# Porous medium

Open-cell ceramic

A porous medium or a porous material is a material containing pores (voids).[1] The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usually a solid, but structures like foams are often also usefully analyzed using concept of porous media.

A porous medium is most often characterised by its porosity. Other properties of the medium (e.g. permeability, tensile strength, electrical conductivity, tortuosity) can sometimes be derived from the respective properties of its constituents (solid matrix and fluid) and the media porosity and pores structure, but such a derivation is usually complex. Even the concept of porosity is only straightforward for a poroelastic medium.

Often both the solid matrix and the pore network (also known as the pore space) are continuous, so as to form two interpenetrating continua such as in a sponge. However, there is also a concept of closed porosity and effective porosity, i.e. the pore space accessible to flow.

Many natural substances such as rocks and soil (e.g. aquifers, petroleum reservoirs), zeolites, biological tissues (e.g. bones, wood, cork), and man made materials such as cements and ceramics can be considered as porous media. Many of their important properties can only be rationalized by considering them to be porous media.

The concept of porous media is used in many areas of applied science and engineering: filtration, mechanics (acoustics, geomechanics, soil mechanics, rock mechanics), engineering (petroleum engineering, bioremediation, construction engineering), geosciences (hydrogeology, petroleum geology, geophysics), biology and biophysics, material science.

## Fluid flow through porous media

Fluid flow through porous media is a subject of common interest and has emerged a separate field of study. The study of more general behaviour of porous media involving deformation of the solid frame is called poromechanics.

The theory of porous flows has applications in inkjet printing[2] and nuclear waste disposal[3] technologies, among others.

## Pore structure models

There are many idealized models of pore structures. They can be broadly divided into three categories:

Porous materials often have a fractal-like structure, having a pore surface area that seems to grow indefinitely when viewed with progressively increasing resolution.[4] Mathematically, this is described by assigning the pore surface a Hausdorff dimension greater than 2.[5] Experimental methods for the investigation of pore structures include confocal microscopy[6] and x-ray tomography.[7]

## Laws for porous materials

One of the Laws for porous materials is the generalized Murray's law. The generalized Murray’s law is based on optimizing mass transfer by minimizing transport resistance in pores with a given volume, and can be applicable for optimizing mass transfer involving mass variations and chemical reactions involving flow proceses, molecule or ion diffusion.[8]

For connecting a parent pipe with radius of r0 to many children pipes with radius of ri , the formula of generalized Murray's law is: ${\displaystyle r_{o}^{a}={1 \over 1-X}\sum _{i=1}^{N}r_{i}^{a}}$, where the X is the ratio of mass variation during mass transfer in the parent pore, the exponent α is dependent on the type of the transfer. For laminar flow α =3; for turbulent flow α =7/3; for molecule or ionic diffusion α =2; etc.