Calculation of the effective permeabilities of field-scale porous media (original) (raw)
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Effective permeability of multifractal porous media
Physica A: Statistical Mechanics and its Applications, 1992
We show how the real space renormalization group method can be used to calculate analytically the scaling exponents of the effective absolute permeability in multifractal porous media. The permeability fields considered are deterministic and random multifractals constructed with multiplicative processes. We discuss the implications of the results on the understanding of fluid ftow in oil reservoirs.
Scaling Invariant Effects on the Permeability of Fractal Porous Media
Transport in porous media
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Finite-size anisotropy in statistically uniform porous media
Physical Review E, 2009
Anisotropy of the permeability tensor in statistically uniform porous media of sizes used in typical computer simulations is studied. Although such systems are assumed to be isotropic by default, we show that de facto their anisotropic permeability can give rise to significant changes of transport parameters such as permeability and tortuosity. The main parameter controlling the anisotropy is a/L, being the ratio of the obstacle to system size. Distribution of the angle α between the external force and the volumetric fluid stream is found to be approximately normal, and the standard deviation of α is found to decay with the system size as (a/L) d/2 , where d is the space dimensionality. These properties can be used to estimate both anisotropy-related statistical errors in large-scale simulations and the size of the representative elementary volume.
A fast algorithm for estimating large-scale permeabilities of correlated anisotropic media
Transport in Porous Media, 1993
The problem of estimating large-scale permeabilities of reservoirs based on knowledge of the small-scale permeabilities is addressed. We present an accurate and fast algorithm to calculate the global permeabilities of two-or three-dimensional correlated and anisotropic block samples, thus providing a fast algorithm for obtaining grid block permeabilities for reservoir simulators from small scale data. The algorithm is tested on both two-and three-dimensional tube networks generated from real images and fractal forgeries modeling porous media. In almost all cases, the algorithm estimates the correct global permeability (calculated using exact but slow algorithms) of the network to better than 5%. The new algorithm is comparable in speed to conventional averaging techniques, such as the geometric mean, but the obtained estimates are always much better.
arXiv (Cornell University), 2021
Understanding porous media properties and their scale dependence have been an active subject of research in the past several decades in hydrology, geosciences and petroleum engineering. The scale dependence of flow in porous media is attributed to small-and large-scale heterogeneities, such as pore size distribution, pore connectivity, long-range correlations, fractures and faults orientations, and spatial and temporal variations. The main objective of this study was to investigate how permeability (k) and formation factor (F) vary with sample dimension at small scales by means of a combination of pore-network modeling and percolation theory. For this purpose, the permeability and formation factor were simulated in twelve threedimensional pore networks with different levels of pore-scale heterogeneities. Simulations were carried out at five different network sizes, i.e., 1130, 2250, 3380, 4510 and 6770 microns (µm). Four theoretical models were also developed based on percolation theory to estimate the scale dependence of permeability and formation factor from the pore-throat radius distribution. In 2 addition, two other theoretical scale-dependent permeability models were proposed to estimate permeability at different scales from the pore-throat radius distribution and formation factor. Comparing theoretical estimations with numerical simulations showed that the proposed models estimate the scale dependence of permeability and formation factor reasonably. The calculated relative error (RE) ranged between-3.7 and 3.8% for the permeability and between 0.21 and 4.04% for the formation factor in the studied pore-networks.
Transport in Porous Media, 2007
We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance σ 2 Y of the local conductivity. Using perturbation expansions up to third order and fourth order in σ 2 Y obtained from the moment equation approach, we construct the general functional dependence of the scalar hydraulic conductivity in the regime where σ 2 Y is of order 1 and larger than 1. Comparison with available numerical simulations show that the proposed method provides reasonable improvements over available expansions.
Conceptual issues in upscaling of permeability of heterogeneous porous formations
We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure. Also, in many applications the interest is in average values of the dependent variables over scales larger or comparable to the conductivity length scales. Assigning values of the conductivity K b to averaging domains, or computational blocks, is the topic of a large body of literature, the problem being of wide interest in various branches of physics and engineering. It is clear that upscaling causes loss of information and at best it can render a good approximation of the fine scale solution after averaging it over the blocks.
Permeability anisotropy and its relations with porous medium structure
J. Geophys. Res, 2008
The complete permeability tensor of 18 porous rock cores was determined by means of X-ray tomography monitoring during the displacement of a salty tracer. To study the effect of the pore space geometry on the anisotropy of permeability, we compared the three-dimensional shape of the invasion front with the X-ray porosity maps obtained before injection. The samples (clean and shale-bearing sandstones, limestones, and volcanic rocks) belong to a broad variety of granulometry and pore space geometry. Their porosity ranges from 12 to 57%, and their permeability ranges from 1.5 Â 10 À14 to 4 Â 10 À12 m 2. For sandstones the permeability anisotropy is well correlated with the presence of bedding. For volcanic rocks it is clearly related to the orientation of vesicles or cracks. However, for limestones, no evident link between the geometry of the porous network and the permeability anisotropy appears, probably because of the influence of the nonconnected porosity that does not contribute to the hydraulic transport. This systematic work evidences the ability and the limitations of the tracer method to characterize the anisotropy of permeability in the laboratory in a simple and rapid way.