Page 196 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 134

∂ 2 p + 1 p = φµ c ∂ p (5 19)
∂
r ∂ 2 r ∂ r k t ∂

which can be more conveniently expressed as

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

Making one final assumption, that the compressibility is constant, means that the
coefficient φµc/k is also constant and therefore, the basic equation has been
effectively linearized.

For the flow of liquids the above assumptions are quite reasonable and have been
2
frequently applied in the past. Dranchuk and Quon , however, have shown that this
simple linearization by deletion must be treated with caution and can only be applied
when the product
cp << 1 (5.21)

This condition makes it necessary to modify the final assumption so that the
compressibility is not just constant but both small and constant. The compressibility
appearing in equ. (5.20) is the total, or saturation weighted, compressibility of the entire
reservoir-liquid system
(5.22)
c t = c o S o + c w S wc + c f

in which the saturations are expressed as fractions of the pore volume. Using typical
figures for the components of equ. (5.22)
−6
c o = 10 × 10 /psi S wc = 0.2
−6
= 3 × 10 /psi p = 3000 psi
c w
−6
= 6 × 10 /psi
c f
−6
then c t in equ. (5.22) has the value 14.6×10 /psi and the product expressed by
equ. (5.21) has the value 0.04, which satisfies the necessary condition for this simple
linearization to be valid. However, when dealing with reservoir systems which have a
higher total compressibility it will be necessary to linearize equ. (5.1); using some form
of integral transformation as detailed by Dranchuk and Quon. Such an approach will be
required when describing the flow of a real gas since, in this case, the compressibility
of the gas alone may, to a first approximation, be expressed as the reciprocal of the
pressure and the cp product, equ. (5.21), will itself be unity. The linearization of
equ. (5.1) under these circumstances will be described in Chapter 8, secs. 2 4.

Before leaving the subject of compressibility, it should be noted that the product of φ
and c in all the equations, in this and the following chapters, is conventionally
expressed as

φ abs ol ut e × (c o S o + c w S wc + c f ) (5.23)