Page 230 - Fundamentals of Reservoir Engineering

OILWELL TESTING 168

which indicate that semi-steady state conditions will occur for the same value of t DA
irrespective of the size of the square. It is generally true for a well situated at the centre
of

- a square
- a circle
- a hexagon

that semi-steady state conditions will prevail after a flowing time such that t DA >0.1.

32
SEMI STEADY STATE
28
24

20
L = 100 ft L = 500 ft
16
p D
12

8
TRANSIENT FLOW
4

0
1 2 3 4 5 6 7 log t D

Fig. 7.5 Dimensionless pressure as a function of dimensionless flowing time for a
well situated at the centre of a square (Exercise 7.4)

7.5 SUPERPOSITION THEOREM: GENERAL THEORY OF WELL TESTING

Mathematically the superposition theorem states that any sum of individual solutions of
a second order linear differential equation is also a solution of the equation.

In practice, this is one of the most powerful tools at the reservoir engineer's disposal for
writing down solutions to complex flow problems in the reservoir without explicitly
solving the full differential equation on each occasion. Applying the superposition
theorem means that individual constant rate wells can be placed in any position in the
reservoir at any time and an expression for the resulting pressure distribution in space
and time derived by inspection. The principle will be illustrated with an example of
superposition in time at a fixed location which is particularly relevant to well test
analysis.