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2 Problems in the Theory of Porous Media

2.1 Physical Problems

[204.1.9] Many physical problems in porous and heterogeneous media can be formulated mathematically as a set of partial differential equations

\displaystyle\boldsymbol{F}_{\mathbb{P}}({\boldsymbol{r}},t,\boldsymbol{u},\partial\boldsymbol{u}/\partial t,\ldots,\boldsymbol{\nabla}\cdot\boldsymbol{u},\boldsymbol{\nabla}\times\boldsymbol{u},\ldots) \displaystyle=0,\qquad{\boldsymbol{r}}\in\mathbb{P}\subset\mathbb{R}^{3},t\in\mathbb{R} (1a)
\displaystyle\boldsymbol{F}_{\mathbb{M}}({\boldsymbol{r}},t,\boldsymbol{u},\partial\boldsymbol{u}/\partial t,\ldots,\boldsymbol{\nabla}\cdot\boldsymbol{u},\boldsymbol{\nabla}\times\boldsymbol{u},\ldots) \displaystyle=0,\qquad{\boldsymbol{r}}\in\mathbb{M}\subset\mathbb{R}^{3},t\in\mathbb{R} (1b)

for a vector of unknown fields \boldsymbol{u}({\boldsymbol{r}},t) as function of position and time coordinates. [204.1.10] Here the two-component porous sample \mathbb{S}=\mathbb{P}\cup\mathbb{M} is defined as the union of two closed subsets \mathbb{P}\subset\mathbb{R}^{3} and \mathbb{M}\subset\mathbb{R}^{3} where \mathbb{P} denotes the pore space (or component 1 in a heterogeneous medium) and \mathbb{M} denotes the matrix space (or component 2). [204.1.11]  In (1) the vector functionals \boldsymbol{F}_{\mathbb{P}} and \boldsymbol{F}_{\mathbb{M}} may depend on the vector \boldsymbol{u} of unknowns and its derivatives as well as on position {\boldsymbol{r}} and time t. [204.1.12] A simple example for (1) is the time independent potential problem

\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{j}({\boldsymbol{r}}) \displaystyle=0,\qquad{\boldsymbol{r}}\in\mathbb{S} (2)
\displaystyle\boldsymbol{j}({\boldsymbol{r}})+C({\boldsymbol{r}})\boldsymbol{\nabla}u({\boldsymbol{r}}) \displaystyle=0,\qquad{\boldsymbol{r}}\in\mathbb{S} (3)

for a scalar field u({\boldsymbol{r}}). [204.1.13] The coefficients

C({\boldsymbol{r}})=C_{\mathbb{P}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}})+C_{\mathbb{M}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{M}}}({\boldsymbol{r}}) (4)

contain the material constants C_{\mathbb{P}}\neq C_{\mathbb{M}}. [204.1.14]  Here the characteristic (or indicator) function \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}({\boldsymbol{r}}) of a set \mathbb{G} is defined as

\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}({\boldsymbol{r}})=\begin{cases}1&\text{\ \ \ \ for }{\boldsymbol{r}}\in\mathbb{G}\\ 0&\text{\ \ \ \ for }{\boldsymbol{r}}\notin\mathbb{G}.\end{cases} (5)

[page 205, §0]    [205.0.1] Hence C({\boldsymbol{r}}) is not differentiable at the internal boundary \partial\mathbb{P}=\partial\mathbb{M}, and this requires to specify boundary conditions

\displaystyle\lim _{{s\searrow 0}}\boldsymbol{n}\cdot\boldsymbol{j}(\boldsymbol{r}+s\boldsymbol{n}) \displaystyle=\lim _{{s\searrow 0}}\boldsymbol{n}\cdot\boldsymbol{j}(\boldsymbol{r}-s\boldsymbol{n}), \displaystyle{\boldsymbol{r}}\in\partial\mathbb{P} (6)
\displaystyle\lim _{{s\searrow 0}}\boldsymbol{n}\times\boldsymbol{\nabla}u(\boldsymbol{r}+s\boldsymbol{n}) \displaystyle=\lim _{{s\searrow 0}}\boldsymbol{n}\times\boldsymbol{\nabla}u(\boldsymbol{r}-s\boldsymbol{n}), \displaystyle{\boldsymbol{r}}\in\partial\mathbb{P} (7)

at the internal boundary. [205.0.2] In addition, boundary conditions on the sample boundary \partial\mathbb{S} need to be given to complete the formulation of the problem. [205.0.3] Inital conditions may also be required. [205.0.4] Several concrete applications can be subsumed under this formulation depending upon the physical interpretation of the field u and the current \boldsymbol{j}. [205.0.5] An overview for possible interpretations of u and \boldsymbol{j} is given in Table 1. [205.0.6] It contains hydrodynamical flow, electrical conduction, heat conduction and diffusion as well as cross effects such as thermoelectric or electrokinetic phenomena.

Table 1: Overview of possible interpretations for the field u and the current \boldsymbol{j} produced by its gradient according to (3).
\boldsymbol{j}\backslash u pressure el. potential temperature concentration
volume Darcy’s law electroosmosis thermal osmosis chemical osmosis
el. charge streaming pot. Ohm’s law Seebeck effect sedim. electricity
heat thermal filtration Peltier effect Fourier’s law Dufour effect
particles ultrafiltration electrophoresis Soret effect Fick’s law

[205.1.1] The physical problems in the theory of porous media may be divided into two categories: direct problems and inverse problems. [205.1.2] In direct problems one is given partial information about the pore space configuration \mathbb{P}. [205.1.3] The problem is to deduce information about the solution u({\boldsymbol{r}},t) of the boundary and/or initial value problem that can be compared to experiment. [205.1.4] In inverse problems one is given partial information about the solutions u({\boldsymbol{r}},t). [205.1.5] Typically this information comes from various experiments or observations of physical processes. [205.1.6] The problem is to deduce information about the pore space configuration \mathbb{P} from these data.

[205.2.1] Inverse problems are those of greatest practical interest. [205.2.2] All attempts to visualize the internal interface or fluid content of nontransparent heterogeneous media lead to inverse problems. [205.2.3] Examples occur in computer tomography. [205.2.4] Inverse problems are often ill-posed due to lack of data [52, 39]. [205.2.5] Reliable solution of inverse problems requires a predictive theory for the direct problem.

2.2 Geometrical Problems

[205.2.6] The geometrical problems arise because in practice the pore space configuration \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}}) is usually not known in detail. [205.2.7] The direct problem, i.e. the solution of a physical boundary value problem, requires detailed knowledge of the internal boundary, and hence of \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}}).

[page 206, §1]    [206.1.1] While it is becoming feasible to digitize samples of several mm{}^{3} with a resolution of a few \ \mu\text{m} this is not possible for larger samples. [206.1.2] For this reason the true pore space \mathbb{P} is often replaced by a geometric model \widetilde{\mathbb{P}}. [206.1.3] One then solves the problem for the model geometry and hopes that its solution \widetilde{u} obeys \widetilde{u}\approx u in some sense. [206.1.4] Such an approach requires quantitative methods for the comparison of \mathbb{P} and the model \widetilde{\mathbb{P}}. [206.1.5] This raises the problem of finding generally applicable quantitative geometric characterization methods that allow to evaluate the accuracy of geometric models for porous microstructues. [206.1.6] The problem of quantitative geometric characterization arises also when one asks which geometrical characteristics of the microsctructure \mathbb{P} have the greatest influence on the properties of the solution u of a given boundary value problem.

[206.2.1] Some authors introduce more than one geometrical model for one and the same microstructure when calculating different physical properties (e.g. diffusion and conduction). [206.2.2] It should be clear that such models make it difficult to extract reliable physical or geometrical information.