Is it Possible to Predict Porosity at Different Scales?

Scaling in general is referred as mathematical transformations that allow calculating object characteristics from one scale to another. In Earth Sciences, we are interested to scale rock properties from the scale of measurement to the scale of modelling, as they are usually different. In particular, when there are only rock fragments or cuttings available, porosity can be extracted from SEM and/or thin sections using image processing. When there are cores available, porosity can also be measured from core plugs using, for example, gas porosimeters (Boyle’s law) or NMR lab measurements. However, the main issue is how to extrapolate these values to the wells and field scales. In other words, is there any scaling law or transformation that can be used for going from lab scale measurements to field scales or vice-versa? Previous works have shown fractal behaviour in pore space of some soils, sandstones and carbonates (e.g. Thompson et al. 1987; Posadas et al. 2001; Xie et al. 2010). As fractal geometry involves self-similarity and its corresponding power laws, it seems that scaling using this mathematical formalism is a promising implication. As a matter of fact, fractal porosity in sandstones has been found to be proportional to the ratio between the minimum and maximum limits of self-similarity to the power of D-Do, where D is the Euclidian dimension (2 for SEM and thins section images and 3 for full core plugs images) and Do is the corresponding fractal dimension or capacity dimension (Thompson et al. 1987). However, this fractal porosity relation implies that self-similarity might be limited or constrain to certain scales. If it is not constrained to the measure scales, which is indeed the need; it is very difficult to find. Few studies have shown multifractal behaviour in carbonate rocks. Multifractal systems are more complex than fractals. The multifractal systems present more than one exponent and one singularity, while the fractals possess only one. Xie et al. (2010) have presented an analysis of SEM images from Permian-Triassic carbonate rocks, showing that all their samples behave as fractal/multifractal. However, their samples have a very small range of porosities – between 0.4 to 8.8 %. On the other hand, it is not clear yet if the found power laws could be used to scale rock properties as porosity and permeability to field scales, except for Muller et al. (1995). Muller et al. (1995) have found a good and clear correlation between the multifractal exponent (D1) from SEM images and permeability from core plugs in chalk samples. In this paper, we aim to investigate if it is acceptable to generalize multifractal behaviour to all type of carbonate rocks, and if it is possible to estimate porosity at different scales using fractal geometry. To accomplish this, we use a set of carbonate samples from the Upper Cretaceous with a considerable range of porosities – between 1 to 31 %, and different type of rocks (mudstones, packstones, grainstones, wackestones, and rudstones).