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
| FPr,t,u,∂u/∂t,…,∇⋅u,∇×u,… | =0,r∈P⊂R3,t∈R | | (1a) |
| FMr,t,u,∂u/∂t,…,∇⋅u,∇×u,… | =0,r∈M⊂R3,t∈R | | (1b) |
for a vector of unknown fields ur,t as function
of position and time coordinates.
[204.1.10] Here the two-component porous sample S=P∪M is
defined as the union of two closed subsets P⊂R3
and M⊂R3 where P denotes the pore space
(or component 1 in a heterogeneous medium) and M
denotes the matrix space (or component 2).
[204.1.11]
In (1) the vector functionals FP and FM
may depend on the vector u of unknowns and its derivatives
as well as on position r and time t.
[204.1.12] A simple example for (1) is the
time independent potential problem
| ∇⋅jr | =0,r∈S | | (2) |
| jr+Cr∇ur | =0,r∈S | | (3) |
for a scalar field ur.
[204.1.13] The coefficients
contain the material constants CP≠CM.
[204.1.14]
Here the characteristic (or indicator) function χGr
of a set G is defined as
| χGr=1 for r∈G0 for r∉G. | | (5) |
[page 205, §0]
[205.0.1] Hence Cr is not differentiable at the internal
boundary ∂P=∂M, and this requires
to specify boundary conditions
| lims↘0n⋅jr+sn | =lims↘0n⋅jr-sn, | r∈∂P | | (6) |
| lims↘0n×∇ur+sn | =lims↘0n×∇ur-sn, | r∈∂P | | (7) |
at the internal boundary.
[205.0.2] In addition, boundary conditions on the sample boundary
∂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 j.
[205.0.5] An overview for possible interpretations of
u and 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
j produced by its gradient
according to (
3).
| j\ 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 P.
[205.1.3] The problem is to deduce information about the
solution ur,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 ur,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 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 χPr
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 χPr.
[page 206, §1]
[206.1.1] While it is becoming feasible to digitize samples
of several
mm3
with a resolution of a few
μm
this is not possible for larger samples.
[206.1.2] For this reason the true pore space P is
often replaced by a geometric model P~.
[206.1.3] One then solves the problem for the model geometry and hopes
that its solution u~ obeys
u~≈u in some sense.
[206.1.4] Such an approach requires
quantitative methods for the comparison
of P and the model 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 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.
