Transport and relaxation processes in a three dimensional
two-component porous medium 𝕊=ℙ∪𝕄 (defined
above in chapter II) may be formulated
very broadly as a system of partial differential equations
| Fℙr1,r2,…,u1r,u2r,…,∂uir∂rj,…,∂2uir∂rj∂rk,… | = | 0r∈ℙ |
|
| F𝕄r1,r2,…,u1r,u2r,…,∂uir∂rj,…,∂2uir∂rj∂rk,… | = | 0r∈𝕄 |
| (4.11) |
| F∂ℙr1,r2,…,u1r,u2r,…,∂uir∂rj,…,∂2uir∂rj∂rk,… | = | 0r∈∂ℙ |
|
for n unknown functions uir with i=1,…,n and
r=r1,…,rd∈ℝd.
Here the unknown functions ui describe properties of the
physical process (such as displacements, velocities, temperatures,
pressures, electric fields etc.), and the given functions
Fℙ,F𝕄 and F∂ℙ depend on a finite number of its
derivatives.
The function F∂ℙ provides a coupling between the
processes in the pore space and in the matrix space.
The main difficulty arises from the irregular structure
of the boundary.
The formulation may be generalized to porous media
with more than two components.
For a stochastic porous medium the solutions uir
of (4.11) depend on the random realization,
and one is usually interested the averages uir.
Some authors [2, 41] have recently emphasized
the difference between continuum descriptions such as
(4.11) and discrete descriptions such as
network models.
The next section will show that discrete formulations
arise as approximations and reduce to the continuum
description of the same phenomenon in an
appropriate limit.
