Page 195 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 133

This condition is appropriate when pressure is being maintained in the reservoir due to
either natural water influx or the injection of some displacing fluid (refer Chapter 10).

It should be noted that the semi-steady state and steady state conditions may never be
fully realised in the reservoir. For instance, semi-steady state flow equations are
frequently applied when the rate, and consequently the position of the no-flow
boundary surrounding a well, are slowly varying functions of time. Nevertheless, the
defining conditions specified by equs. (5.7) and (5.15) are frequently approximated in
the field since both production and injection facilities are usually designed to operate at
constant rates and it makes little sense to unnecessarily alter these. If the production
rate of an individual well is changed, for instance, due to closure for repair or
increasing the rate to obtain a more even fluid withdrawal pattern throughout the
reservoir, there will be a brief period when transient flow conditions prevail followed by
stabilized flow for the new distribution of individual well rates.

5.4 THE LINEARIZATION OF EQUATION 5.1 FOR FLUIDS OF SMALL AND
CONSTANT COMPRESSIBILITY

A simple linearization of equ. (5.1) can be obtained by deletion of some of the terms,
dependent upon making various assumptions concerning the nature of fluid for which
solutions are being sought. In this section the fluid considered will be a liquid which, in
a practical sense, will apply to the flow of undersaturated oil. Expanding the left hand
side of equ. (5.1), using the chain rule for differentiation gives

1 ∂ k ∂ p k ∂ ρ ∂ p kρ ∂ p kρ ∂ 2 p ∂ p
r ρ + r + + r 2 = φ cρ (5.16)
r ∂ r µ
r ∂ µ r ∂ r ∂ µ r ∂ µ r ∂ t ∂

and differentiating equ. (5.4) with respect to r gives

∂ p ∂ ρ
cρ = (5.17)
r ∂ r ∂

which when substituted into equ. (5.16) changes the latter to

1 ∂ k ∂ p k ∂ p 2 kρ ∂ p kρ ∂ 2 p ∂ p
r ρ + c r ρ + + r = φ cρ (5.18)
r ∂ r µ
r ∂ µ r ∂
µ r ∂ µ r ∂ 2 t ∂

For liquid flow, the following assumptions are conventionally made

- the viscosity, µ is practically independent of pressure and may be regarded as a
constant

2
- the pressure gradient ∂p/∂r is small and therefore, terms of the order (∂p/∂r) can
be neglected.

These two assumptions eliminate the first two terms in the left hand side of equ. (5.18),
reducing the latter to