Page 212 - Fundamentals of Reservoir Engineering

OILWELL TESTING 150

mathematics and for this solution the boundary and initial conditions may be stated as
follows

a) p = p at t = 0, for all r
i

b) p = p at r = ∞ , for all t (7.1)
i

∂ p qµ
c) lim r = , for t > 0
π
r → 0 r ∂ 2 kh

Condition (a) is merely the initial condition that, before producing, the pressure
everywhere within the drainage volume is equal to the initial equilibrium pressure p i.

Condition (b) ensures the condition of transience, namely that the pressure at the
outer, infinite boundary is not affected by the pressure disturbance at the wellbore and
vice versa.

Condition (c) is the line source inner boundary condition.

In addition, the assumptions made in deriving the radial diffusivity equation in
Chapter 5 are retained. That is, that the formation is homogeneous and isotropic, and
drained by a fully penetrating well to ensure radial flow; the fluid itself must have a
constant viscosity and a small and constant compressibility. The solution obtained will,
therefore, be applicable to the flow of undersaturated oil. Having developed the simple
theory of pressure analysis based on these assumptions, many of the restrictions will
be removed by considering, for instance, the effects of partial well completion, the flow
of highly compressible fluids, etc. These modifications to the basic theory will be
gradually introduced in this and the following chapter.

Under the above conditions the diffusivity equation

1 ∂ r ∂ p = φµ c ∂ p (5.20)
r ∂ r r ∂ k t ∂

can be solved by making use of Boltzmann's transformation

r 2 φµ cr 2
s = =
4 (Diffusivity constant)t 4kt

so that

∂ s φµ cr
= (7.2)
t ∂ 2k t

and

∂ s φµ cr 2
= (7.3)
r ∂ 4k t 2