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# III Preliminaries

[page 3, §1]

## A Notation

[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

 (3a) (3b) (3c) (3d) (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

 (4)

where

 (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

 (6)

holds and the volume fraction

 (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.

 (8)

holds for all . [3.1.2.6] Their volume fractions

 (9)

are called saturations. [3.1.2.7] The volumetric injection rates of the two immiscible and incompressible fluids are denoted for as

 (10)

and have units of ms. [3.2.0.1] Matrix rigidity implies

 (11)

where is the total volumetric flux.

## B Scale separation

[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

 (12)

in the scaling limit . [3.2.1.7] To illustrate its meaning, consider the microscopic saturations

 (13a) (13b)

that are trivially defined in terms of characteristic functions. [3.2.1.8] Let

 (14)

[3.2.1.9] The macroscopic saturations are to be understood formally as the scaling limit

 (15a) (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 .