A model is developed that allows accurate prediction of the permeability of a core sample of sedimentary rock, based on two-dimensional image analysis of its pore structure. The pore structure is idealized as consisting of a cubic network of pore tubes, with the tubes having an arbitrary distribution of cross-sectional areas and shapes. The areas and perimeters of the individual pores are estimated from image analysis of scanning electron micrographs of thin sections, with appropriate stereological corrections introduced to account for the angle between axis of the pore tube and plane of the thin section. The individual conductances of each tube are estimated from the measured areas and perimeters, using the hydraulic radius approximation. Variations in the pore diameter along the length of the tube are accounted for with a “constriction factor” whose derivation is based on laminar flow through an irregular tube. Effective-medium theory is used to find the effective single-tube conductance, based on the measured distribution of individual conductances. This procedure is applied to several consolidated North Sea reservoir sandstones, and some outcrop sandstones, with permeabilities ranging from 20 to 1400 mD. The predicted permeabilities are typically within a factor of 2 of the measured values, with an average error in logk of only 0.168.
Several methods are compared for estimating the core-scale permeability of a rock whose pore space is idealized as consisting of a cubic network of pore tubes having a distribution of pore-scale hydraulic conductances. The conductance distribution of the pores is estimated from image analysis of scanning electron micrographs of rock sections. An explicit solution of the network equations is used as a benchmark to establish the "exact" macroscopic permeability. The other three upscaling methods used are Kirkpatrick's isotropic effective medium approximation, Bernasconi's anisotropic effective medium approximation, and the generalized perturbation ansatz (GPA) proposed by Gelhar and Axness. The analysis is carried out on a suite of petroleum reservoir sandstones from the North Sea, with measured core-scale permeabilities ranging from 20 to 500 mD. The log-variances of the pore-scale hydraulic conductance distribution were in the range of 2-3. The predictions of both the Kirkpatrick equation and the GPA are in each case within 10% of those computed by explicit network calculation, and all permeability predictions are generally within a factor of two of the core-scale values measured in the laboratory. RÉSUMÉPlusieurs methodes sont comparées pour estimer la perméabilité d'une roche dont l'espace poreux est idealise par un réseau cubique de tubes poreux ayant une distribution connue de Ia conductivité hydraulique a l'échelle des pores. La distribution de la conductivité des pores est estimée a partir d'une analyse d'image conduite sur les micro-images électroniques obtenues par balayage de sections de roche. Une solution explicite des equations representatives du réseau est utilisée comme repère pour établir la perméabilité macroscopique "exacte" du milieu. Les trois autres methodes employees sont 1'approximation moyenne efficace isotrope de Kirkpatrick, 1'approximation moyenne efficace anisotrope de Bernasconi, et la methode de perturbation généralisée (GPA) propose par Gelhar et Axness. L analyse est appliquée a une série de gres réservoirs pétrolifères de la Mer du Nord, présentant des perméabilités de masse mesurées s'étendant de 20 a 500 millidarcys. Les variations notées de la distribution de la conductivité hydraulique a l'échelle des pores étaient dans la gamme de 2 ou 3. Les previsions de l'équation de Kirkpatrick et du GPA sont dans chaque cas a moins de 10% de celles données par Ie calcul explicite de réseau, et toutes les previsions de perméabilité sont généralement proches, dans un facteur inférieur a 2, des valeurs a l'échelle de l'échantillon mesurées en laboratoire.
A model is developed that allows accurate prediction of the permeability of a core sample of sedimentary rock, based solely on two-dimensional image analysis of its pore structure. The only required input data are the areas and perimeters of the pores observed in, for example, an SEM image. The hydraulic radius approximation is used to estimate the individual pore conductances. Prior to this, stereological corrections are used to convert apparent pore geometries, and various hydraulic corrections are applied to account for converging-diverging flow paths. These various corrections are the principal and crucial differences between our approach and previous methods based on two-dimensional images. Finally, Kirkpatrick's effective medium approximation is used to find the effective value of the hydraulic conductances of the individual pores. The method has been applied to several data sets, including consolidated North Sea reservoir sandstones, outcrop sandstones, outcrop carbonates, and carbonates from Middle East oil and gas fields. The permeabilities of this entire data set range from 0.5–1377 mD, which covers a significant portion of the range of permeabilities that are relevant in reservoir engineering. In 80% of the cases, the permeabilities predicted by our method are within a factor of two of the measured values, and the predictions are within a factor of three in over 90% of the cases. The method requires minimal data manipulation and computation when compared to approaches that require three-dimensional imaging and/or full solution of the Navier-Stokes equations, and is much more accurate than primitive empirical methods such as the Kozeny-Carman equation. Aside from giving insight into the influence of pore structure on permeability, our method offers the potential of permitting permeability predictions to be made using drill cuttings, in situations where it is not possible to recover intact core. Another possible future application is to use downhole borehole imaging technology to provide an image with the appropriate resolution, thereby allowing in situ permeability estimation, without the need for core samples. Introduction Permeability is arguably the most important petrophysical property of a reservoir rock, and the ability to predict its value without time-consuming and expensive laboratory measurements would obviously be of great practical value. As the permeability is entirely controlled by the pore geometry, the possibility arises of estimating the permeability from quantifiable attributes of the pore space. This problem can be addressed at various levels of detail, with the resulting models requiring varying amounts of microstructural data. At one extreme, simple models such as that of Kozeny-Carman attempt to predict the permeability using knowledge only of the porosity and a single "characteristic length" such as the mean pore diameter, mean grain size, or specific surface (Berryman and Blair, 1986; Blair et al., 1996). Although easy to implement, the Kozeny-Carman equation is usually found to be insufficiently accurate for reservoir characterization purposes. The Katz-Thompson equation (Katz and Thompson, 1986) can yield accurate estimations of the permeability, using the porosity and the electrical formation factor. However, the requirement of having a measured value of the electrical formation factor is a disadvantage of this method.
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