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IV.A Examples

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

    \frac{\partial T({\bf r},t)}{\partial t}=\nabla^{T}\cdot({\bf D}({\bf r})\nabla T({\bf r},t)) (4.1)

    where {\bf r}\in\mathbb{S}, {\bf D}({\bf r})=\kappa({\bf r})/(c_{p}({\bf r})/\rho({\bf r}))>0 is the thermal diffusivity tensor, T({\bf r},t) is the space time dependent temperature field, \rho({\bf r}) is the density, \kappa({\bf r}) the thermal conductivity and c_{p}({\bf r}) the specific heat at constant pressure. The superscript T denotes transposition. If the tensor field {\bf D}({\bf r}) is sufficiently often differentiable the equations are completed with boundary conditions at the sample boundary \partial\mathbb{S}. For the microscopic description of diffusion in a two component porous medium \mathbb{S}=\mathbb{P}\cup\mathbb{M} whose components have diffusivities {\bf D}_{\mathbb{P}} and {\bf D}_{\mathbb{M}} the diffusivity field{\bf D}({\bf r}) has the form {\bf D}({\bf r})={\bf D}_{\mathbb{P}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\bf r})+{\bf D}_{\mathbb{M}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{M}}}({\bf r}) which is not differentiable at \partial\mathbb{P}. In such cases additional boundary conditions are required at the internal interface \partial\mathbb{P}, and the equation is interpreted in the sense of distributions [259]. Typical values for sedimentary rocks are \kappa _{\mathbb{M}}\approx 1..6 Wm{}^{{-1}}K{}^{{-1}}, \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{M}}}\approx 1..3 g cm{}^{{-3}} and c_{{p\mathbb{M}}}\approx 0.8..1.2 kJ kg{}^{{-1}}K{}^{{-1}}.

  • the Laplace equation with variable coefficients

    \nabla^{T}\cdot({\bf C}({\bf r})\nabla u({\bf r}))=0 (4.2)

    where {\bf C}({\bf r}) is again a second rank tensor field of local transport coefficients, u({\bf r}) is a scalar field, {\bf r}\in\mathbb{S} and the same remarks apply as for the diffusion equation with respect to differentiability of {\bf C}({\bf r}) and boundary conditions. For constant {\bf C}({\bf r}) the equation reduces to the Laplace equation \Delta u=0. If the medium is random the coefficient matrix {\bf C} is a random function of {\bf r}. 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 u is the electric potential and {\bf C} is the matrix of local (spatially varying) dielectric permittivity. In diffusion problems or heat flow u is the concentration field or temperature, and {\bf C} the local diffusivity. In Darcy flow through porous media u is the pressure and {\bf C} is the tensor of locally varying absolute permeabilities.

  • the elastic wave equation is a system of equations for the three components u_{i}({\bf r},t)(i=1,2,3) of a vector displacement field

    \frac{\partial^{2}u_{i}({\bf r},t)}{\partial t^{2}}=v_{s}^{2}\Delta u_{i}({\bf r},t)+(v_{p}^{2}-v_{s}^{2})\frac{\partial}{\partial r_{i}}\left(\sum _{{j=1}}^{3}\frac{\partial u_{j}({\bf r},t)}{\partial r_{j}}\right) (4.3)

    where v_{p} is the compressional and v_{s} the shear wave velocity of the material.

  • Maxwells equations in SI units for a medium with real dielectric constant \epsilon^{\prime}, magnetic permeability \mu^{\prime} and real conductivity \sigma^{\prime} and charge density \rho

    \displaystyle\nabla\cdot(\epsilon^{\prime}\epsilon _{0}{\bf E}({\bf r},t)) \displaystyle= \displaystyle\rho({\bf r},t) (4.4)
    \displaystyle\nabla\cdot(\mu^{\prime}\mu _{0}{\bf H}({\bf r},t)) \displaystyle= \displaystyle 0 (4.5)
    \displaystyle\nabla\times{\bf E}({\bf r},t) \displaystyle= \displaystyle-\frac{\partial}{\partial t}(\mu^{\prime}\mu _{0}{\bf H}({\bf r},t)) (4.6)
    \displaystyle\nabla\times{\bf H}({\bf r},t) \displaystyle= \displaystyle\sigma^{\prime}{\bf E}({\bf r},t)+\frac{\partial}{\partial t}(\epsilon^{\prime}\epsilon _{0}{\bf E}({\bf r},t)) (4.7)

    for the electric field {\bf E}({\bf r},t), magnetic field {\bf H}({\bf r},t) supplemented by boundary conditions and the continuity equation

    \frac{\partial\rho({\bf r},t)}{\partial t}\nabla\cdot\sigma^{\prime}{\bf E}({\bf r},t)=0. (4.8)

    Here \epsilon _{0}=8.8542\times 10^{{-12}}F/m is the permittivity and \mu _{0}=4\pi\times 10^{{-7}}H/m is the magnetic permeability of empty space.

  • the Navier-Stokes equations for the velocity field {\bf v}({\bf r},t) and the pressure field P({\bf r},t) of an incompressible liquid flowing through the pore space

    \displaystyle\rho\frac{\displaystyle\partial{\bf v}}{\displaystyle\partial t}+\rho({\bf v}^{T}\cdot\mbox{\boldmath$\nabla$}){\bf v} \displaystyle= \displaystyle\eta\Delta{\bf v}+\rho g\:\mbox{\boldmath$\nabla$}z-\mbox{\boldmath$\nabla$}P (4.9)
    \displaystyle\mbox{\boldmath$\nabla$}^{T}\cdot{\bf v} \displaystyle= \displaystyle 0 (4.10)

    where \rho is the density and \mu the viscosity of the liquid. The coordinate system was chosen such that the acceleration of gravity g points in the z-direction. These equations have to be supplemented with the no slip boundary condition {\bf v}=0 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].