2 Definition of Phases

[142.3.1] A porous sample $\mathbb{S}=(\mathbb{P}\cup\mathbb{M})\subset\mathbb{R}^{3}$ consists of a subset $\mathbb{P}$ (called pore space) and a subset $\mathbb{M}$ (called matrix). [142.3.2] The pore space $\mathbb{P}$ contains two immiscible fluids, namely a wetting fluid, called water and denoted as $\mathbb{W}$ , plus a nonwetting fluid, called oil and denoted as $\mathbb{O}$ .

[142.4.1] Each of the two fluids $\mathbb{W},\mathbb{O}$ consists of disjoint and pathconnected subsets (regions) $\mathbb{W}_{i},\mathbb{O}_{i}$ . [142.4.2] More precisely one has

$\displaystyle\mathbb{W}$	$\displaystyle=\bigcup _{{i=1}}^{{{N_{\mathbb{W}}}}}\mathbb{W}_{i}$		(1a)
$\displaystyle\mathbb{O}$	$\displaystyle=\bigcup _{{i=1}}^{{{N_{\mathbb{O}}}}}\mathbb{O}_{i}$		(1b)

where the subsets $\mathbb{W}_{i},\mathbb{O}_{i}$ 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 $\mathbb{O}_{i}\cap\mathbb{O}_{j}=\emptyset$ and $\mathbb{W}_{i}\cap\mathbb{W}_{j}=\emptyset$ holds for all $i\neq j$ . [142.4.5] The integers ${N_{\mathbb{W}}},{N_{\mathbb{O}}}$ give the total number of pathconnected subsets for water resp. oil. [142.4.6] These numbers vary with time, as do the regions $\mathbb{W}_{i},\mathbb{O}_{i}$ .

[142.5.1] Now define percolating ( $\mathbb{F}_{1},\mathbb{F}_{3}$ ) and nonpercolating ( $\mathbb{F}_{2},\mathbb{F}_{4}$ ) fluid regions by classifying the subsets $\mathbb{W}_{i},\mathbb{O}_{i}$ as to whether they have empty or nonempty intersection with the sample boundary $\partial\mathbb{S}$ . [page 143, §0] [143.0.1] More formally, define

$\displaystyle\mathbb{F}_{1}$	$\displaystyle=\bigcup^{{N_{\mathbb{W}}}}_{{\substack{i=1\\ \partial\mathbb{W}_{i}\cap\partial\mathbb{S}\neq\emptyset}}}\mathbb{W}_{i}$	(2a)
$\displaystyle\mathbb{F}_{2}$	$\displaystyle=\bigcup^{{N_{\mathbb{W}}}}_{{\substack{i=1\\ \partial\mathbb{W}_{i}\cap\partial\mathbb{S}=\emptyset}}}\mathbb{W}_{i}$	(2b)
$\displaystyle\mathbb{F}_{3}$	$\displaystyle=\bigcup^{{N_{\mathbb{O}}}}_{{\substack{i=1\\ \partial\mathbb{O}_{i}\cap\partial\mathbb{S}\neq\emptyset}}}\mathbb{O}_{i}$	(2c)
$\displaystyle\mathbb{F}_{4}$	$\displaystyle=\bigcup^{{N_{\mathbb{O}}}}_{{\substack{i=1\\ \partial\mathbb{O}_{i}\cap\partial\mathbb{S}=\emptyset}}}\mathbb{O}_{i}$	(2d)

so that $\mathbb{F}_{1}$ is the union of all regions $\mathbb{W}_{i}$ , and $\mathbb{F}_{3}$ is the union of all regions $\mathbb{O}_{i}$ , that have nonempty intersection with the sample boundary $\partial\mathbb{S}$ . [143.0.2] Similarly $\mathbb{F}_{2}$ is the union of all regions $\mathbb{W}_{i}$ that have empty intersection with $\partial\mathbb{S}$ , and similarly for $\mathbb{F}_{4}$ . [143.0.3] In this way each point in $\mathbb{P}$ belongs to one of four regions $\mathbb{F}_{i}$ , $i=1,2,3,4$ . [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 $\mathbb{M}$ .

Author:	R. Hilfer
from:	...
originally submitted:	2009