What is the characteristic length scale for permeability? Direct analysis from microtomographic data

Rock formation permeability is arguably the most important flow parameter associated with subsurface production and injection. Its importance is reflected by the number of techniques (well-log evaluation and correlation, core measurement and well testing) used to estimate it. Clearly permeability should be linked to other porous media properties (e.g., surface area, porosity, pore/grain size). There have been numerous attempts over the last sixty years to establish a relationship between the permeability of a rock and other characteristic rock properties. Most empirical approaches for the prediction of permeability, which has units of length squared, propose a function of a characteristic length scale, formation factor (tortuosity) and porosity. The most widely used is the Carmen-Kozeny equation where the length scale is equated to the hydraulic radius (pore volume / pore surface area). Other length scales used include a critical pore radius associated with mercury injection experiments (Katz-Thompson), lengths associated with NMR relaxation (e.g., T₂) and grain size and rock fabric measures. To uncover the relationship between permeability and other pore scale properties requires directly measuring the geometric and transport properties of the pore system. This is now possible with 3D microtomographic imaging (Knackstedt et.al. SPE 87009, Arns et.al. SPE 90368). In this paper we describe a comprehensive study of permeability correlation across a range of rock types. We directly compute permeability, formation factor, NMR response, hydraulic radius, rock fabric and texture, pore size and capillary pressure on 3D microtomographic images of 39 porous materials including over 30 clastic and carbonate samples from a wide range of reservoirs. Subsampling enables one to generate more than 6500 "independent" samples. Empirical correlations between permeability and various length scales are tested for a range of lithotypes including unconsolidated sands, homogeneous sands, consolidated reservoir sands, limestones and reservoir carbonates. We find that the most robust length scale correlation is based on the critical pore radius. All correlations which use the Formation factor as a measure of tortuosity give good predictions. Empirical correlations for permeability based on grain size perform well for permeabilities greater than one Darcy.