Page 356 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 291

8.13 SUMMARY OF PRESSURE ANALYSIS TECHNIQUES

For those who have struggled through this and the previous three chapters, it is
worthwhile presenting a brief summary of their contents in an effort to simplify and
generalise the theory of well testing.

To start with, the application of the principle of mass conservation, together with
Darcy's law and the definition of isothermal compressibility, lead to the non-linear,
radial, second order, partial differential equation for the flow of a single phase fluid (in
Darcy units) as

1 ∂ kρ ∂ p ∂ p
c
r = φρ (5.1)
rr ∂ µ r ∂ t ∂
Prior to obtaining useful solutions of this equation it must first be linearized (or partially
linearized) and the method by which this can be achieved depends on the nature of the
fluid under consideration, as follows.

Undersaturated oil

2
Linearization by deletion of terms, assuming that (p / r)∂ ∂ ≈ 0;µ ≈ constant and
cp << 1.

Real gas

Partial linearization using the integral transformation

p pdp
m(p) = 2 (8.7)
p b µΖ

Gas-oil

Partial linearization using the integral transformation

p k(S )
′
m(p) = ro o dp (8.62)
p b µ o B o

or, strictly speaking, in Darcy units, the correct form of this transformation should be

p k(S )ρ
′
m(p) = ro o o dp
p b µ o

Application of any of the above methods leads to the re-formulation of equ. (5.1) as

1 ∂ ∂ β = φµ c ∂ β (8.66)
r ∂ r r ∂ k t ∂

which has the form of the radial diffusivity equation and in which:

for undersaturated oil β = p