Chemical Engineering Journal 111 (2005) 225–236 Permeability–porosity relationship from a geometrical model of shrinking and lattice Boltzmann and Monte Carlo simulations of flow in two-dimensional pore networks Javier R. Quispe a , Roberto E. Rozas b , Pedro G. Toledo c,∗ a School of Environmental Science and Engineering, Catholic University of Temuco, PO Box 15-D, Temuco, Chile b Institut f ¨ ur Physikalische Chemie, Universit¨ at zu K ¨ oln, Luxemburger Str. 116, 50939 K¨ oln, Germany c Chemical Engineering Department, Surface Analysis Laboratory (ASIF), University of Concepci´ on, PO Box 160-C, Correo 3, Concepci´ on, Chile Abstract For a broad range of applications, the most important transport property of porous media is permeability. Here we calculate the permeability of pore network approximations of porous media as simple diagenetic or shrinking processes reduces their pore spaces. We use a simple random bond-shrinkage mechanism by which porosity is decreased; a tube is selected at random and its radius is reduced by a fixed factor, the process is repeated until porosity is reduced either to zero or a preset value. For flow simulations at selected porosity levels, we use precise Monte Carlo calculations and the lattice Boltzmann method with a 9-speed model on two-dimensional square lattices. Calculations show a simple power-law behavior, k ∝ φ m , where k is the permeability and φ the porosity. The value of m relates strongly to the shrinking process and extension, and hence to the skewness of the pore size distribution, which varies with shrinking, and weakly to pore sizes and shapes. Smooth shrinking produces pore space microstructures resembling the starting primitive material; one value of m suffices to describe k versus φ for any value of porosity. Severe shrinking however produces pore space microstructures that apparently forget their origin; the k–φ curve is only piecewise continuous, different values of m are needed to describe it in the various porosity intervals characterizing the material. The power-law thus is not universal, a well-known fact. An effective pore length or critical pore size parameter, l c , characterizes pore space microstructures at any level of porosity. For severe shrinking l c becomes singular, indicating a change in the microstructure controlling permeability, and thus flow, thus explaining k–φ power-law transitions. Continuation of the various k–φ pieces down to zero permeability reveals pseudo-percolation thresholds φ ′ c for the porosity of the controlling microstructures. New graphical representations of k/l 2 c versus φ - φ ′ c for the various φ intervals display straight and parallel lines, with a slope of 1. Our results confirm that a universal relationship between k/l 2 c and φ should not be discarded. © 2005 Published by Elsevier B.V. Keywords: Critical pore length; Pore space microstructure transitions; Pore-level flow; Permeability–porosity relation 1. Introduction As our conceptual understanding and numerical expertise to simulate more and more complex flow and transport sys- tems increase, the accuracy of current simulations hinges on the quality and completeness of input and system parame- ters. This is true in modeling flow in oil and gas rocks, de- termining flow in underground reservoirs and fate of chem- ical contaminants in the vadose zone, assessing the effec- ∗ Corresponding author. Tel.: +56 41 204534; fax: +56 41 247491. E-mail address: petoledo@udec.cl (P.G. Toledo). tiveness of leaching processes, and optimizing filtration and sedimentation operations. For a broad range of applications, the most important transport property of porous media is per- meability. Until the 1980s, numerous relations between per- meability (k) and porosity (φ) were proposed, starting with the Kozeny–Carman relation based on capillary tube models [1,2], however all seem to lack universality and predictabil- ity. For a review of the history of the k–φ relationships, see for instance, [3]. In the 1980s, the diagenetic origin of rock pore space and the geometric-topologic similarities between sedimentary rocks led Katz and Thompson (K&T) [4] to the idea than certain common elements of pore formation may 1385-8947/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.cej.2005.02.003