2 Definition of Phases

[142.3.1] A porous sample S=P∪M⊂R3 consists of a subset P (called pore space) and a subset M (called matrix). [142.3.2] The pore space P contains two immiscible fluids, namely a wetting fluid, called water and denoted as W, plus a nonwetting fluid, called oil and denoted as O.

[142.4.1] Each of the two fluids W,O consists of disjoint and pathconnected subsets (regions) Wi,Oi. [142.4.2] More precisely one has

where the subsets Wi,Oi are mutually disjoint, and each of them is pathconnected. [142.4.3] A set is called pathconnected if any two of its points can be connected by a path contained inside the set. [142.4.4] The sets are called mutually disjoint if Oi∩Oj=∅ and Wi∩Wj=∅ holds for all i≠j. [142.4.5] The integers NW,NO give the total number of pathconnected subsets for water resp. oil. [142.4.6] These numbers vary with time, as do the regions Wi,Oi.

[142.5.1] 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. [page 143, §0]    [143.0.1] More formally, define

so that F1 is the union of all regions Wi, and F3 is the union of all regions Oi, that have nonempty intersection with the sample boundary ∂⁡S. [143.0.2] Similarly F2 is the union of all regions Wi that have empty intersection with ∂⁡S, and similarly for F4. [143.0.3] In this way each point in P belongs to one of four regions Fi, i=1,2,3,4. [143.0.4] This results in a total of four fluid phases called percolating resp. nonpercolating water, and percolating resp. nonpercolating oil. [143.0.5] The index i=5 will be used for the rigid matrix M.