In an attempt to clarify these concepts, Richards' equation will be derived using the homogenization technique. To simplify the problem, the derivation is based on the assumptions that the isotropic porous material is an infinitely rigid solid (no significant deformations) kept under isothermal conditions (i.e., the transport of water is solely due to capillary suction). Other assumptions will arise during the development of the model.
The mathematical rules of the averaging technique can be found in textbooks  and . Only the basic definitions will be exposed in the following paragraphs. More information on the technique can also be found in Ref. . The technique is outlined here because it is at the core of development of all the transport equations.
As previously mentioned, the homogenization technique starts with a conservation and a transport equation at the microscopic level (i.e., at the scale of the pore). These equations are then integrated over a Representative Elementary Volume (REV), such as the one depicted in Fig. 1. The size of the volume depends on the intrinsic properties of the material. For instance, for concrete and mortar mixtures, the size of the REV depends on the maximum diameter of the aggregate particles. For the hydrated cement paste, the REV is typically a few cubic centimeters.
Fig. 1. The representative elementary volume (REV).
The total volume of the REV is given by Vo. The volume occupied by the liquid phase is designated by VoL. The volumetric fraction of liquid θL is the ratio of the liquid volume to the total volume:
The gaseous phase occupies a volume VoG. It is a mixture of air and water vapor. It is assumed that both air and the water vapor fill the whole gaseous phase volume. As for the liquid phase, the volumetric fraction of gas θG is the ratio of the gas volume to the total volume:
In the remainder of the text, the superscripts L and G will designate the liquid and gaseous phases, respectively. Furthermore, the superscript V will represent the water vapor phase within the total gaseous phase.
Let aα denote the amount per unit volume of some extensive quantity A in the phase α, either solid, liquid, or gas. Concentration or mass density can serve as examples for aα. Two averages can be defined. The volumetric phase average is given by:
The volumetric intrinsic phase average is defined as:
The two values are related by the following relationship: