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III Preliminaries

[page 3, §1]

A Notation

[] Time is denoted as tR and position as xR3. [] The geometry of the porous medium and its fluid content is represented mathematically by several closed subsets of R3. [] These are

R3S:=sample core(3a)
SP:=pore space(3b)
SSP=M:=matrix space(3c)
PWt:=wetting fluid at time t(3d)
PPWt=Ot:=non-wetting fluid at time t(3e)

where W stands for water (wetting) and O for oil (non-wetting). [] The wetting and nonwetting fluid are assumed to be immiscible. [] The Euclidean position space R3 carries the usual topology, metric structure and Lebesgue measure d3x. [] The volume of a subset GR3 is defined and denoted as



χGx:=1for xG0for xG(5)

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


holds and the volume fraction


is called porosity. [] The surface, or boundary, of a set G is denoted as G. [] All boundaries are assumed to be sufficiently smooth. [] The set PM=PS=MS is the rigid internal boundary between P and M.

[] It will be assumed throughout, that the sets P and M are pathconnected. [] A set is called pathconnected if any two of its points can be connected by a path contained inside the set. [] This excludes isolated disconnected pores and grains [28, Problem 6.(a), p. 120]. [] Moreover, it will be assumed that PS and MS. [] Analogous to eq. (6) also the fluid regions are disjoint except for their boundary, i.e.


holds for all t. [] Their volume fractions


are called saturations. [] 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. [] Matrix rigidity implies


where Qt is the total volumetric flux.

B Scale separation

[] Porous media physics requires to distinguish the microscopic pore scale from the macroscopic sample scale L. [] The two scales are related by coarse graining or, mathematically, by a scaling limit as emphasized in [29, 30]. [] The two scales will be distinguished by a tilde ~ for microscopic pore scale quantities when necessary. [] Thus x~R~3 represents a position in the pore scale description, while xR3 refers to a macroscale position. [] The relation between R~3 and R3 may be symbolically written as R3R~3R~3. [] This expression symbolizes the formal relation


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


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


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


where xSR3 whenever this limit exists and is independent of G. [] Note that eq. (15) is formal, because the integrand is understood as a function of y~=εyR~3. [] For more mathematical rigour on homogenization see e.g. [31]. [] 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.