Linear problem of the dissociation of the hydrates of a gas in a porous medium

Self-similar solutions are obtained for the linearized equations of the flow of a gas in porous regions with a moving interface that describe the dissociation of gas hydrates when the pressure is reduced.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 174–177, January–February, 1980.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

Self-similar solution to a problem of axisymmetric motion of gas through a porous medium in the case of a quadratic law of resistance

The results of solution of the self-similar problem of planar flow of gas through a porous medium in the case of a quadratic law of resistance [1] are generalized to the case of axisymmetric motion. The equation in similarity variables for the velocity of isothermal gas flow is reduced to an equation having cylindrical functions as solution. Analytic dependences of the pressure and the gas velocity on the coordinate and time are obtained for a given flow rate of the gas at the coordinate origin and for zero Initial gas pressure in the porous medium.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 168–171, July–August, 1982.     

Dissociation of hydrates in a porous medium under depressive effects

Institute of Multiphase System Mechanics, Russian Academy of Sciences, Siberian Division, Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 4, pp. 120–130, July–August, 1995.     

Diffusion and adsorption of a radioactive gas in a porous medium

The results of solving the one-dimensional problem of the motion of a pulse of radioactive gas, carried through a porous medium by a stream of inert carrier with constant velocity, are generalized by the case of taking diffusion processes into account. For a delta-shaped input pulse, the solution is obtained of a system of equations which describes the migration of the pulse, taking diffusion washout and nonequilibrium adsorption into account. It is shown that in the case of equilibrium adsorption the time of appearance of the concentration maximum at the adsorber outlet depends on the decay constant and the coefficient of diffusion. Approximate solutions for strong- and weak-nonequilibrium adsorption and in the case of weak diffusion are considered. An estimate is carried out of the maximum magnitude of the diffusion coefficient, when its effect can be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–90, September–October, 1976.The authors thank A. N. Gudkova for discussing the work.     

An asymptotically self-similar solution of the problem of three-dimensional gas flow through a porous medium

An exact solution of the Cauchy problem is constructed for the equation describing the three-dimensional molecular diffusion of a gas. The result obtained is a natural generalization of the solution of the analogous Boussinesq problem. Dnepropetrovsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 178–180, September–October, 1988.     

Nonequilibrium adsorption of a radioactive gas in a porous medium

An analytical solution is obtained to a system of equations describing the nonequilibrium adsorption of a radioactive gas, which is moving with a constant velocity in a semi-infinite, sorbing medium, for a boundary condition depending arbitrarily on the time. An investigation is made of the case of the nonequilibrium adsorption of a motionless gas. The solution is also investigated for a boundary condition in the form of a δ-function. It is shown that, in distinction from equilibrium adsorption, in the medium there are propagated, generally speaking, two maxima of the concentration, one of which arrives at a given point without a lag, while the second lags. Approximate solutions are obtained for strongly washed-out and δ-shaped pulses.     

Numerical investigation of the decomposition of gas hydrates coexisting with gas in natural reservoirs

The problem of methane hydrate decomposition in a reservoir saturated with a gas and hydrate mixture is investigated numerically. The results of the numerical simulation and an analytic solution obtained in the linear approximation are compared. It is shown that for high-permeability rocks the convective heat transfer in the near-well space of the reservoir predominates over the conductive transfer. This makes the use of intra-well heaters ineffective. It is found that an increase in the reservoir and well pressures and a decrease in the permeability suppress the formation of an extended hydrate dissociation region. Critical diagrams of existence of the frontal decomposition regime are constructed.     

Problem of propagation of a gas fracture in a porous medium

The problem of gas fracture formation in a porous medium is investigated. An inertialess viscous polytropic gas flow along the fracture is considered. The assumption of small fracture width with respect to the height and length makes it possible to adopt the vertical plane cross-section hypothesis on the basis of which the dependence of the gas pressure inside the fracture on its width can be reduced to a linear law. Initially, the soil surrounding the fracture is soaked with oil-bearing fluid. During fracturing the reservoir gas penetrates into the soil mass and displaces the fluid. A closed system of equations, which describes the evolution of the fracture opening, the depth of gas penetration into the reservoir, and the gas velocities inside the fracture, is constructed. The limiting regimes of gas seepage into the surrounding reservoir are considered and a one-parameter family of self-similar solutions of the system is given for each. The asymptotics of the solution in the neighborhood of the fracture nose is investigated and analytic expressions for the fracture length are obtained. The solution of the problem of gas fracture is compared with the hydraulic fracturing problem in an analogous formulation within the framework of the plane cross-section hypothesis.     

Numerical study of single-phase gas filtration in a porous medium

This paper describes a method for numerical solution of single-phase nonstationary gas filtration in a porous medium. Linearization of the classical Leibenzon equation is performed. To solve the resulting linear equation, an efficient numerical algorithm was constructed without saturation over spatial variables and time.     

Self-similar axisymmetric thermal in a variable-density medium

A model of a turbulent axisymmetric thermal rising in a medium whose density varies with height is proposed. An analytic solution describing the self-similar stage of ascent of the thermal is constructed. The effect of the density stratification of the atmosphere on the structure of the thermal during its motion is analyzed, and relations describing the characteristic points of the thermal and the smearing of its leading edge are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 45–53, July–August, 1991.     

Self-similar implosion of a continuous stratified medium

Abstract. The subject of this paper is the non perturbed self-similar implosion of a continuous stratified medium in cylindrical and spherical geometry. The main contribution is twofold. The first one is a condition on the initial density profile which guarantees that the integral of the internal energy over a fixed volume stays finite at the collapse. The second one is the proof that the self-similar implosion model partly restitutes the dynamics of an experiment of cylindrical implosion. Numerical values of the self-similar parameter are also given for various kind of stratified media in the two geometries. Received 8 March 1999 / Accepted 9 April 2001     

Motion of a gas mixture through a porous medium with sorption

Solutions are investigated of a system of linear partial differential equations describing the motion of a gaseous (liquid) mixture through an undeformable homogeneous porous medium with sorption at interfaces between gaseous (liquid) and solid phases, the kinetics of which are described by a linear equation. If the porous medium consists of spherical granules, the problem is solved in quadratures. For the case of symmetric granules with arbitrary symmetry parameter, various approximate solutions are obtained; first and central moments are used as criteria for the accuracy of the approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–100, September–October, 1970.