[3.0.1.1] The necessity to distinguish between percolating
and nonpercolating fluid regions arises from the
fact that in static equilibrium the pressure can become
hydrostatic only in those fluid regions
that are connected (or percolating) to the sample
boundary [14, 16, 15].
[3.0.1.2] Each of the two fluids W,O
consists of disjoint and pathconnected subsets
(regions) Wi,Oi.
[3.0.1.3] More precisely one has

W | =⋃i=1NWWi |
| (14a) |

O | =⋃i=1NOOi |
| (14b) |

where the subsets Wi,Oi are mutually disjoint,
and each of them is pathconnected.
[3.0.1.4] A set is called pathconnected if any two of its points can be
connected by a path contained inside the set.
[3.0.1.5] The sets are called mutually disjoint if
Oi∩Oj=∅ and Wi∩Wj=∅
holds for all i≠j.
[3.0.1.6] The integers NW,NO give the total number of
pathconnected subsets for water resp. oil.
[3.0.1.7] Of course, these numbers change with time
as do the regions Wi,Oi.

[3.0.2.1] Now define percolating (F1,F3) and nonpercolating
(F2,F4) fluid regions by classifying the subsets
Wi,Oi as
to whether they have empty or nonempty intersection with the
sample boundary ∂S.
[3.1.0.1] More formally, define

F1 | =⋃NWi=1∂Wi∩∂S≠∅Wi |
| (15a) |

F2 | =⋃NWi=1∂Wi∩∂S=∅Wi |
| (15b) |

F3 | =⋃NOi=1∂Oi∩∂S≠∅Oi |
| (15c) |

F4 | =⋃NOi=1∂Oi∩∂S=∅Oi |
| (15d) |

so that F1 is the union of all regions Wi, and
is F3 the union of all regions Oi, that have nonempty
intersection with the sample boundary ∂S.
[3.1.0.2] Similarly F2 is the union of all regions Wi
that have empty intersection with ∂S, and
similarly for F4.
[3.1.0.3] In this way each point in P belongs
to one of four regions Fi, i=1,2,3,4.
[3.1.0.4] This results in a total of four fluid phases called
percolating resp. nonpercolating water,
and percolating resp. nonpercolating oil.
[3.1.0.5] The index i=5 will be used for the rigid matrix (=rock).