Self-similar solutions of the problem of displacement of one gas by another in an axisymmetric case with a quadratic drag law

A problem of piston-induced displacement of one gas by another in cracks (porous media) in an axisymmetric case with a quadratic drag law is studied. Self-similar solutions for determining the dynamic characteristics (velocity and pressure) of the displacing and displaced gases are constructed in quadratures. The velocity and pressure are studied as functions of a self-similar variable for several initial conditions and parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 87–92, September–October, 2008.     

初始压力对多孔介质中气体水合物生成的影响

利用自制的一维天然气水合物生成与开采模拟实验系统,实验研究多孔介质中天然气水合物生成时不同初始压力对生成量、生成时间的影响.分别用相同气水比注入、相同注气量不同注水量、相同注水量不同注气量三种方式来控制初始压力.结果表明:在砂粒粒径300μm～500μm,盐水质量浓度2%,系统温度为2℃、初始压力为5MPa～9MPa的条件下进行水合物的等容生成实验时,初始压力越大,生成的水合物量越多,水合物开始生成的时间越早;但初始压力越大,实验系统中水合物生成最终稳定所需的时间越长.本实验系统采用的三种不同的控制初始压力的方式都可以得到上述结果.由此,可以为今后室内进行天然气水合物的生成实验提供科学指导.     

Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress—strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 95–104, June 2005.     

Investigation of the flow stability of liquids with incipient gas bubbles in porous media

The stability of the steady-state flow regimes of a liquid with dissolved gas in a porous medium is investigated in the region of the saturation pressure. It is shown that under certain conditions periodic and stochastic self-oscillations caused by the accumulation in the porous medium and subsequent entrainment of the very small gas bubbles formed as a result of pressure reduction may arise. Experimental data that confirmed the theoretical results are presented. Ufa. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 66–73, March–April, 1994.     

Analytical Model for The Capillary Pressure Gradient in Oil–Water–Rock System

A simplified analytical expression for the capillary pressure gradient in homogeneous porous media is proposed. Basic assumptions are: (1) the three phase contact angle between two fluids and porous rock is finite, (2) the surface area of contact between two fluids is small in comparison with contact surface areas between each fluid and porous rock in the unit volume of the system under consideration, (3) the model corresponds to conditions when both phases are continuous.     

Effect of distributed gas injection on aerodynamic characteristics of a body of revolution in a supersonic flow

Results of experimental and numerical investigations of the effect of gas injection through a permeable porous surface on the drag coefficient of a cone-cylinder body of revolution in a supersonic flow with the Mach number range M _h = 3–6 are presented. It is demonstrated that gas injection through a porous nose cone with gas flow rates being 6–8% of the free-stream flow rate in the mid-section leads to a decrease in the drag coefficient approximately by 5–7%. The contributions of the decrease in the drag force acting on the model forebody and of the increase in the base pressure to the total drag reduction are approximately identical. Gas injection through a porous base surface with the flow rate approximately equal to 1% leads to a threefold increase in the base pressure and to a decrease in the drag coefficient. Gas injection through a porous base surface with the flow rate approximately equal to 5% gives rise to a supersonic flow zone in the base region.     

Coarse Scale Capillary Pressure in Heterogeneous Reservoirs

The problem of capillary pressure upscaling and generation of initial water saturation data on the simulation scale consistent with the underlying geological model is addressed analytically. The approach is based on the concept of random spatial variations of permeability and porosity. We have revised the previously published expression for the coarse scale capillary pressure (Desbarats, Water Resour Res 31, 281–288) and rigorously derived a new one avoiding the unnecessary assumptions. Both expressions are evaluated by their comparison to the directly averaged realizations of the corresponding random fields. Generally, the new expression is superior to the one previously published. The important features of the analytical coarse scale capillary pressure expression are the dependence of the endpoint water saturation on the drainage pressure, or equivalently, on the elevation over the free water level and an additional multiplier taking into account the variation of rock properties within the coarse scale grid-block. George A. Virnovsky was the main editor of this article and we, the co-writers, want to honour his memory. He died suddenly on 12 March 2008 before this article was published. George was an internationally recognized scientist especially in the areas of multi-phase upscaling, interpretation of special core analysis and pore scale modelling.     

An Analytical Model for Capillary Pressure–Saturation Relation for Gas–Liquid System in a Packed-Bed of Spherical Particles

Capillary pressure is considered in packed-beds of spherical particles. In the case of gas–liquid flows in packed-bed reactors, capillary pressure gradients can have a significant influence on liquid distribution and, consequently, on the overall reactor performance. In particular, capillary pressure is important for non-uniform liquid distribution, causing liquid spreading as it flows down the packing. An analytical model for capillary pressure–saturation relation is developed for the pendular and funicular regions and the factors affecting capillary pressure in the capillary region are discussed. The present model is compared to the capillary pressure models of Grosser et al. (AIChE J., 34:1850–1860, 1988) and Attou and Ferschneider (Chem. Eng. Sci., 55:491–511, 2000) and to the experiments of Dodds and Srivastava (Part Part Syst. Charact., 23:29–39, 2006) and Dullien et al. (J. Colloid Interface Sci., 127:362–372, 1989). The non-homogeneity of real packings is considered through particle size and porosity distributions. The model is based on the assumption that the particles are covered with a liquid film, which provides hydrodynamic continuity. This makes the model more suitable for porous or rough particles than for non-porous smooth particles. The main improvements of the present model are found in the pendular region, where the liquid dispersion due to capillary pressure gradients is most significant. The model can be used to improve the hydrodynamic models (e.g., CFD and cellular automata models) for packed-bed reactors, such as trickle-bed reactors, where gas, liquid, and solid phases are present. Models for such reactors have become quite common lately (Sáez and Carbonell, AIChE J., 31:52–62, 1985; Holub et al., Chem. Eng. Sci, 47, 2343–2348, 1992; Attou et al., Chem. Eng. Sci., 54:785–802, 1999; Iliuta and Larachi, Chem. Eng. Sci., 54:5039–5045, 1999, IJCRE 3:R4, 2005; Narasimhan et al., AIChE J., 48:2459–2474, 2002), but they still lack proper terms causing liquid dispersion.     

Experimental Study of the Productive Bed Filtration Mode Changed by an Unsteady Action on the Host Rock Blocks

A laboratory study including physical modeling of filtration processes in porous beds with similarity criteria satisfied is performed. It is demonstrated that weak dynamic actions on stressed host rock blocks can initiate repacking of the system of blocks, leading to an increase in pressure in the productive bed and in oil recovery. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 131–138, January–February, 2006.     

In-Situ Capillary Trapping of CO<Subscript>2</Subscript> by Co-Injection

Co-injection of water with CO₂ is an effective scheme to control initial gas saturation in porous media. A fractional flow rate of water of approximately 5–10% is sufficient to reduce initial gas saturations. After water injection following the co-injection, most of the gas injected in the porous media is trapped by capillarity with a low fractional volume of migrating gas. In this study, we first derive an analytical model to predict the gas saturation levels for co-injection with water. The initial gas saturation is controlled by the fractional flow ratio in the co-injection process. Next, we experimentally investigate the effect of initial gas saturation on residual gas saturation at capillary trapping by co-injecting gas and water followed by pure water injection, using a water and nitrogen system at room temperature. Depending on relative permeability, initial gas saturation is reduced by co-injection of water. If the initial saturation in the Berea sandstone core is controlled at 20–40%, most of the gas is trapped by capillarity, and less than 20% of the gas with respect to the injected gas volume is migrated by water injection. In the packed bed of Toyoura standard sand, the initial gas saturation is approximately 20% for a wide range of gas with a fractional flow rate from 0.50 to 0.95. The residual gas saturation for these conditions is approximately 15%. Less than approximately 25% of the gas migrates by water injection. The amount of water required for co-injection systems is estimated on the basis of the analytical model and experimental results.     

Linearized Burnett Equation for the Dynamic Pressure of a Relativistic Gas

The linearized Burnett equation for the dynamic pressure of a relativistic gas of hadrons is calculated from a relativistic kinetic theory. It is shown, as in a previous paper [1], that the coefficient of the term with a non-homogeneous temperature field, – the heating term – is bigger than the one with the divergence of the four-velocity, – the bulk viscosity term.     

Dynamic Analysis of an Infinite Cylindrical Hole in a Saturated Poroelastic Medium

In this paper, the dynamic response of an infinite cylindrical hole embedded in a porous medium and subjected to an axisymmetric ring load is investigated. Two scalar potentials and two vector potentials are introduced to decouple the governing equations of Biot’s theory. By taking a Fourier transform with respect to time and the axial coordinate, we derive general solutions for the potentials, displacements, stresses and pore pressures in the frequency-wave-number domain. Using the general solutions and a set of boundary conditions applied at the hole surface, the frequency-wave-number domain solutions for the proposed problem are determined. Numerical inversion of the Fourier transform with respect to the axial wave number yields the frequency domain solutions, while a double inverse Fourier transform with respect to frequency as well as the axial wave number generates the time-space domain solution. The numerical results of this paper indicate that the dynamic response of a porous medium surrounding an infinite hole is dependant upon many factors including the parameters of the porous media, the location of receivers, the boundary conditions along the hole surface as well as the load characteristics.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.     

Kinetics of degradation of a molecular beam in a gas at rest

Conclusion Based on a direct simulation, comprehensive and systematic studies of the degradation kinetics of a molecular beam at rest made it possible to reveal the qualitative and quantitative relaxation characteristics of the molecular beam to thermal equilibrium with the background gas within the ranges of mass ratios of the injected and background gases 0.1–20 and within velocity ratios of the injection for model molecules as hard spheres. The conclusions on the qualitative character of degradation are valid for a wider spectrum of parameters as well. As is established, at an energy of the injected molecules exceeding that of the background gas, during relaxation, a shock effect, i.e., the exceeding of the thermal energy of the injected gas over that of the background gas, is observed. Having been evaluated by analogy with supersonic-flow deceleration, the shock effect is the stronger, the closer the molecules of the injected and background gases in their masses. An initial increase in the energy of the injected molecules that is more considerable for light molecules has been discovered. The dependence of the energy relaxation length on the ratio of the masses and initial velocities of injected molecules has been established. The data obtained allow us to estimate the dimensions of the region of nonequilibrium mixing of injected and background gases with small values of the intensity of molecular flow in the beam. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 38, No. 4, pp. 103–110, July–August, 1997.     

Relaxation Effects of a Saturated Porous Media Using the Two-Dimensional Generalized Thermoelastic Theory

A model of the equations of a generalized thermoelasticity (GT) with relaxation times for a saturated porous medium is given in this article. The formulation can be applied to the GT theories: Lord–Shulman theory, Green–Lindsay theory, and Coupled theory for the porous medium. A two-dimensional thermoelastic problem that is subjected to a time-dependent thermal/mechanical source is investigated with the model of the generalized porous thermoelasticity. By using the Laplace transform and the Fourier transform technique, solutions for the displacement, temperature, pore pressure, and stresses are obtained with a semi-analytical approach in the transform domain. Numerical results are also performed for portraying the nature of variations of the field variables. In addition, comparisons are presented with the corresponding four theories.     

Self-similar problem of decomposition of gas hydrates in a porous medium upon depression and heating

We consider the specifics of decomposition of gas hydrates under thermal and depressive action on a porous medium completely filled with a solid hydrate in the initial condition. The existence of volumetric-expansion zones, in which the hydrate coexists in equilibrium with water and gas, is shown to be possible in high-permeable porous media. The self-similar problems of hydrate decomposition upon depression and heating are studied. Ii is shown that there are solutions according to which hydrate decomposition can occur both on the surface of phase transitions and in the volumetric region. We note that, in the first case, decomposition is possible without heat supply to a medium and even with heat removal. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 111–118, May–June, 1998.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part II. Applications of Model and Comparison of Interacting and Non-Interacting Capillary Bundle Models

In the first part of this work (Dong et al., Transport Porous Media, 59, 1–18, 2005), an interacting capillary bundle model was developed for analysing immiscible displacement processes in porous media. In this paper, the second part of the work, the model is applied to analyse the fluid dynamics of immiscible displacements. The analysis includes: (1) free spontaneous imbibition, (2) the effects of injection rate and oil–water viscosity ratio on the displacement interface profile, and (3) the effect of oil–water viscosity ratio on the relative permeability curves. Analysis of a non-interacting tube bundle model is also presented for comparison. Because pressure equilibration between the capillaries is stipulated in the interacting capillary model, it is able to reproduce the behaviour of immiscible displacement observed in porous media which cannot be modelled by using non-interacting tube bundle models.     

Time-Dependent Shape Factors for Uniform and Non-Uniform Pressure Boundary Conditions

Matrix–fracture transfer functions are the backbone of any dual-porosity or dual-permeability formulation. The chief feature within them is the accurate definition of shape factors. To date, there is no completely accepted formulation of a matrix–fracture transfer function. Many formulations of shape factors for instantly-filled fractures with uniform pressure distribution have been presented and used; however, they differ by up to five times in magnitude. Based on a recently presented transfer function, time-dependent shape factors for water imbibing from fracture to matrix under pressure driven flow are proposed. Also new matrix–fracture transfer pressure-based shape factors for instantly-filled fractures with non-uniform pressure distribution are presented in this article. These are the boundary conditions for a case for porous media with clusters of parallel and disconnected fractures, for instance. These new pressure-based shape factors were obtained by solving the pressure diffusivity equation for a single phase using non-uniform boundary conditions. This leads to time-dependent shape factors because of the transient part of the solution for pressure. However, approximating the solution with an exponential function, one obtains constant shape factors that can be easily implemented in current dual-porosity reservoir simulators. The approximate shape factors provide good results for systems where the transient behavior of pressure is short (a case commonly encountered in fractured reservoirs).     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.