[page 3, §1]
[3.1.1.1] Time is denoted as and
position as
.
[3.1.1.2] The geometry of the porous medium and its fluid content
is represented mathematically by several closed subsets of
.
[3.1.1.3] These are
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(3a) | |
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(3b) | |
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(3c) | |
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(3d) | |
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(3e) |
where stands for water (wetting) and
for oil (non-wetting).
[3.1.1.4] The wetting and nonwetting fluid are assumed to be immiscible.
[3.1.1.5] The Euclidean position space
carries
the usual topology, metric structure and Lebesgue measure
.
[3.1.1.6] The volume of a subset
is defined and denoted as
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(4) |
where
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(5) |
is the characteristic (or indicator) function of a set .
[3.1.1.7] The interior
of a set
is the
union of all open sets contained in the set
.
[3.1.1.8] It is assumed that
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(6) |
holds and the volume fraction
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(7) |
is called porosity.
[3.1.1.9] The surface, or boundary, of a set
is denoted as
.
[3.1.1.10] All boundaries are assumed to be sufficiently smooth.
[3.1.1.11] The set
is the rigid internal boundary between
and
.
[3.1.2.1] It will be assumed throughout, that
the sets and
are pathconnected.
[3.1.2.2] A set is called pathconnected if any two of its points can be
connected by a path contained inside the set.
[3.1.2.3] This excludes isolated disconnected pores and
grains [28, Problem 6.(a), p. 120].
[3.1.2.4] Moreover, it will be assumed that
and
.
[3.1.2.5] Analogous to eq. (6) also the fluid regions
are disjoint except for their boundary, i.e.
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(8) |
holds for all .
[3.1.2.6] Their volume fractions
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(9) |
are called saturations.
[3.1.2.7] The volumetric injection
rates of the two immiscible and
incompressible fluids
are denoted for as
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(10) |
and have units of ms.
[3.2.0.1] Matrix rigidity implies
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(11) |
where is the total volumetric flux.
[3.2.1.1] Porous media physics requires to distinguish the microscopic
pore scale from the macroscopic sample scale
.
[3.2.1.2] The two scales are related by coarse graining or,
mathematically, by a scaling limit as emphasized in [29, 30].
[3.2.1.3] The two scales will be distinguished
by a tilde
for microscopic
pore scale quantities when necessary.
[3.2.1.4] Thus
represents a position
in the pore scale description, while
refers to a macroscale position.
[3.2.1.5] The relation between
and
may be
symbolically written as
.
[3.2.1.6] This expression symbolizes the formal relation
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(12) |
in the scaling limit .
[3.2.1.7] To illustrate its meaning, consider the microscopic saturations
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(13a) | |||
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(13b) |
that are trivially defined in terms of characteristic functions. [3.2.1.8] Let
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(14) |
[3.2.1.9] The macroscopic saturations are to be understood formally as the scaling limit
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(15a) | ||
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(15b) |
where
whenever this limit exists and is independent of
.
[3.2.1.10] Note that eq. (15) is formal, because
the integrand is understood as a function of
.
[3.2.1.11] For more mathematical rigour on homogenization see e.g. [31].
[3.2.1.12] Existence of the scaling limit is tantamount
to the existence of an intermediate
“representative elementary volume”
large compared to
and small compared to
.