Page 182 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 120

In this case, the normalised relative permeability curves are defined as

k (S) k (S)
K(S ) = o w and K (S ) = w w (4.32)
ro
w
w
rw
k(S = S ) k (S = S )
o w wc o w wc
To describe the simultaneous flow of oil and water in the reservoir, applying Darcy's
law, the absolute permeability k, implicitly used in the earlier sections of this chapter,
must be replaced by the effective permeabilities k o(S w) and k w(S w) respectively. Using
the alternative methods of normalising the effective permeability curves, the required
permeabilities can be expressed as either

k o (S w) = kk ro (S w) or k o (S w) = k o (S w = S wc) K ro (S w) (4.33)

and
k w (S w) = kk rw (S w) or k w (S w) = k o (S w = S wc) K rw (S w)

both interpretations, naturally, giving the same values of the effective permeabilities. As
already mentioned, in many equations describing the displacement of one immiscible
fluid by another it is the ratio of effective permeabilities which is required, and from equ.
(4.33) this can be expressed as
k ro K ro (4.34)
k rw = K rw

To complicate matters further, in the literature, it is not normal to distinguish between
the two ways of presenting relative permeability curves by assigning one of them a
capital letter; both interpretations are denoted by the symbol k r. In this text, the relative
permeabilities used will be those obtained by normalising the effective permeability
curves with the absolute permeability (fig. 4.8(b)).
Relative permeabilities are measured in the laboratory by studying the displacement of
oil by water (or gas) in very thin core plugs, in which it is safe to assume that the fluid
saturations are uniformly distributed with respect to thickness. Therefore, these
laboratory-measured, or rock-relative permeability relationships, can only be used
directly to describe flow in a reservoir in which the saturations are also uniformly
distributed with respect to thickness. In the majority of practical cases, however, there
is a non-uniform water saturation distribution in the vertical direction which is governed
by capillary and gravity forces and, therefore, there must also be a relative permeability
distribution with respect to thickness. Because of this, the rock-relative permeabilities
can seldom be used directly in field displacement calculations.

Practically the whole of Chapter 10 is devoted to describing methods of generating
averaged (or pseudo) relative permeabilities, as functions of the thickness averaged
water saturation. These are used to describe the displacement of oil by water in a more
realistic fashion, taking account of the manner in which the fluid saturations are
distributed, with respect to thickness, as they simultaneously move through the
reservoir.