III Preliminaries

[3.1.1.1] Time is denoted as t∈R and position as x∈R3. [3.1.1.2] The geometry of the porous medium and its fluid content is represented mathematically by several closed subsets of R3. [3.1.1.3] These are

where W stands for water (wetting) and O 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 R3 carries the usual topology, metric structure and Lebesgue measure d3⁢x. [3.1.1.6] The volume of a subset G⊂R3 is defined and denoted as

where

is the characteristic (or indicator) function of a set G. [3.1.1.7] The interior in⁢G of a set G is the union of all open sets contained in the set G. [3.1.1.8] It is assumed that

holds and the volume fraction

is called porosity. [3.1.1.9] The surface, or boundary, of a set G is denoted as ∂⁡G. [3.1.1.10] All boundaries are assumed to be sufficiently smooth. [3.1.1.11] The set P∩M=∂⁡P∖∂⁡S=∂⁡M∖∂⁡S is the rigid internal boundary between P and M.

[3.1.2.1] It will be assumed throughout, that the sets P and M 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 ∂⁡P∩∂⁡S≠∅ and ∂⁡M∩∂⁡S≠∅. [3.1.2.5] Analogous to eq. (6) also the fluid regions are disjoint except for their boundary, i.e.

holds for all t. [3.1.2.6] Their volume fractions

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

and have units of ms-1. [3.2.0.1] Matrix rigidity implies

where Q⁢t is the total volumetric flux.

[3.2.1.1] Porous media physics requires to distinguish the microscopic pore scale ℓ from the macroscopic sample scale L. [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 x~∈R~3 represents a position in the pore scale description, while x∈R3 refers to a macroscale position. [3.2.1.5] The relation between R~3 and R3 may be symbolically written as R3∼R~3R~3. [3.2.1.6] This expression symbolizes the formal relation

in the scaling limit ε→0. [3.2.1.7] To illustrate its meaning, consider the microscopic saturations

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

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

where x∈S⊂R3 whenever this limit exists and is independent of G. [3.2.1.10] Note that eq. (15) is formal, because the integrand is understood as a function of y~=ε⁢y∈R~3. [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” G large compared to ℓ and small compared to L.