Mechanical model of lung parenchyma as a two-phase porous medium

The anatomy and geometry of the lung at the micro- and macroscopic level have been described briefly. A notion of lung parenchyma — a macroscopically continuous medium whose mechanical properties result from those of microstructural components — has been adapted. Simplifying assumptions propounded in the constitutive model have been discussed. Two phases have been distinguished in the medium: the solid phase — a highly deformable, nonlinearly elastic skeleton in the form of a thin-walled tissue structure on the micro-scale — and the fluid phase — perfect gas (air) filterating through the structure. General constitutive relations for both phases and their mechanical interactions have ben formulated. Further, the fundamental set of differential equations of the quasi-static coupled problem has been developed. Large deformations, material nonlinearities, and dependence of permeability on skeleton deformation have been included. Matrix formulation of the problem has been presented from the point of view of the finite element method. An implicit iterative time integration scheme has been proposed. The algorithm has been illustrated with results of simple numerical tests.     

Flow through geometrically irregular media with lattice gas automata

Summary The determination of the permeability is an interesting problem of fluid dynamics of wide interdisciplinary concern. Many authors approached this subject by developing numerical models of flows through porous media at either macro-scale and micro-scale. According to the latter point of view, we present in this paper a verification of Darcy's law and a first determination of the permeability starting by the knowledge of the microstructure of a three-dimensional random medium. The flow is here reproduced by using a particular class of cellular automata with the Boltzmann approximation.

---

Sommario La determinazione del coefficiente di permeabilità rappresenta un interessante problema comune a molte discipline. Tale determinazione viene affrontata dai diversi autori o tramite la conoscenza di grandezze macroscopiche — come la resistività o la porositá — ovvero partendo dalla conoscenza della microstruttura del mezzo poroso e cercando di simulare il processo di moto fluido che si svolge lungo i canalicoli del mezzo stesso. Seguendo quest'ultimo punto di vista, viene qui di seguito presentata una tecnica alternativa per la verifica della legge di Darcy e, quindi, per una prima valutazione del coefficiente di permeabilità per un mezzo tridimensionale complesso. La simulazione dei moti fluidi viene effettuata tramite una particolare classe di gas reticolari che evolvono secondo le regole degli automi cellulari.

     

Mathematical model of the dissociation of gas hydrates coexisting with ice in natural reservoirs

The dissociation of gas hydrate coexisting with ice in a low-temperature natural reservoir is investigated. A mathematical model of the process consisting of a generalization of the Stefan problem and containing two unknown moving phase transition boundaries — the hydrate dissociation and ice melting fronts — is constructed. It is shown that in high-permeability reservoirs the velocity of the dissociation surface is higher than that of the ice melting surface. As the permeability decreases, the fronts change places. The problem is solved in the self-similar approximation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 84–92, March–April, 1993.     

Solution to problem of strong evaporation of a monatomic gas by the Monte Carlo method

The Monte Carlo method has been used to obtain a numerical solution to the problem of strong evaporation of a monatomic gas in which the molecules are modeled by pseudo-Maxwellian and hard spheres. A comparison with the results of other authors is made. The results agree well with the solution of the problem obtained on the basis of the model Bhatnagar—Gross—Krook kinetic equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 185–188, January–February, 1984.I should like to thank M. N. Kogan for discussing the results.     

The symmetry approximation for nonsymmetric permeability tensors and its consequences for mass transport

It is well known that the permeability has a tensor character. In practical applications, this is accounted for by the introduction of three principal permeabilities — three scalars — and three mutually orthogonal principal axes. In this paper, it is investigated whether this is always the exact way of describing anisotropy and, if not, what the consequences of the principal axes approximation are for flow and transport. First, it is shown that spatial upscaling may result in nonsymmetric large-scale permeability tensors, for which principal axes do not exist. However, it is possible to define generalized principal axes: three principal axes for the flux and three for the pressure gradient, with only three principal permeabilities. Since nonsymmetric permeability tensors are undesirable in practical applications, an approximation method making the nonsymmetric permeability symmetric is introduced. The important conclusion is then that the exact large-scale flux and large-scale pressure gradient do not have the same directions as the approximate flux and approximate pressure gradient. A practical consequence is that the principal axes approximation results in a difference between flux and transport direction. When considering miscible displacement or transport of mass dissolved in groundwater, the velocity component normal to the flux direction may be considered as a contribution to the transverse macro dispersion.     

Structure of weak shock waves with phase transitions of the first kind

An analytic solution is found to the problem of the structure of weak shock waves with phase transitions of the first kind; it is based on the use of the Mandel ' shtam—Leontovich—D ' yakov concept of second viscosity. The kinetic and thermodynamic characteristics of the problem are investigated, and also the condition of applicability of the second viscosity concept. It is shown that the obtained solution simplifies considerably if the vapor phase is regarded as a perfect gas. The results of calculations for the water—water vapor system are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–62, July–August, 1980.     

Study of fracture permeability using Lattice Gas Automata

We study the problem of flow permeability of fracture joints using Lattice-Gas Automata simulations. We model the fracture as a rough channel bounded by a self-affine surface. Changing the surface roughness exponent, rough walls having different microstructures are obtained. Different relative roughnesses — defined as the height of the largest surface asperity divided by the mean aperture — are obtained pulling apart the two surfaces that constitute the rough walls of the channel. We calculate the macroscopic variables volume flow rate and pressure difference using microscopic balances. In the low Reynolds number regime the pressure difference and the flow rate are linearly related (the behavior is described by Darcy's law). In this regime, we study the effect of geometry on the permeability. We have found that permeability is independent of the surface roughness exponentH and it is fully determined in terms of the relative roughness and mean aperture of the fracture joint. For larger Reynolds numbers a transition to a regime in which pressure difference and flow rate are not longer linearly related is observed. This transition is observed in a domain of Reynolds numbers for which the behavior in a smooth channel remains linear. We discuss this transition.     

Influence of anisotropy of the permeability on the convection and heat transfer in a porous annular layer

Investigations into the convective transport of heat in porous materials are of interest for many applications in connection with the problem of increasing the efficiency of thermal insulation. In [1–5], convection in Isotropic porous media was considered. However, in many cases porous materials have an essential anisotropy of their permeability. Convective heat transfer has been inadequately studied for this case. In [6], the linearized equations were used to study the convection between infinite horizontal planes with a filling of an anisotropic material; the value of the critical Rayleigh number was found, and this agreed satisfactorily with experimental data. In the present paper, we investigate numerically convection between two infinite coaxial cylinders with an anisotropic porous filling, using the equations of convection in the Darcy—Boussinesq approximation [1–3]. The permeability tensor in the annular region is constructed from its principal values, which can be found experimentally. A method of calculation is developed and a parametric study made of the structure of the flow and of the local and averaged characteristics of the heat transfer, which are of interest for the design of thermal insulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–64, January–February, 1980.     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

Experimental investigations of the unsteady rotating flow field in a cylindrical vessel

The rotating flow field in a cylindrical vessel — the so-called whirlpool — is widely used in food engineering as a method for separating particles out of a suspension (Cup-of-tea-method). However many of these whirlpools do not operate adequately or fail entirely. In order to solve this problem, the first step was to investigate the flow field and its time dependency which has not been sufficiently understood until now.The rotating flow in a cylindrical vessel — induced by a fluid jet during the filling period of this vessel — is slowed down by fluid friction after the closing of the inlet valve. The velocity fields to be found mainly near, and pressure distributions at the bottom of the vessel, are measured during this unsteady flow. The results, especially those which describe vortex systems, are used to improve the separation system. This paper is restricted to the hydrodynamic aspect. Therefore success in industrial applications can only be indicated.     

On the propagation of pressure waves in saturated porous media

The problem of the propagation of pressure waves through compressible porous media saturated with a slightly compressible fluid is considered. By using Darcy's law the problem is reduced to a mixed initial-boundary value problem for an equation of the heat conduction type with a nonlinear term. The method of quasi-characteristics is used to solve this equation numerically. Solutions of the wave propagation problem for media with different permeability coefficients are presented. A solution of the inverse problem of determining the permeability coefficient using wave-pulse test data is constructed on the basis of a set of solutions of the direct problem.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 81–84, November–December, 1996.     

Construction of two-phase equilibria in a non-elliptic hyperelastic material

This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable, two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps — which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution — unique up to an additive constant — for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.     

Nonlinear waves and localized structures in two-phase flow through porous media

Three problems of two-phase flow through porous media are considered. In the first two flows in the region of near-critical saturations are investigated. Since under these conditions the active saturation and hence the phase permeability of one of the phases are small, it is important to take into account the delay in phase redistribution — such types of flow as nonlinear waves and localized structures become important. In the third problem it is assumed that the capillary jump is insignificant as compared with the phase pressures. It is shown that in this case localized structures may also occur.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 71–76, January–February, 1988.     

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.     

Numerical Estimation of the Degree of Reservoir Permeability Heterogeneity Using Pressure Drawdown Tests

This study aims to correlate the response of pressure transient test to permeability distribution type. For this purpose, correlated permeability distributions in x–y direction are generated using fractional Brownian motion (fBm) as it has been shown in literature that permeability in carbonate reservoirs exhibits an fBm type distribution horizontally. 2-D fBm permeability distributions created using mid point displacement method are employed as data to a black oil simulator. The intermittence exponent, H or fractal dimension of the distribution, D, as defined by D=2 – H, characterizes the distribution type. All permeability distributions are normalized to represent the same arithmetic mean (20, 100, and 500 mD) and uniform variance so that only their fractal dimension that underlies the smoothness of the distribution distinguishes them. Many different realizations of permeability distributions are generated based on the random number seeds used and pressure transient (drawdown) tests are simulated using a black oil simulator package (ECLIPSE 100). Pressure transient analysis is performed using PanSystem package. As a base case and for the comparison purpose, the same procedure is repeated for the totally homogeneous case (the same permeability for all grids) and a random (normally distributed) permeability distribution with the same mean and uniform variance. The effects of permeability distribution type on the pressure response are clarified. A strong impact of heterogeneity is observed as an increase in skin effect with increasing fractal dimension of permeability distribution. This additional (or pseudo) skin effect due to heterogeneity is correlated to the fractal dimension of the permeability distribution. As a further step, the procedure is repeated for different flow rates applied during the drawdown test. The correlation between the fractal dimension of permeability distribution and additional skin is improved by incorporating the rate into it. The methodology followed can be used in the assessment of reservoir heterogeneity quantitatively using pressure transient response.     

Dependence of the Critical Load on the Geometric Characteristics of a Hinged Plate with a Crack

The plane problem of three-dimensional stability of a hinged plate with a central crack under uniaxial loading along the crack is considered. The net approach is used to solve the problem. The variational difference and gradient methods are used, respectively, to construct a difference scheme and to solve difference problems. The dependence of the critical load on two parameters — the crack length and the thickness ratio — is derived. Formulas for calculation of the critical load are given     

The Complex-Moduli Solution of an Axisymmetric Dynamic Problem of Coupled Thermoviscoplasticity Under Harmonic Loading

An approximate formulation is given to the axisymmetric dynamic thermoviscoplastic problem with coupled mechanical and thermal fields. The formulation employs an approximate cyclic thermoviscoplastic model based on the concept of complex moduli. The distribution of the field quantities and the temperature— and amplitude—frequency resonant characteristics of a circular disk under forced flexural vibrations are studied. The capabilities of the approximate model in solving this class of problems are also studied     

Rheological influences on the consolidation process of soils

Summary Consolidation of soils occurs when the imposed loading produces strains accompanied by the slow escape of pore-water from the voids between the particles. This straining is accompanied by rolling and sliding of the particles into a more dense packing.Terzaghi realized that the plastic resistance to deformation, in the case of clay consolidation, was so small that the consolidation process could be considered as a hydrodynamic problem; in this way, he used simplifying assumptions which led to the development of a time-settlement relationship for clays based entirely on the escape of pore-water; this is a linear equation.Consolidation theory gives a unique stress-strain-time relationship — provided that the strains are not sufficiently large to overcome the shear strength of the soil, with the consequence that rheological considerations govern the deformation process.In the case of soft soils, especially organic soils, the volume of water expelled during consolidation is very large and the shear strength of the soils small. The experimental results indicated that, at low stresses, the strain rates were low indicating plastic deformation of the material with the process governed by the hydrodynamic theory — while at higher stresses, the strain rates increased causing the soil to flow in viscous form and the rheological behaviour predominated over the hydrodynamic process.A further complication to theTerzaghi theory arises with soft soils having high water contents; during the consolidation process, the permeability is reduced as the particles assume a more dense packing.A Flow-Loading parameter was obtained for the soils which incorporated the changes in permeability as a function of loading. This parameter was incorporated in the consolidation theory and provided a non-linear second-order differential equation which can predict the settlement-time behaviour as a function of both loading and changing soil properties.With 6 figures     

Effect of the shape of a blunt nose on the drag coefficient of a body in a hypersonic rarefied gas flow

The effect of the shape of a blunt nose of a body located in a hypersonic rarefied gas flow on the field of flow and on the aerodynamic characteristics is studied in the example of flow round ellipsoids of revolution at a zero angle of attack. The problem of the flow in the transition regime is solved on the basis of numerical analysis of the model kinetic Bhatnagar—Gross—Krook (BGK) equation for a monatomic gas. The good agreement of the results of the numerical calculations with the experimental data in a broad range of Mach numbers has shown [1, 2] that the numerical solution of the model kinetic equations is a reliable and effective means for studying flow problems. In the case when the problem is posed of determining the laws of the purely force interaction of a flow with the body, sufficiently good accuracy is given by the use of the model BGK equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–192, March–April, 1985.     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.