p vs. t
π
*
)
qµ DA
where t DA is the dimensionless flowing time. These charts have been included in this
text as figs. 7.11-15. The individual plots are for different geometries and different
asymmetries of the producing well with respect to the no-flow boundary.

The MBH charts were originally designed to facilitate the determination of p from
*
pressure buildup data by first determining p , by the extrapolation of the Horner plot,
and k from the slope of the straight line. If an estimate is made of the area being
drained, t DA = kt/φµcA can be calculated for the actual flowing time t. Then, using the
*
appropriate MBH chart the value of 4πkh (p −p)/qµ is read off the ordinate from which
p can be calculated. The details of this important technique will be described in
sec. 7.7. For the moment, the MBH charts will be used in a more general manner to
determine p D (t D) functions for the range of geometries covered by the charts and for
any value of the flowing time.