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

(1a) | |||

(1b) |

for a vector of unknown fields as function of position and time coordinates. [204.1.10] Here the two-component porous sample is defined as the union of two closed subsets and where denotes the pore space (or component 1 in a heterogeneous medium) and denotes the matrix space (or component 2). [204.1.11] In (1) the vector functionals and may depend on the vector of unknowns and its derivatives as well as on position and time . [204.1.12] A simple example for (1) is the time independent potential problem

(2) | |||

(3) |

for a scalar field . [204.1.13] The coefficients

(4) |

contain the material constants . [204.1.14] Here the characteristic (or indicator) function of a set is defined as

(5) |

[page 205, §0] [205.0.1] Hence is not differentiable at the internal boundary , and this requires to specify boundary conditions

(6) | ||||

(7) |

at the internal boundary. [205.0.2] In addition, boundary conditions on the sample boundary 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 and the current . [205.0.5] An overview for possible interpretations of and 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.

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 . [205.1.3] The problem is to deduce information about the solution 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 . [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 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.

[205.2.6] The geometrical problems arise because in practice the pore space configuration 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 .

[page 206, §1] [206.1.1] While it is becoming feasible to digitize samples of several mm with a resolution of a few this is not possible for larger samples. [206.1.2] For this reason the true pore space is often replaced by a geometric model . [206.1.3] One then solves the problem for the model geometry and hopes that its solution obeys in some sense. [206.1.4] Such an approach requires quantitative methods for the comparison of and the model . [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 have the greatest influence on the properties of the solution 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.