The study of transport and relaxation in porous media is scattered throughout many fields of science and technology ranging from mathematics [9, 10] through solid state physics [11, 12, 13] and materials science [14, 15, 16] to applications in in geology [17, 18, 19], hydrology [20, 21], geophysics [22, 23], environmental technology [24, 25, 26], petroleum engineering [27, 28, 29], or separation technology [30]. In recent years a large number of books [31, 2, 5, 32, 33, 34, 35, 36, 37, 38, 38a] and comprehensive reviews [39, 22, 40, 41, 42, 43, 44, 8, 17, 45] have discussed transport and relaxation in porous media. Therefore the present review will try to emphasize those aspects of the central question which are complementary to the existing discussions.

As pointed out by Landauer [46] more emphasis is often placed on calculational schemes for effective transport properties than on finding general geometric characterizations of the medium which can be used as input for such calculations. Consequently the present review puts more emphasis on the first subproblem of geometric characterization than on the second subproblem of solving equations of motion for media with correlated disorder. Geometrical characterizations of porous media are discussed in chapter III, and they fall into two categories discussed in sections III.A and III.B. The first category are general theories which attempt to identify general and well defined geometric quantities that can be used to distinguish between different classes of porous media. The second category are specific models which attempt to idealize one particular class of porous media by abstracting its most essential geometrical features and incorporating them into a detailed model. The main difference between the two categories is the degree to which they specify the geometric microstructure. In general theories the microstructure remains largely unspecified while it is specified completely in modeling approaches.

Dielectric relaxation and fluid transport are discussed in chapter V as representative examples for more general physical processes in porous media. Transport and relaxation processes in porous media invariably involve the disordered Laplacian operator where is the Nabla operator, superscript denotes transposition, and the second rank tensor field gives the fluctuating local transport coefficients. Dielectric relaxation and single phase fluid transport in porous media are problems of practical and scientific interest which show unexpected experimental behaviour such as permeability–porosity correlations, Archie’s law or dielectric enhancement. The discussion in chapter V will specifically address these issues. Methodically the discussion in chapter V will emphasize homogenization theory and local porosity theory because they allow to control the macroscopic limit.

The upscaling problem as the controlled transition from microscopic to macroscopic length scales will then become the focus of attention in chapter VI discussing two phase fluid transport. The upscaling problem for two phase flow is largely unresolved. Recent work [47, 48] has revisited the fundamental dimensional analysis [49] dating back more than 50 years, and uncovered a tacit assumption in the analysis which could help to resolve the upscaling difficulties. Thus the upscaling problem is treated in chapter VI merely by comparing the microscopic and macroscopic dimensional analysis, and not by calculating effective relative permeabilities. Despite its simplicity the revisited dimensional analysis allows quantitative estimates of fluid transport rates and gravitational relaxation times based on the balance of viscous, capillary and gravitational forces in the macroscopic limit.