[143.1.1] Let denote the sample volume,
denote the volume of pore space,
the volume filled with water,
the volume filled with oil,
the volume occupied by matrix,
and
the volumes of the subsets
.
[143.1.2] The volumes are defined as
![]() |
(3) |
where , and
![]() |
(4) |
is the characteristic function of a set .
[143.1.3] Then volume conservation implies
![]() |
![]() |
(5a) | |
![]() |
![]() |
(5b) | |
![]() |
![]() |
(5c) |
where .
[page 144, §0]
[144.0.1] The volume fraction
is
called total or global porosity.
[144.0.2] The volume fraction
is the total or global water saturation, and analogous
intensive quantities can be defined for the other phases.
[144.1.1] Often the saturations are not constant but vary on
macroscopic scales.
[144.1.2] Local volume fractions are defined by introducing a one parameter
family of functions by defining
on the diagonal and then extending it as
![]() |
(6) |
to the full space.
[144.1.3] Here is the scale separation parameter,
and
is the fast variable.
[144.1.4] For an infinite sample
the local volume
fractions may be defined as
![]() |
(7) |
where , and
is a sphere of radius
centered at
with volume
.
[144.1.5] In the following it is assumed that the
limit exists, but may in general depend also on time
so that the local volume fractions
become
position and time dependent.
[144.1.6] Local volume conservation implies the relations
![]() |
![]() |
(8a) | |
![]() |
![]() |
(8b) | |
![]() |
![]() |
(8c) |
where
are
volume fractions, and
are saturations.
[144.1.7] The water saturation is defined as
,
and the oil saturation as
.
[144.2.1] The general law of mass balance in
differential form reads ()
![]() |
(9) |
where
denote mass density, volume fraction and velocity
of phase
as functions of position
and time
.
[144.2.2] Exchange of mass between the two phases is
described by mass transfer rates
giving
the amount of mass by which phase
changes per
unit time and volume.
[144.2.3] The rate
is the rate of mass transfer from
phase
into phase
.
[144.3.1] The law of momentum balance is formulated as ()
![]() |
(10) |
where is the stress tensor in the
th phase,
is
the body force per unit volume acting on the
th phase,
is the momentum transfer into phase
from
all the other phases, and
![]() |
(11) |
denotes the material derivative for phase .