The present chapter begins the discussion of physical processes in porous media involving the transport or relaxation of physical quantities such as energy, momentum, mass or charge. As discussed in the introduction physical properties require equations of motion describing the underlying physical processes. Recurrent examples of experimental, theoretical and practical importance include :
the disordered diffusion equation
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(4.1) |
where ,
is the thermal diffusivity tensor,
is the space time
dependent temperature field,
is the density,
the thermal conductivity and
the specific heat at constant
pressure.
The superscript
denotes transposition.
If the tensor field
is sufficiently often differentiable
the equations are completed with boundary conditions at the sample
boundary
.
For the microscopic description of diffusion in a two component porous
medium
whose components have diffusivities
and
the diffusivity field
has the
form
which is not differentiable at
.
In such cases additional boundary conditions are required at the
internal interface
, and the equation is interpreted
in the sense of distributions [259].
Typical values for sedimentary rocks are
Wm
K
,
g cm
and
kJ kg
K
.
the Laplace equation with variable coefficients
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(4.2) |
where is again a second rank tensor field of local
transport coefficients,
is a scalar field,
and the same remarks apply as for the diffusion equation
with respect to differentiability of
and boundary
conditions.
For constant
the equation reduces to the
Laplace equation
.
If the medium is random the coefficient matrix
is a random function of
.
Equation (4.2) is the basic
equation for the next chapter.
It is frequently obtained as the steady state limit
of the timedependent equations such as the diffusion
equation (4.1).
Other examples of (4.2) occur in
fluid flow, dielectric relaxation, or dispersion in
porous media.
In dielectric relaxation
is the electric potential
and
is the matrix of local (spatially varying)
dielectric permittivity. In diffusion problems or heat
flow
is the concentration field or temperature,
and
the local diffusivity. In Darcy flow through
porous media
is the pressure and
is the
tensor of locally varying absolute permeabilities.
the elastic wave equation is a system of equations for the three
components of a vector displacement field
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(4.3) |
where is the compressional and
the shear wave
velocity of the material.
Maxwells equations in SI units for a medium with real dielectric
constant , magnetic permeability
and real
conductivity
and charge density
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(4.4) | |
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(4.5) | |
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(4.6) | |
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(4.7) |
for the electric field , magnetic field
supplemented by boundary conditions and the continuity equation
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(4.8) |
Here F/m is the permittivity
and
H/m is the magnetic permeability
of empty space.
the Navier-Stokes equations for the velocity field
and the pressure field
of an incompressible liquid flowing through
the pore space
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(4.9) | |
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(4.10) |
where is the density and
the viscosity of the
liquid.
The coordinate system was chosen such that the acceleration
of gravity
points in the
-direction.
These equations have to be supplemented with the no slip
boundary condition
on the pore space boundary.
In the following mainly the equations for fluid transport and Maxwells equation for dielectric relaxation will be discussed in more detail. Combining fluid flow and diffusion into convection-diffusion equations yields the standard description for solute and contaminant transport [260, 261, 24, 262, 21, 26].