Percolation models of the permeability properties of media

Models that describe the permeability of media with allowance for the structure of the pore space are considered. It is proposed to use percolation theory to describe the topology of the pore space. If the distribution of the pore channels in the medium is random, percolation theory makes it possible to determine the percolation threshold, and also to estimate the fluid conductivity of the cluster that then results. Results obtained for models of granular, porous, and cracked media are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 81–86, May–June, 1984.     

Effect of the microinhomogeneity of the medium on the flow law in porous media

A lattice percolation model of an inhomogeneous medium [6] introducing f(r), the probability density function of the microcapillaries with respect to their radius r, is employed. The calculation scheme described makes it possible to determine the dependence of the permeability K on the pressure gradient G for media with arbitrary f(r). It is shown that for inhomogeneous media the behavior of K(G) is mainly determined by the form of f(r). The question of the effect of the state of stress on the permeability of the medium is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 84–94, March–April, 1989.     

Percolation model of two-phase equilibrium flow in a porous medium with microheterogeneous wettability

The percolation model of two-phase flow described in [1, 2] is used as a basis for examining the problem of the behavior of the characteristics of two-phase equilibrium flow in a porous medium when the capillaries have a radius distribution and differ with respect to the wettability properties of their surfaces. Analytic expressions describing the dependence of the relative phase permeability coefficients on the saturation of the medium by the displacing phase and the microinhomogeneous wettability parameters are obtained. A qualitative comparison shows the theoretical results to be consistent with the data of a direct numerical computer calculation of a grid model [3]. The effect of the microinhomogeneity parameters and the form of the capillary radius distribution function on the phase permeabilities is analyzed within the framework of the approach developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–93, September–October, 1989.     

Percolation model of an inhomogeneous anisotropic medium

A model of the variation in capillary conductivity is proposed. The change in the permeability of an inhomogeneous medium under load is investigated on the basis of the percolation model [3] and is numerically modeled for cases of hydrostatic compression and nonisotropic loading. The validity of the percolation approach to the determination of the change in flow properties under load is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1986.     

Vorticity of the velocity field for flow in a porous medium with random inhomogeneities

The vorticity field of the flow velocity in a porous medium with random inhomogeneities is considered in the correlation approximation of perturbation theory. The correlation tensor of the vorticity, the correlation between the vorticity and the permeability field, and the circulation of the velocity are calculated for three- and two-dimensional flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–160, July–August, 1982.     

Problems of pressureless nonlinear filtration

The article discusses pressureless flows for filtration with a limiting pressure gradient. In the plane of the hodograph of the filtration rate, the search for the stream function reduces to the solution of a boundary-value problem for an elliptical equation in a region with curvilinear boundaries. A numerical solution of this problem is constructed by the method of finite elements, after which the whole picture of the flow is reconstructed by integration. An analysis is made of the effect of a limiting gradient on the integral characteristics of the flow. The fundamental special characteristics of problems of pressureless filtration in the presence of a limiting gradient are noted: the nonsingularity of the solution, the formation of stagnant zones adjacent to the free surface, etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 178–181, March–April, 1978.     

A Statistical Correlation Between Permeability,Porosity, Tortuosity and Conductance

A statistical evaluation of 13,000 numerical simulations of random porous structures is used to establish a correlation between permeability, porosity, tortuosity and conductance. The random structures are generated with variable porosities, and parameters such as the permeability and the tortuosity are determined directly from the structures. It is shown that the prevalent definition of tortuosity, as the ratio of length of the real flow path to the projected path in the overall flow direction, does not correlate with permeability in the general case. Also, the correlation between the conductance of the medium, as an indicator of the accessible cross section of a flow path and permeability is no more reliable than the permeability–porosity correlation. However, if the definition of tortuosity is corrected using the cross-sectional variations, the resulting parameter (i.e., the minimum-corrected tortuosity) has a reliable correlation with permeability and can be used to estimate permeability with an acceptable error for most of the simulations of the random porous structures. The feasibility of extending the conclusions from 2-dimensional to 3-dimensional configurations and the numerical percolation thresholds for random structures are also discussed.     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Mechanism of turbulent boundary layer separation from a smooth wall

The mechanism of turbulent boundary layer separation under the influence of a positive pressure gradient is analyzed. The process of turbulent separation from a smooth wall in a plane diffuser channel has been experimentally investigated. It is shown that separation is determined by the nature of the flow in a certain inner part of the boundary layer, where the friction effect is unimportant. This region of the boundary layer is most exposed to the action of the positive pressure gradient and it is there that the stagnant zone primarily appears.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 69–77, November–December, 1990.     

Surface porosity and permeability of porous media with a periodic microstructure

Various ways of determining the surface porosity, the relation between the porosity and the surface porosity and the representation of the permeability in terms of the characteristics of the microstructure of the porous medium are analyzed with reference to model porous media with a periodic microstructure. It is shown that it is necessary to distinguish between the geometric (scalar) and physical (tensor) suface porosities and that the geometric surface porosity, the physical surface porosity and the porosity are different characteristics of the porous medium.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 79–85, January–February, 1995.     

Nonstationary model of immiscible fluid flow through porous media

Unsteady two-phase flow through a microinhomogeneous porous medium is considered. A forest growth model — a percolation model that enables nonequilibrium effects to be taken into account — is proposed for describing the dynamics of the process. In the context of the plane problem expressions are obtained for determining the saturation and the characteristic dimensions of the stagnation zones of trapped phase behind the displacement front.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1993.     

Dispersion coefficient with the displacement of a gas from a porous medium by a gas

The article gives the dependence of the dispersion coefficient on the Péclet number, obtained with the analysis of experimental data on the displacement of a gas by a gas from a porous medium of varying permeability.Translated from Zhurnal Prikladkoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 142–145, July–August, 1975.The authors are grateful to N. P. Baturina and L. N. Demidova for their aid in carrying out and analyzing the experiments, and to V. M. Ryzhik for evaluating the results of the work.     

Unsteady flows of a viscoplastic medium in channels

We numerically study the nonstationary Poiseuille problem for a Bingham-Il’yushin viscoplastic medium in ducts of various cross-sections. The medium acceleration and deceleration problems are solved by using the Duvaut-Lions variational setting and the finite-difference scheme proposed by the authors. The dependence of the stopping time on internal parameters such as density, viscosity, yield stress, and the cross-section geometry is studied. The obtained results are in good agreement with the well-known theoretical estimates of the stopping time. The numerical solution revealed a peculiar characteristic of the stagnant zone location, which is specific to unsteady flows. In the annulus, disk, and square, the stagnant zones arising shortly before the flow cessation surround the entire boundary contour; but for other domains, the stagnant zones go outside the critical curves surrounding the stagnant zones in the steady flow. The steady and unsteady flows are studied in some domains of complicated shape.     

Solution of a problem of flow in a porous medium with limiting gradient

For the law of flow in a porous medium with limiting gradient studied previously in [1], an exact solution is found for the problem formulated in [2] of the plane steady motion of an incompressible fluid in a channel with a rectangular step. Particular cases of the solution obtained are given; these represent the solutions of the problem of flow past a broken wall and of motion from a point source in a strip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–78, January–February, 1985.     

Three-dimensional percolation from a bounded stratum to a single borehole

Three-dimensional percolation of a liquid from a nonuniform bounded stratum to a single borehole is considered. It is shown that the method leads to insignificant errors, even in those cases when the percolation zone differs substantially from the circular and the percolation differs from radial flow.Translated from Ivestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–127, November–December, 1971.     

Convective instability of a horizontal layer of fluid between permeable membranes

The equilibrium stability is investigated of a system consisting of two semi-infinite isothermal masses of fluid divided by a horizontal layer of finite thickness of the same fluid with a vertical temperature gradient directed downwards. The transition layer is separated by thin permeable membranes. Neutral stability curves are constructed for different membrane resistances. In the case of high permeability, the equilibrium is absolutely unstable with respect to monotonic-type longwave perturbations. For low permeability membranes, instability with respect to monotonic finite-wavelength perturbations is characteristic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–173, July–August, 1985.     

Liquid and gas flows in porous media with allowance for induced anisotropy

The changes in the permeability of a porous medium resulting from the reorientation of the solid matrix particles under the influence of the percolating flow are considered. The mathematical model also contains the angular momentum equation, including the moment of the viscous flow forces. The state of the elastic matrix is characterized not only by the repacking strains but also by the particle orientation vector. The latter determines the anisotropy of the permeability tensor. The effective stress, strain, pore pressure and orientation vector fields in the neighborhood of an operating well are constructed. The effect of the induced permeability anisotropy on well productivity is noted.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 96–103, May–June, 1992.The authors are grateful to G. A. Zotov and V. A. Chernykh for their interest and support.     

Cross-Properties Relations in 3D Percolation Networks: II. Network Permeability

An efficient method to estimate the absolute permeability of three-dimensional percolation networks was proposed. It uses a Kozeny–Carman relationship in the form of a scaling law to relate the network permeability to its hydraulic characteristic length. This characteristic length was determined at the network percolation threshold using a three-dimensional extension of the Hoshen–Kopelman algorithm. For developing the scaling laws, the network permeability was calculated by solving the Kirchoff’s law for all sample spanning clusters that had been identified by the three-dimensional version of the Hoshen–Kopelman algorithm. The method was tested with simple cubic site-bond network models with and without spatial correlations. The universality of the exponents in the scaling laws were also investigated. It was shown that, once the scaling law has been derived, the permeability value can be estimated 3–9 times faster using the present method.     

Stagnant zones in viscoplastic media

The problem of the flow of a viscoplastic medium with linear viscosity, resulting from the motion of a rigid cylinder of arbitrary cross section has been considered by Oldroyd [1]. In [2] the author finds several exact particular solutions of this problem. Problems of a similar nature have also been examined in [3–4], In the following we consider the formation of stagnant zones in viscoplastic media.     

Steady Herschel–Bulkley fluid flow in three-dimensional expansions

In this paper we study steady flow of Herschel–Bulkley fluids in a canonical three-dimensional expansion. The fluid behavior was modeled using a regularized continuous constitutive relation, and the flow was obtained numerically using a mixed-Galerkin finite element formulation with a Newton–Raphson iteration procedure coupled to an iterative solver. Results for the topology of the yielded and unyielded regions, and recirculation zones as a function of the Reynolds and Bingham numbers and the power-law exponent, are presented and discussed for a 2:1 and a 4:1 expansion ratio. The results reveal the strong interplay between the Bingham and Reynolds numbers and their influence on the formation and break up of stagnant zones in the corner of the expansion and on the size and location of core regions.