THE FRACTAL MENGER SPONGE AND SIERPINSKI CARPET AS MODELS FOR RESERVOIR ROCK PORE SYSTEMS .4. RELATIONSHIP OF FRACTAL DIMENSION TO THE MEASURED PERMEABILITY OF NATURAL FRACTAL RESERVOIR ROCKS

The rate at which a fluid can travel through the pore system of a rock is controlled by the path along which it must travel. This path is a subset of the overall geometry of the pore system. The geometry of the pore/rock system can be described by apparent fractal dimensions. The apparent surface fractal dimension D(s)' obtained from the diameter-number distribution exhibits a wide range of values for any particular rock type, contains information about pore lacunarity, and contains information about multiple fractal processes. For natural fractal reservoir rocks, the apparent dimension D(s)' and the pore cross-sectional area shape factor S(a) are highly correlated with measured permeability. A single empirical equation is valid for the Arun Limestone, the San Andres Dolomite, the Spraberry Sandstone, and the Norphlet Sandstone, suggesting that D(s)' and Sa are quantitative descriptors of pore geometry and are independent of rock type. The validity of these two parameters can be shown from Poiseuille's and Darcy's equations. This relationship may be valid for a range of rock types for which there is a general correlation of pore cross-sectional area and pore throat cross-sectional area. In multifractal sandstone pore systems, only the fractal dimension of the large diameter pores is necessary in the empirical equation, suggesting that below some threshold length, the pores do not contribute substantially to the flow path of the fluid through the rock and can be considered ineffective.