Page 192 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 130

This condition is only applicable for a relatively short period after some pressure
disturbance has been created in the reservoir. In terms of the radial flow model this
disturbance would be typically caused by altering the well's production rate at r = r w.
In the time for which the transient condition is applicable it is assumed that the
pressure response in the reservoir is not affected by the presence of the outer
boundary, thus the reservoir appears infinite in extent. The condition is mainly applied
to the analysis of well tests in which the well's production rate is deliberately changed
and the resulting pressure response in the wellbore is measured and analysed during a
brief period of a few hours after the rate change has occurred. Then, unless the
reservoir is extremely small, the boundary effects will not be felt and the reservoir is,
mathematically, infinite.

This gives rise to a complex solution of equ. (5.1) in which both the pressure and
pressure derivative, with respect to time, are themselves functions of both position and
time, thus
p = g(r,t)
and ∂ p
= f(r,t)
t ∂
Transient analysis techniques and their application to oil and gas well testing will be
described in Chapters 7 and 8, respectively.

b) Semi-Steady State condition

q = constant
p e
Pressure
= constant
∂ p = 0, at r = r
∂ r e
p wf

r w r r e

Fig. 5.2 Radial flow under semi-steady state conditions

This condition is applicable to a reservoir which has been producing for a sufficient
period of time so that the effect of the outer boundary has been felt. In terms of the
radial flow model, the situation is depicted in fig. 5.2. It is considered that the well is
surrounded, at its outer boundary, by a solid "brick wall" which prevents the flow of
fluids into the radial cell. Thus at the outer boundary, in accordance with Darcy's law

∂ p
= 0at r = r e (5.6)
r ∂

Furthermore, if the well is producing at a constant flow rate then the cell pressure will
decline in such a way that

∂ p
≈ constant, for all r and t. (5.7)
t ∂