Predicting transport properties in porous and fractured media, how fractal-based models can help petrophysicists?

Since the great paradigmatic revolution initiated by Mandelbrot, we know that fractals are ubiquitous in nature. From coastlines to plant growth, fractal mathematics help us to describe and quantify many of nature’s properties. In the same way, the fractal theory can be applied to porous and fractured media. In recent decades, numerous research studies have shown that fractal theory provides a solid framework to describe the properties of geological media. Based on advanced physical knowledge at the microscale, it is possible to use fractal patterns to describe transport properties in porous and fractured media. Fractal laws can be applied to describe the size distribution of pores and fractures, fracture widths, and pore irregularities, but also to relate these pore sizes to pore tortuosities. In this contribution, we review the significant advances that have been made in the field of petrophysics by applying fractal mathematics to describe fundamental petrophysical properties such as porosity, permeability, electrical conductivity, thermal conductivity, and electrokinetic and electroosmotic coupling coefficients. These new petrophysical models are based on the upscaling procedure applied to different fractal objects such as the Sierpinski carpet, Koch curves, Pigeon holes, and Menger sponge, among others. Among the interesting results obtained by means of fractal-based petrophysics, one can derive transport properties of saturated or partially saturated media, above and below freezing temperature, and considering hysteretic behavior and reactive media dissolution/precipitation processes. Integrating these fractal-based petrophysical relationships into the laboratory or field-scale, numerical simulations are now opening a wide range of potential avenues for progress in near-surface and reservoir geophysics.