Mass fractal dimension of soil macropores using computed tomography: from the box-counting to the cube-counting algorithm

Transport phenomena in porous media depend strongly on three-dimensional pore structures. Macropore networks enable water and solute to move preferentially through the vadose zone. A complete representation of their geometry is important for understanding soil behaviour such as preferential flow. Once we know the geometrical, topological and scaling attributes of preferential flow paths, we can begin computer simulations of water movement in the soil. The box-counting method is used in three dimensions (i.e. cube-counting algorithm) to characterize the mass fractal dimension of macropore networks using X-ray computed tomography (CT) matrices. We developed an algorithm to investigate the mass fractal dimension in three dimensions and to see how it compares with the co-dimensions obtained using the box-counting technique in two dimensions. For that purpose, macropore networks in four large undisturbed soil columns (850 mm x 77 mm diameter) were quantified and visualized, in both two and three dimensions, using X-ray CT. We observed an increasing trend between the fractal dimension and macroporosity for the four columns. Moreover, similar natural logarithm functions were obtained for the four cores by a least squares fit through plots of mass fractal dimension against macroporosity.