The transport of heat and mass through porous structures has been the focus of extensive research for many decades. Porous materials have excellent characteristics like large contact area, controllable pore sizes and low density, which are widely exploited in chemistry, biomechanics and fluid mechanics. Cellular heat exchangers utilise porous materials and are of particular interest in this research. These types of heat exchangers combine high conductivity materials with good enhancement of fluid mixing to increase heat transfer rates. However, the use of porous media presents challenges in the form of extremely complex geometries, which are difficult to accurately represent and analyse. This research focuses on the use of fractals (or more correctly pre-fractals) for the representation of porous media and a new numerical analysis method to enable the application of continuum thermal analysis. This is achieved by tessellating the continuum and extending classical continuum mechanics by a procedure coined tessellated continuum mechanics for the study of the thermo-mechanical response of porous media. The new procedure for the representation of porous materials involves pre-fractals which can produce extraordinarily complex porous geometries using a relatively small number of linear affine contraction maps. This approach is mirrored by an almost identical approach for the creation of tessellations but in this case affine expansion maps are employed. Elements on a pre-fractal are placed in a one-to-one correspondence with tiles in a tessellation and the associated bijection map is termed a hole-fill map. With tiles doubling up as elements, numerical analysis can be performed on the tessellation and the results immediately "lifted" to the corresponding pre-fractals. The whole approach is shown to be extremely accurate with discontinuous physics on tessellations being accounted for with a new concept termed the discontinuity network. Results obtained by the new approach are contrasted with direct analysis using a commercial package and high accuracy is recorded.