[page 3, §1]

[3.1.1.1] Time is denoted as

(3a) | |||

(3b) | |||

(3c) | |||

(3d) | |||

(3e) |

where

(4) |

where

(5) |

is the characteristic (or indicator) function of a set

(6) |

holds and the volume fraction

(7) |

is called porosity.
[3.1.1.9] The surface, or boundary, of a set

[3.1.2.1] It will be assumed throughout, that
the sets

(8) |

holds for all

(9) |

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

(10) |

and have units of ms

(11) |

where

[3.2.1.1] Porous media physics requires to distinguish the microscopic
pore scale

(12) |

in the scaling limit

(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