Barothermal effect in three-phase flow through a porous medium with phase transitions

The formation of a temperature field in a porous medium due to the barothermal effect and phase transitions associated with oil, water, and gas flow is investigated. Degassing of the liquid has an important influence on the formation of the temperature field, the presence of stratal water tending to reduce the cooling anomaly.Ufa. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 113–117, November–December, 1994.     

Micromechanics of flow through porous media

The present state of development of the micromechanics of (primarily two-phase) flow through porous media is briefly reviewed: the aims, approaches, results achieved and promising research trends are discussed.Based on a paper presented to the Fluid Mechanics Section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 90–102, November–December, 1992.     

Approximations for solute transport through porous media with flow transverse to layering

Analytical approaches for the prediction of solute transport in layered porous media are investigated for the case of flow perpendicular to the direction of layering. One approach involves the use of averaging techniques to treat the profile as an equivalent homogeneous medium. The method is demonstrated on hypothetical and laboratory-measured data sets and a criterion for validity of the method is given. The second approach involves the use of time convolution to predict breakthrough curves for layered systems on the assumption that layer interactions have no significant effect on transport. Accuracy criteria are derived by comparing moments of the exact and approximate solutions and it is found that the convolution method has broader applicability than the equivalent single-layer analysis. An extension of the convolution method to include consideration of nonequilibrium transport due to the presence of mobile-immobile regions is presented and demonstrated by analysis of laboratory breakthrough data from a two-layer system exhibiting mobile-immobile regions.     

Nonisothermal viscoplastic fluid flow through porous media

The Alishaev model [1] is extended to the case of nonisothermal flow. Neglecting conductive heat transfer, it is shown that for the model in question in the plane of the complex potential not only are the problems linear but the decoupling of the thermal and hydrodynamic problems is also allowed. The latter is reduced to a mixed problem for an analytic function. This makes it possible to use the wellknown methods and results of the theory of limiting equilibrium pillars for isothermal flow [2–5]. It is also established that the solutions of the unsteady problems tend asymptotically to the solutions of the corresponding steady-state problems and can be obtained from the latter by simpler conversion. The effectiveness of the approach proposed is illustrated with reference to the problem of a source-sink system [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–122, July–August, 1990.     

Mathematical model of nonisothermal multiphase binary mixture flow through a porous medium

A model of nonisothermal binary mixture flow through a porous medium, applicable over a wide range of thermobaric conditions, including temperatures higher than the critical mixture temperature, is proposed. A nonclassical approach used for modeling the mixture properties makes it possible to uniformly describe its single-, two- and three-phase thermodynamic equilibria and the corresponding flows under sub- and supercritical thermodynamic conditions. The wide application of thermodynamic methods to determining the real mixture properties leads to a nonstandard mathematical model in which the conservation laws are closed with a conditional extremum problem, not finite or differential equations. A dispersion analysis of the model equations is performed and the characteristic velocities in zones of different mixture phase states are determined.     

Percolation model of two-phase flow through porous media

A physical model of the process of two-phase flow of immiscible fluids through a porous medium is developed and used to make an analytical calculation of the dependence of the relative phase permeabilities on the saturation of the medium by one of the phases. The theory is compared qualitatively with experiment for a model capillary radius frequency function and quantitatively with numerical calculations made on a computer. In both cases good agreement is obtained. The pressure dependences of the phase permeabilities are analyzed. The question of residual saturation with the wetting fluid after completion of the displacement process is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1987.     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

Model of flow through porous media with sources of hereditary type

A phenomenological approach is made to the study of transport processes in a medium whose structural blocks are separated by boundaries with high conductivity, so that the pressure head around the particles in a representative volume can be assumed to be the same. The essence of the approach consists in the fact that the effect of blocks on the flow through the cracks is taken into account by introducing into the usual flow equations distributed sources of hereditary type. A complete system of equations is presented, methods of solving it are discussed, and an example, in which the flow of methane at a coal face is described and conclusions are drawn concerning the relations between the various outburst criteria, is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 174–177, May–June, 1989.     

A resistance model for flow through porous media

A new model for resistance of flow through granular porous media is developed based on the average hydraulic radius model and the contracting–expanding channel model. This model is expressed as a function of tortuosity, porosity, ratio of pore diameter to throat diameter, diameter of particles, and fluid properties. The two empirical constants, 150 and 1.75, in the Ergun equation are replaced by two expressions, which are explicitly related to the pore geometry. Every parameter in the proposed model has clear physical meaning. The proposed model is shown to be more fundamental and reasonable than the Ergum equation. The model predictions are in good agreement with the existing experimental data.     

The transient two-dimensional flow through double porous media

This paper presents the analytical solutions in Laplace domain for two-dimensionalnonsteady flow of slightly compressible liquid in porous media with double porosity by usingthe methods of integral transforms and variables separation.The effects of the ratio ofstorativities ω,interporosity flow parameter λ,on the pressure behaviors for a verticallyfractured well with infinite conductivity are investigated by using the method of numericalinversion.The new log-log diagnosis graph of the pressures is given and analysed.     

Calculations on the flow of heterogeneous mixtures through porous media

Summary Muskat and Meres¹) have formulated basic differential equations governing the motion of heterogeneous fluid mixtures through porous media. They obtained a solution of these equations by numerical integration for the problem of a column initially filled with liquid saturated with gas, which is closed at one end and kept at a constant low pressure at the other¹).Buckley and Leverett²) obtained an analytical solution for the problem of a column initially filled with liquid which is flooded with a second immiscible liquid. They found a solution in which the saturation is a three-valued function of the coordinate along the column.In our paper in the first place a discussion is given of Buckley and Leverett's solution. It appeared that the true solution which contains a discontinuity may be derived from the three-valued solution by a discussion of the integral relation which represents the total liquid recovery from the column. This discussion bears a formal resemblance to that occurring in the theory of van der Waal's equation of state.The second problem treated in our paper is that of a vertical column initially filled with liquid saturated with gas under a high pressure, which is opened at its lower end. For this problem we succeeded in finding an analytical solution for low values of the pressure gradient. Here again the paradox of a three-valued solution occurred and led to a discontinuity in the saturation.A general argument is given to the effect that three-valued solutions are unavoidable for a theoretical treatment based on Muskat's equations of problems which lead to discontinuities in the saturation.Paper presented at the VIIth Congres of Applied Mechanics, London 1948.     

Transition to steady-state solutions of the equations of multicomponent flow through porous media

The question of the dynamics of the approach to steady-state solutions is examined in relation to the system of equations of equilibrium multicomponent two-phase flow through porous media [1, 2]. The equations are analyzed in the linear approximation for the case of small fluctuations. An expression is obtained for the characteristic transition time. Numerical modeling is carried out for substantial deviations from the steady-state solution. The problem of the plane linear displacement of a ten-component gas-condensate mixture by an enriched gas with subsequent transition to the extraction regime is solved. It is shown that the change in phase compositions and pressure proceeds with characteristic times of the order of the time required to create a spatially nonuniform distribution of the mixture properties during injection. However, the change in the saturations and overall composition takes place in times approximately 200 times greater than the injection time. The question of the existence of discontinuous steady-state solutions of the system of equations investigated is considered, and in the case of a binary mixture it is shown that such solutions cannot be realized.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 130–135, November–December, 1988.The author is grateful to V. N. Nikolaevskii for discussing his work.     

Mathematical modelling of flow through consolidated isotropic porous media

A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.     

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.