Permeability from porosimetry measurements: Derivation for a tortuous and fractal tubular bundle

Abstract Permeability modeling of complex carbonate reservoirs is difficult. Porosity–permeability relationships are weak in carbonates and conventional porosity–permeability transforms give poor results. Even though the link between porosity and permeability in carbonates persists, other pore system properties, such as the largest connected pore-throat diameter, are more strongly linked to permeability. This important pore-throat diameter, as well as related porosity and other pore system architectural information, is determined by the analysis of mercury injection capillary pressure (MICP) porosimetry experiments. This paper explores the use of porosimetry data for the calculation of permeability as originally demonstrated by Purcell in 1949. We return to the tubular bundle model of Purcell and Burdine with a general mathematical form for the porosimetry data and a new tortuous and fractal relative tubular bundle. Using mathematical reasoning, without fitting parameters, we obtain a new formula for the computation of permeability based on the pore system architectural information of highly connected systems using the MICP porosimetry data. Moreover, we include the observation that the flow paths and the related tortuosity have a fractal aspect. The result is compared to an extensive porosimetry data set of the highly connected Arab D limestone, where vugs are absent. For porous media characterized by porosimetry data, the following expression emerges: κ ≈ 506 B v ∞ P d 2 e − 4.43 G , which is the permeability for a monomodal carbonate pore system characterized by a single Thomeer hyperbola with associated Thomeer parameters ( κ is in Darcy; B v ∞ in fractional bulk volume, and P d is the minimum entry pressure in psi and G is the pore-geometrical factor). The nearly equal sign is used here only because of an approximation used for the modified Bessel function of the second kind present in the general solution and approximate knowledge of the fractal exponent and the percolation path length ratio. There are no fitting factors. The exponents on the variables in our permeability formula demonstrate the significant shift in emphasis from porosity to the diameter of the largest connected pore throats, P d . Note that the presence of vugs are not considered in this work, since they do not form part of the Arab-D limestone matrix. This mathematical effort emphasizes the relative importance of pore system attributes on permeability as commonly found in carbonate porosimetry data. The approach can be readily extended to multimodal carbonate pore systems, to other sources of pore system architectural data and is shown to be equivalent to the operation of an incomplete Laplace transform on the porosimetry data. Importantly, and in contrast to previous permeability models to which we compare, this new formulation sets the stage for a complete and scale independent understanding of permeability in carbonate pore systems commonly encountered in the Arab D limestone and similar pore systems.