[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

(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.

[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 .