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.     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.     

Hydrodynamic averaging of highly inhomogeneous porous media with linear flow

A method is proposed for constructing homogeneous anisotropic models of new classes of highly inhomogeneous nondeformable porous media consisting of arbitrarily oriented systems of layers representing fractures and impermeable barriers (screens) embedded in each other with an arbitrary depth of embedment. It is assumed that the permeability functions of the elementary cells of the media can be represented in some Cartesian coordinate system (proper to each cell) in the form of a product of three integrable functions that depend on the corresponding coordinates. As distinct from known methods of averaging differential operators, the method in question is based on porous media flow considerations and reduces to replacing the highly inhomogeneous soils with homogeneous anistropic soils, so that on the boundaries of the domain considered the basic flow parameters remain the same.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika, Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1993.     

Multicontinuum description of percolating flow through periodic inhomogeneous porous media

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 112–119, November–December, 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.     

Upper limit of the linear law of gas flow through a porous medium

The dependence of the critical Reynolds number Re₁ characterizing the upper limit of applicability of the linear flow law, on the other characteristics of the porous medium is considered. It is shown that Re₁ decreases with increase in the dimensionless inertial resistance coefficient on the developed inertial flow interval _*. Most of the known experimental data can be quite closely approximated by the expression Re₁=7_* ^–1.16. The effect of the error in determining Re₁ by means of the relation proposed on the error in finding the resistance coefficient of the porous medium is analyzed. It is concluded that the relation obtained can be used for determining Re₁ in engineering calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–190, January–February, 1991.     

On the Riemann-Hilbert's boundary value problem, for flow through porous inhomogeneous media

The present paper is an extension of other results concerning Emile Picard's Great Theorem [2], [3] used for the study of plane, stationary flow with free and seepage surfaces in porous inhomogeneous media of second type.     

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.     

Mathematical simulation of two-phase fluid flow through a water-flooded porous reservoir with sediment formation

The results of numerical investigations of the problem of flow through a porous reservoir in the process of its water flooding with addition of a sediment-forming component are given. The mathematical model is based on the mass conservation laws for each of the considered phases and components supplemented with the equations of motion and constitutive relations necessary to close the system of equations. In solving the problem, an empirical dependence of the sediment formation intensity on the content of the sediment-forming component in the aqueous solution with allowance for variation in the effective porosity of the medium is used. The main features in solving the sediment-formation problem are distinguished using the empirical dependence and a contrastive analysis of the effect of choosing this dependence on the solution results is carried out. It is shown that the neglect of the experimental results in the mathematical formulation can lead to not only unjustified overestimated results in realizing the method but also give a distorted pattern of the entire process of sediment formation in fluid flow through a water-flooded porous reservoir.     

Laws of flow with a limiting gradient in anisotropic porous media

The equations of viscoplastic fluid flow through a porous medium are written for all types of anisotropy. It is shown that in anisotropic media the flows with a limiting gradient are characterized by two material tensors: the tensor of permeability (flow resistance) coefficients and the tensor of limiting gradients. A complex of laboratory measurements for determining the tensors of permeability coefficients and limiting gradients is considered for all types of anisotropic media. It is shown that the tensors of permeability coefficients and limiting gradients are coaxial. Conditions of flow onset and fluid flow laws are formulated for media with monoclinic and triclinic symmetries of flow characteristics.     

Axial dispersion in fluid flow through porous plates in parallel and through a porous tube

Summary Effects of axial diffusion on liquid-liquid displacement in fluid flow through porous plates in parallel and through a porous tube are considered as problems of two zones in unsteady state mass transfer. The solutions of the differential equations of the system in terms of the Laplace transformed variable contain an infinite number of essential singularities in a complicated form. Therefore approximate solutions are obtained by numerical inversion of the Laplace transform. Some of the numerical results are presented and discussed.Nomenclature C ₁ concentration of solute in Zone 1 - C ₂ concentration of solute in Zone 2 - C ₀ initial concentration of solute in Zone 2 - D _e effective diffusivity - D* axial dispersion (mixing) coefficient - K ratio D*/D _e - P _e Péclet number, Xv/D _e, Rv/D _e - P _e * longitudinal Péclet number, Xv/D*, Rv/D* - R inner radius of a porous tube - t time - v average velocity of fluid flow through Zone 1 - W width of a porous plate - Y length of a porous plate (tube) - porosity - ₁ dimensionless concentration of solute in Zone 1, C ₁/C ₀ - ₂ dimensionless concentration of solute in Zone 2, C ₂/C ₀ - Laplace transform of ₁ - Laplace transform of ₂ - ₁ dimensionless distance in porous plate, x/X - ₂ dimensionless distance in a porous tube, r/R - ₁ dimensionless axial distance in porous plate, y/X - ₂ dimensionless axial distance in a porous tube, y/R - ₁ dimensionless time in porous plate, tD _e/X ² - ₂ dimensionless time in a porous tube, tD _e/R ² - Units CGS system     

Temperature field of heat sources during fluid injection in an anisotropic inhomogeneous reservoir

The problem of the temperature field produced by sources whose position does not depend on the vertical coordinate and which are concentrated in a horizontal permeable layer surrounded by a heat-conducting medium with radial steady-state fluid flow. The problem is solved using an averagely accurate asymptotic method. Analytical expressions for the zero-order approximation and the first coefficient of the expansion. A condition is determined under which the averaged problem for the remainder term has a trivial solution.     

Curve of the reestablishment of the pressure in a borehole in a fractured porous reservoir

It is shown that the slope of the initial segment of the curve for reestablishment of the pressure cannot be less than half of the asymptotic slope over a long period of time; the article gives the limits of the errors in determination of the characteristic size of the block from the known lag time for reestablishment of the pressure.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 137–145, September–October, 1971.The authors thank V. P. Stepanov for his discussion of the article and for his encouragement.     

Frictional passage of fluid through inhomogeneous porous system: a variational principle

A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in inhomogeneous isotropic porous media where fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an optimal nature of nonlinear paths in steady Darcy’s flows of fluids. An expression for the total path resistance leads to a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for the frictional fluid flux in Lagrange coordinates. This law shows that—by minimizing the total resistance—a ray spanned between two given points takes the shape assuring that a relatively large part of it resides in the region of lower flow resistance (a ‘rarer’ region of the medium).     

Calculation of the concentration fields of solid and fluid phases in the flow of a chemically active fluid through a porous medium

The flow of a chemically active fluid through a porous medium is considered. Under certain assumptions, an analytical solution is obtained which describes the distribution of the concentration of the chemically active component and the variation of the solidphase porosity. On the basis of the results obtained, the optimization of the hydrochloric acid treatment of wells is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 78–81, March–April, 1985.     

Creeping flow through a model fibrous porous medium

Pressure losses and velocity distributions were measured for creeping flow through three model fibrous porous media. The three models consisted of square arrays of circular rods with solid volume fractions of 2.5, 5 and 10%. Measurements of flow resistances are in good agreement with theoretical predictions after wall effects are accounted for using Brinkman’s equation. Two-dimensional velocity vector maps were obtained in each array using particle image velocimetry. The velocity distributions are necessary for identifying non-Newtonian effects in flows with viscoelastic fluids.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

Two-phase three-component flow through a porous medium with variable total flow

In analyzing the processes of the displacement of oil, in which intensive interphase mass transfer takes place, it is normally assumed that the partial volumes of the components as they mix are additive (Amagat's Law) [1, 2]. Then the equations of motion have an integral, which is the total volume flow rate through the porous medium, and the basic problems of frontal displacement, if there are not too many components in the system, permit an exact analytical study to be made [3–5]. If this assumption is rejected, the total flow becomes variable [3, 6, 7]. It appears that the consequences of this as applied to the processes of the displacement of oil by high pressure gases have not previously been considered. The results of such a study, developing the approach outlined in [4], are given below. The initial multicomponent system is simulated by a three-component one which contains oil (the component being displaced), gas (the neutral or main displacing component), and intermediate hydrocarbon fractions or solvent (the active component). It is shown that instead of the triangular phase diagram (TPD) normally used where the partial volumes of the components are additive, in this case it is convenient to use a special spatial phase diagram (SPD) of the apparent volume concentrations of the components to construct the solutions and to interpret them graphically. The method of constructing the SPD and its main properties are explained. A corresponding graphoanalytical technique is developed for constructing the solutions of the basic problems of frontal displacement which correspond to motions with variable total flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1985.