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# III Preliminaries

[page 3, §1]

## A Notation

[3.1.1.1] Time is denoted as tR and position as xR3. [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

 R3⊃S :=sample core (3a) S⊃P :=pore space (3b) S⊃S∖P=M :=matrix space (3c) P⊃W⁢t :=wetting fluid at time ⁢t (3d) P⊃P∖W⁢t=O⁢t :=non-wetting fluid at time ⁢t (3e)

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 d3x. [3.1.1.6] The volume of a subset GR3 is defined and denoted as

 G:=∫R3χ⁢G⁢x⁢d3⁢x=∫Gd3⁢x (4)

where

 χ⁢G⁢x:=1⁢for ⁢x∈G0⁢for ⁢x∉G (5)

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

 S=P∪M,in⁢P∩in⁢M=∅ (6)

holds and the volume fraction

 ϕ=PS=1-MS (7)

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 PM=PS=MS 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 PS and MS. [3.1.2.5] Analogous to eq. (6) also the fluid regions are disjoint except for their boundary, i.e.

 P=W⁢t∪O⁢t,in⁢W⁢t∩in⁢O⁢t=∅ (8)

holds for all t. [3.1.2.6] Their volume fractions

 SW⁢t=W⁢tP=P∖O⁢tP=1-SO⁢t (9)

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

 Qi=volumetric injection rate of fluid ⁢i (10)

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

 Q⁢t=QW⁢t+QO⁢t (11)

where Qt 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 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 xR3 refers to a macroscale position. [3.2.1.5] The relation between R~3 and R3 may be symbolically written as R3R~3R~3. [3.2.1.6] This expression symbolizes the formal relation

 x~ =εx=ℓLx, x~∈R~3,x∈R3 (12)

in the scaling limit ε0. [3.2.1.7] To illustrate its meaning, consider the microscopic saturations

 SW~⁢x~,t=χ⁢W⁢t⁢x~, x~∈R~3 (13a) SO~⁢x~,t=χ⁢O⁢t⁢x~, x~∈R~3 (13b)

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

 ε⁢G=ε⁢x:x∈G. (14)

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

 SW⁢x,t=limε→0⁡1ε⁢G⁢∫Gχ⁢W⁢t⁢x+ε⁢y⁢d3⁢y, (15a) SO⁢x,t=limε→0⁡1ε⁢G⁢∫Gχ⁢O⁢t⁢x+ε⁢y⁢d3⁢y, (15b)

where xSR3 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~=εyR~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 and small compared to L.