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III Preliminaries

[page 3, §1]

A Notation

[3.1.1.1] Time is denoted as t\in\mathbb{R} and position as \mathbf{x}\in\mathbb{R}^{3}. [3.1.1.2] The geometry of the porous medium and its fluid content is represented mathematically by several closed subsets of \mathbb{R}^{3}. [3.1.1.3] These are

\displaystyle\mathbb{R}^{3}\supset\mathbb{S} \displaystyle:=\text{sample core} (3a)
\displaystyle\mathbb{S}\supset\mathbb{P} \displaystyle:=\text{pore space} (3b)
\displaystyle\mathbb{S}\supset\mathbb{S}\setminus\mathbb{P}=\mathbb{M} \displaystyle:=\text{matrix space} (3c)
\displaystyle\mathbb{P}\supset\mathbb{W}(t) \displaystyle:=\text{wetting fluid at time~}t (3d)
\displaystyle\mathbb{P}\supset\mathbb{P}\setminus\mathbb{W}(t)=\mathbb{O}(t) \displaystyle:=\text{non-wetting fluid at time~}t (3e)

where \mathbb{W} stands for water (wetting) and \mathbb{O} 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 \mathbb{R}^{3} carries the usual topology, metric structure and Lebesgue measure \mathrm{d}^{3}x. [3.1.1.6] The volume of a subset \mathbb{G}\subset\mathbb{R}^{3} is defined and denoted as

\displaystyle|\mathbb{G}|:=\int\limits _{{\mathbb{R}^{3}}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}(\mathbf{x})\,\mathrm{d}^{3}\mathbf{x}=\int\limits _{{\mathbb{G}}}\mathrm{d}^{3}\mathbf{x} (4)

where

\displaystyle\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}(x):=\begin{cases}1\qquad\text{for~}x\in\mathbb{G}\\ 0\qquad\text{for~}x\notin\mathbb{G}\end{cases} (5)

is the characteristic (or indicator) function of a set \mathbb{G}. [3.1.1.7] The interior \mathrm{in}(\mathbb{G}) of a set \mathbb{G} is the union of all open sets contained in the set \mathbb{G}. [3.1.1.8] It is assumed that

\displaystyle\mathbb{S}=\mathbb{P}\cup\mathbb{M},\qquad\mathrm{in}(\mathbb{P})\cap\mathrm{in}(\mathbb{M})=\emptyset (6)

holds and the volume fraction

\displaystyle\phi=\frac{|\mathbb{P}|}{|\mathbb{S}|}=1-\frac{|\mathbb{M}|}{|\mathbb{S}|} (7)

is called porosity. [3.1.1.9] The surface, or boundary, of a set \mathbb{G} is denoted as \partial\mathbb{G}. [3.1.1.10] All boundaries are assumed to be sufficiently smooth. [3.1.1.11] The set \mathbb{P}\cap\mathbb{M}=\partial\mathbb{P}\setminus\partial\mathbb{S}=\partial\mathbb{M}\setminus\partial\mathbb{S} is the rigid internal boundary between \mathbb{P} and \mathbb{M}.

[3.1.2.1] It will be assumed throughout, that the sets \mathbb{P} and \mathbb{M} 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 \partial\mathbb{P}\cap\partial\mathbb{S}\neq\emptyset and \partial\mathbb{M}\cap\partial\mathbb{S}\neq\emptyset. [3.1.2.5] Analogous to eq. (6) also the fluid regions are disjoint except for their boundary, i.e.

\displaystyle\mathbb{P}=\mathbb{W}(t)\cup\mathbb{O}(t),\qquad\mathrm{in}(\mathbb{W}(t))\cap\mathrm{in}(\mathbb{O}(t))=\emptyset (8)

holds for all t. [3.1.2.6] Their volume fractions

\displaystyle{S_{{\mathbb{W}}}}(t)=\frac{|\mathbb{W}(t)|}{|\mathbb{P}|}=\frac{|\mathbb{P}\setminus\mathbb{O}(t)|}{|\mathbb{P}|}=1-{S_{{\mathbb{O}}}}(t) (9)

are called saturations. [3.1.2.7] The volumetric injection rates of the two immiscible and incompressible fluids are denoted for i=\mathbb{W},\mathbb{O} as

Q_{i}=\text{volumetric injection rate of fluid~}i (10)

and have units of ms{}^{{-1}}. [3.2.0.1] Matrix rigidity implies

\displaystyle Q(t)=Q_{\mathbb{W}}(t)+Q_{\mathbb{O}}(t) (11)

where Q(t) is the total volumetric flux.

B Scale separation

[3.2.1.1] Porous media physics requires to distinguish the microscopic pore scale \ell from the macroscopic sample scale L. [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 \widetilde{\quad} for microscopic pore scale quantities when necessary. [3.2.1.4] Thus \widetilde{\mathbf{x}}\in\widetilde{\mathbb{R}}^{3} represents a position in the pore scale description, while \mathbf{x}\in\mathbb{R}^{3} refers to a macroscale position. [3.2.1.5] The relation between \widetilde{\mathbb{R}}^{3} and \mathbb{R}^{3} may be symbolically written as \mathbb{R}^{3}\sim(\widetilde{\mathbb{R}}^{3})^{{\widetilde{\mathbb{R}}^{3}}}. [3.2.1.6] This expression symbolizes the formal relation

\displaystyle\widetilde{\mathbf{x}} \displaystyle=\varepsilon\mathbf{x}=\frac{\ell}{L}\mathbf{x}, \displaystyle\widetilde{\mathbf{x}}\in\widetilde{\mathbb{R}}^{3},\ \mathbf{x}\in\mathbb{R}^{3} (12)

in the scaling limit \varepsilon\to 0. [3.2.1.7] To illustrate its meaning, consider the microscopic saturations

\displaystyle\widetilde{{S_{{\mathbb{W}}}}}(\widetilde{\mathbf{x}},t)=\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{W}(t)}}(\widetilde{\mathbf{x}}), \displaystyle\widetilde{\mathbf{x}}\in\widetilde{\mathbb{R}}^{3} (13a)
\displaystyle\widetilde{{S_{{\mathbb{O}}}}}(\widetilde{\mathbf{x}},t)=\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{O}(t)}}(\widetilde{\mathbf{x}}), \displaystyle\widetilde{\mathbf{x}}\in\widetilde{\mathbb{R}}^{3} (13b)

that are trivially defined in terms of characteristic functions. [3.2.1.8] Let

\displaystyle\varepsilon\mathbb{G}=\left\{\varepsilon\mathbf{x}:\mathbf{x}\in\mathbb{G}\right\}. (14)

[3.2.1.9] The macroscopic saturations are to be understood formally as the scaling limit

\displaystyle{S_{{\mathbb{W}}}}(\mathbf{x},t)=\lim _{{\varepsilon\to 0}}\frac{1}{|\varepsilon\mathbb{G}|}\int\limits _{\mathbb{G}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{W}(t)}}(\mathbf{x}+\varepsilon\mathbf{y})\,\mathrm{d}^{3}\mathbf{y}, (15a)
\displaystyle{S_{{\mathbb{O}}}}(\mathbf{x},t)=\lim _{{\varepsilon\to 0}}\frac{1}{|\varepsilon\mathbb{G}|}\int\limits _{\mathbb{G}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{O}(t)}}(\mathbf{x}+\varepsilon\mathbf{y})\,\mathrm{d}^{3}\mathbf{y}, (15b)

where \mathbf{x}\in\mathbb{S}\subset\mathbb{R}^{3} whenever this limit exists and is independent of \mathbb{G}. [3.2.1.10] Note that eq. (15) is formal, because the integrand is understood as a function of \widetilde{\mathbf{y}}=\varepsilon\mathbf{y}\in\widetilde{\mathbb{R}}^{3}. [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” \mathbb{G} large compared to \ell and small compared to L.