# Basics of porous media

Transport in porous media is applicable to a wide range of fields, including mechanical, chemical, environmental, and petroleum engineering, as well as geology. Porous media can be found naturally in rocks and sand beds, and also can be fabricated as wicks and catalytic pellets. They are essential component in high-technology devices such as fuel cells and heat pipes. A fundamental formulation of the governing equations within a porous media will be presented here. The variation of some of the assumptions made has been quantified in recent studies. For these analyses of the variants within a particular porous media model, see Alazmi and Vafai (2000, 2001), and Nield and Bejan (1999). A porous medium is a solid matrix with a several voids, or pores, which are continuously connected. The voids are filled with one or more fluids that can pass through the medium because the voids are interconnected. The void fraction in the solid matrix is most frequently referred to as the porosity, $\varepsilon$. Conversely, $1-\varepsilon$ is the volume fraction of the solid matrix, ${{\varepsilon }_{sm}}$, that makes the porous medium. If g(x,y,z,t) is a geometric function that is identically equal to 1 in a void, and 0 in the solid matrix, the porosity is defined as:

$\varepsilon =\frac{1}{\Delta V}\int_{\Delta V}{gdV}\qquad \qquad(1)$

ΔV is an elemental volume of the porous zone. When there is more than one phase present in a porous medium, there is a porosity of a single phase, ${{\varepsilon }_{k}}$, which is simply the volume fraction of a particular phase in an elemental volume.

${{\varepsilon }_{k}}=\frac{1}{\Delta V}\int_{\Delta {{V}_{k}}}{gdV}\qquad \qquad(2)$

The subscript, k, pertains only to phase k. The volume fraction of the solid matrix in an elemental volume is represented by ${{\varepsilon }_{sm}}$. Therefore,

$1=\varepsilon +{{\varepsilon }_{sm}}=\sum\limits_{k=1}^{\Pi -1}{{{\varepsilon }_{k}}}+{{\varepsilon }_{sm}}\qquad \qquad(3)$

To model a porous zone, it is imperative that one be familiar with the different length scales of the zone. The first dimension is the particle or void length scale, d, and the second dimension, L, is the system or porous zone length scale. If d is on the order of L, such as in a very thin porous layer on a heat transfer surface, where the bulk flow is normal to the surface, the porous zone can be directly modeled with minimal assumptions. However, this situation is usually not the case, and more often $d\ll L$. When $d\ll L$, direct simulation of transport characteristics in an individual pore is not practical, therefore the local mean velocity is used. Because of this difference in scale, the volume averages defined in averaging approaches:

$\left\langle {{\Phi }_{k}} \right\rangle =\frac{1}{\Delta V}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}$ and $\left\langle {{\Phi }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}$

are often used to describe a porous system.

Figure 1: Comparison of velocity distributions in an elemental volume in a porous medium

Figure 1 (a) shows how the flow in an individual pore can be very difficult to model when a larger scale is needed, and therefore a volume-averaged velocity, seen in Fig. 1 (b), is often more useful. For analyses in a porous system, one of two velocities are often used: the intrinsic average velocity, ${{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}$, which is the volume-averaged velocity over the volume filled with a particular phase, or the extrinsic average velocity, $\left\langle {{\mathbf{V}}_{k}} \right\rangle$, which is the extrinsic averaged velocity over an entire volume including a solid. These two velocities are related by the Dupuit-Forchheimer relationship:

$\left\langle {{\mathbf{V}}_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}\qquad \qquad(4)$

The governing equations for the fluid phase in a porous medium can now be developed. It is assumed that the porous medium is saturated with the fluid phase and that the relative velocity, Vf,rel, only refers to the relative motion of a fluid phase f with respect to the solid matrix. In this section, the solid matrix is assumed to be stationary, therefore ${{\mathbf{V}}_{f}}={{\mathbf{V}}_{f,rel}}$. The conservation laws will be derived for a porous zone by using a volume-averaging technique for a single fluid occupying the pores. The momentum equations will be derived in a manner in which the viscous interactions between a fluid and the solid matrix are modeled with a property of the porous wick called the permeability, K. The energy equation will be derived in such a way that local thermal equilibrium between the solid matrix and the fluid can be assumed, or local thermal nonequilibrium conditions can be assumed.

## References

Alazmi, B. and Vafai, K., 2000, “Analysis of Variants within the Porous Media Transport Models,” ASME Journal of Heat Transfer, Vol. 122, pp. 303-326.

Alazmi, B. and Vafai, K., 2001, “Analysis of Fluid Flow and Heat Transfer Interfacial Conditions between a Porous Medium and a Fluid Layer,” International Journal of Heat and Mass Transfer, Vol. 44, pp. 1735-1749.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Nield, D.A., and Bejan, A., 1999, Convection in Porous Media, 2nd ed., Springer-Verlag, New York.