Laws of viscoplastic fluid flow in anisotropic porous and fractured media

In invariant tensor form, the laws of viscoplastic fluid flow are formulated for capillary and fractured media with a periodic microstructure that has orthotropic and transversely isotropic symmetry in the flow properties. An analysis of the laws of viscoplastic fluid flow in transversely isotropic and orthotropic porous and fractured media shows that in formulating the equations it is necessary to distinguish between the permeability tensor and the limiting gradient tensor, which may differ in the symmetry of the flow characteristics, and that the flow law is multivariant and admits one-, two-, and three-dimensional flows.     

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.     

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

Diffusion in anisotropic porous media

An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. Roman Letters A interfacial area between - and -phases for the macroscopic system, m² - A _e area of entrances and exits of the -phase for the macroscopic system, m² - A interfacial area contained within the averaging volume, m² - a characteristic length of a particle, m - b average thickness of a particle, m - c _A concentration of species A, moles/m³ - c _o reference concentration of species A, moles/m³ - c _A intrinsic phase average concentration of species A, moles/m³ - c _a c _A–c _A, spatial deviation concentration of species A, moles/m³ - C c _A/c ₀, dimensionless concentration of species A - binary molecular diffusion coefficient, m²/s - D _eff effective diffusivity tensor, m²/s - D _xx component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m²/s - D _yy component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m²/s - D _eff effective diffusivity for isotropic systems, m₂/s - f vector field that maps c _A on to c _a , m - h depth of the mixing chamber, m     

Comparison theorems for problems of nonlinear anisotropic flow through porous media

Flow law constraints that make it possible to establish comparison theorems (analogs of the theorems of [1, 2]) for nonlinear flows in an anisotropic inhomogeneous medium are formulated. In the theorems obtained the changes in the values of the pressure head and, moreover, the flow rate, filter velocity and pressure head gradients for such perturbations of the problem as the depression of individual surfaces, changes in the given boundary values of the head, etc., are established. The strict monotonicity of the relation between the flow rate and the pressure head difference in a region of the enlarged stream tube type and the possibility of an increase in flow rate with increase in flow resistance are demonstrated. The question of the correspondence between the constraints introduced and certain common models of porous media is discussed. Kazan'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 45–51, September–October, 1988.     

Upscaling the flow of generalised Newtonian fluids through anisotropic porous media

The homogenisation method with multiple scale expansions is used to investigate the slow and isothermal flow of generalised Newtonian fluids through anisotropic porous media. From this upscaling it is shown that the first-order macroscopic pressure gradient can be defined as the gradient of a macroscopic viscous dissipation potential, with respect to the first-order volume averaged fluid velocity. The macroscopic dissipation potential is the volume-averaged of local dissipation potential. Using this property, guidelines are proposed to build macroscopic tensorial permeation laws within the framework defined by the theory of anisotropic tensor functions and by using macroscopic isodissipation surfaces. A quantitative numerical study is then performed on a 3D fibrous medium and with a Carreau–Yasuda fluid in order to illustrate the theoretical results deduced from the upscaling.     

The effect of anisotropic permeability on free convective boundary layer flow in porous media

The effect of an anisotropic permeability on thermal boundary layer flow in porous media is studied. The convective flow is induced by a vertical, uniformly heated surface embedded in a fluid-saturated medium. A leading-order boundary layer theory is presented. It is shown that the thickness of the resulting boundary layer flow is different from that obtained in an isotropic porous medium. In general, an anisotropic permeability induces a fluid drift in the spanwise direction, the strength of which depends on the precise nature of the anisotropy. Conditions are found which determine whether or not the boundary layer flow is three-dimensional.     

Equations of nonlinear anisotropic flow through a porous medium

A general law of nonlinear anisotropic flow through a porous medium is proposed. A corresponding equation for the pressure of the fluid is obtained in velocity hodograph variables. The conditions of ellipticity of this equation are expressed in terms of the dissipative function.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1980.I thank V. M. Entob for discussing the work.     

Material instabilities of anisotropic saturated multiphase porous media

This paper analyses the material instability of fully saturated multiphase porous media. On account of the fact that anisotropic mechanical behaviours are widely observed in saturated and partially saturated geomaterials, the anisotropic constitutive model developed by Rudnicki for geomaterials is used to model the anisotropic mechanical behaviour of the solid skeleton of saturated porous geomaterials in axisymmetric compression test. The inertial coupling effect between solid skeleton and pore fluid is also taken into account in dynamic cases. Conditions for static instability (strain localisation) and dynamic instability (stationary discontinuity and flutter instability) of fully saturated porous media are derived. The critical modulus, shear band angle for strain localisation, and the bound within which flutter instability may occur are given in explicit forms. The effects of material parameters on material instability are investigated in detail by numerical computations.     

Nonlinear flow in fractured porous media

Conventional models of filtration in fractured porous bodies involve certain unwarrantable assumptions related to the definition of basic equations and the underestimation of a connection between the effective properties of a body and both the stress system and the pressure of a flowing fluid. A new theory is developed with the help of reconsidering those underlying assumptions and of a conception of the body being subject to elastic deformations. The theory is illustrated by means of studying a particular problem of stationary filtration.     

Vertical two-phase flow in porous media

New concepts are introduced to describe single-component two-phase flow under gravity. The phases can flow simultaneously in opposite directions (counterflow), but information travels either up or down, depending on the sign of the wavespeedC. Wavespeed, saturation and other quantities are defined on a two-sheeted surface over the mass-energy flow plane, the sheets overlapping in the counterflow region. A saturation shock is represented as an instantaneous displacement along a line of constant volume fluxJ _Q in the flow plane. Most shocks are of the wetting type, that is, they leave the environment more saturated after their passage. When flow is horizontal all shocks are wetting, but it is a feature of vertical two-phase flow that for sufficiently small mass and energy flows there also exist drying shocks associated with lower final saturations.     

Mixing with first-order decay in variable-velocity porous media flow

The transport and mixing of solutes undergoing first-order decay is central to many problems in groundwater hydrology. Mixing in porous media flow occurs due to advective dilution, hydrodynamic dispersion, and molecular diffusion. Mixing is stronger in regions of higher velocity, and weaker in slower-moving regions. Two-dimensional numerical experiments show that concentration profiles normal to the flow direction are displaced toward regions of slower flow in a flow field with a velocity gradient. Variable-velocity flow fields occur in subsurface flow around permeability heterogeneities, between recirculation cells, and in flow driven by natural convection. We examine two-dimensional solute concentration fields rather than breakthrough curves since for many complicated flow patterns, the breakthrough curve cannot discern important details of the concentration field. For the case of a species undergoing first-order decay, the effect of parent accumulation in regions of low velocity is enhanced for the daughter species because (i) the rate of daughter production is proportional to the local concentration of parent, and (ii) mixing is proportional to the local velocity. The resulting displacement of concentration profiles toward low-velocity regions may have important consequences for subsurface radionuclide transport and also for flows in chemically reactive systems and strongly coupled systems.Nomenclature d daughter component - d, ^(k) molecular diffusivity m² s^-1 - D dispersion coefficient m² s^-1 - dispersion tensor m²s^-1 - g acceleration of gravity vector m s^-2 - F Darcy flux vector kg m² s^-1 - identity matrix - k permeability m² - Kd distribution coefficient m³ kg^-1 - M mass accumulation term kg m^-3 - MW molecular weight kg mol^-1 - n outward unit normal vector - NK number of mass components (species) - p parent component - P total pressure Pa - q source term kg m^-3 s^-1 - R retardation factor - t time s, years - t _1/2 half-life s, years - u Darcy velocity in Y-direction m s^-1 - u Darcy velocity vector m s^-1 - v pore velocity m s^-1 - V volume m³ - X mass fraction - Y Y-coordinate - Z Z-coordinate (positive upward) - intrinsic dispersivity m - x distance interval m - surface area m² - decay constant s^-1 - dynamic viscosity kg m^-1 s^-1 - Ø porosity - fluid density kg m^-3 - tortuosity factor - d dispersion - L longitudinal - 0 reference value - T transverse - k mass component index     

Two-circle theorem of flow in porous media

In [1], using the method of conformai mapping, an attempt was made to extend the known circle theorem of flow in porous media [2, 3] to the case of two circular regions of the same radius occupied by the same medium. Below, using bipolar coordinates, the theorem is further extended to cover the problem of circular inclusions of different radii and different permeabilities. A corresponding theorem for two contiguous inclusions is presented as a limiting case. The exposition is illustrated with examples from which the results described in the literature follow as special cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 76–82, July–August, 1986.