SLIPPAGE SOLUTION OF GAS PRESSURE DISTRIBUTION IN PROCESS OF LANDFILL GAS SEEPAGE

A mathematical model of landfill gas migration was established under presumption of the effect of gas slippage. The slippage solutions to the nonlinear mathematical model were accomplished by the perturbation and integral transformation method. The distribution law of gas pressure in landfill site was presented under the conditions of considering and neglecting slippage effect. Sensitivity of the model input parameters was analyzed. The model solutions were compared to observation values. Results show that gas slippage effect has a large impact on gas pressure distribution. Landfill gas pressure and pressure gradient considering slippage effect is lower than that neglecting slippage effect, with reasonable agreement between model solution and measured data. It makes clear that the difference between considering and neglecting slippage effect is obvious and the effects of coupling cannot be ignored. The theoretical basis is provided for engineering design of security control and decision making of gas exploitation in landfill site. The solutions give scientific foundation to analyzing well test data in the process of low-permeability oil gas reservoir exploitation.     

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.     

Exact solutions of the system of the equations of thermal motion of gas in the rarefied space

We study the model describing thermal motion of gas in the rarefied space. This model can be used, in particular, in the study of the state of the medium behind the front of shock wave after very strong blast, in the study of the processes taking place inside of tornado, in the study of the motion of the gas in outer space. For any given initial distribution of the pressure a specific selection of mass Lagrange variables leads to reduction of the system of differential equations describing this motion to the system, for which the number of independent variables is less on the unit. For the obtained system we found all nontrivial conservation laws of the first order. In addition to the classical conservation laws the system has other conservation laws, which generalizes the energy conservation law. We obtained the exact solutions of this system. These solutions describe a variety of different physical processes taking place in the rarefied medium. Using the symmetry properties of the system we got the generating formulas for the receipt of the new solutions using already found earlier solutions of the system.     

Formation of a gas hydrate due to injection of a cold gas into a porous reservoir partly saturated by water

Specific features of formation of gas hydrates due to injection of a gas into a porous medium initially filled by a gas and water are considered. Self-similar solutions of an axisymmetric problem, which describe the distributions of the basic parameters in the reservoir, are constructed. The existence of solutions is demonstrated, which predict gas hydrate formation both on the frontal surface and in the volume zone. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 137–150, May–June, 2008.     

Hydraulic model for oil and gas reservoirs

This paper considers a two-dimensional hydraulic model which describes gas or oil flow to a horizontal well with hydraulic fractures and takes into account the reservoir geometry and fluid flow between the reservoir and the well. A computational algorithm is proposed, and calculations for gas and oil reservoirs are performed. A comparison of the calculation results and the solutions of the corresponding problems in a three-dimensional formulation show that the calculations using the approximate hydraulic model yield reasonably accurate results.     

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.     

Similarity solutions for the flow behind an exponential shock in a non-ideal gas

Similarity solutions for the flow of a non-ideal gas behind a strong exponential shock driven out by a piston (cylindrical or spherical) moving with time according to an exponential law are obtained. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic. It is found that the assumption of zero temperature gradient brings a profound change in the density distribution as compare to that of the adiabatic case. Effects of the non-idealness of the gas on the flow-field between the shock and the piston are investigated. The variations of density-ratio across the shock and the location of the piston with the parameter of non-idealness of the gas are also obtained.     

Partially invariant solutions for a submodel of radial motions of a gas

All partially invariant solutions in terms of the group of extensions for a model of radial motions of an ideal gas are found. The solutions are obtained by the method of separation of variables in an equation containing functions of one variable but different functions of different independent variables. The solutions predict different continuous unsteady convergence or expansion of the gas under the action of a piston with a point sink or source. If the sink or source affects all particles simultaneously, a collapse or an explosion occurs. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 26–34, September–October, 2007.     

Wave propagation in a non-ideal dusty gas

In this paper we have studied the behavior of wave motion as propagating wavelets and their culmination into shock waves in a non-ideal gas with dust particles. In the absence of non-ideal effect the gas satisfies an equation of state of Mie–Gruneisen type. An expansion wave resulting from the action of receding piston is considered and the solutions to this problem showing effects of dust particles and non-idealness are obtained. The propagation of weak waves is considered and the flow variables in the region bounded by the piston and the characteristic wave front are found out. The expansive action of a receding piston undergoing an abrupt change in velocity is discussed. Cases of central expansion fan and shock fronts are studied and the solutions up to first order in the physical plane are obtained. The effects of non-idealness and dust particles are discussed in each case.     

Analysis on nonlinear effect of unsteady percolation in the inhomogeneous shale gas reservoir

The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.     

On Vertical Diffusion of Gases in a Horizontal Reservoir

Exact and approximate solutions to vertical diffusion in gravity-stable, ideal gas mixtures in gas reservoirs, depleted oil reservoirs, or drained aquifers are presented, and characteristic times of diffusion are obtained. Our solutions also can be used to test numerical simulators that model diffusion after gas injection. First, we consider isothermal, countercurrent vertical diffusion of carbon dioxide and methane in a horizontally homogeneous reservoir. Initially, the bottom part of the reservoir, with no flow boundaries at the top and bottom, is filled with CO₂ and the upper part with CH₄. At time equal zero, the two gases begin to diffuse. We obtain the exact solution to the initial and boundary-value problem using Fourier series method. For the same problem, we also obtain an approximate solution using the integrated mass balance method. The latter solution has a particularly simple structure, provides a good approximation and retains the important features of the exact solution. Its simplicity allows one to perform calculations that are difficult and non-transparent with the Fourier series method. It also can be used to test numerical algorithms. Furthermore, we consider diffusion of CO₂ with partitioning into connate water. We show that at reservoir pressures the CO₂ retardation by water cannot be neglected. The diffusion-retardation problem is modelled by a non-linear diffusion equation whose self-similar solution is obtained. Finally, we obtain a self-similar solution to a nonlinear diffusion problem. This solution is a good approximation at early times, before the diffusing gases reach considerable concentrations at the top and bottom boundaries of the reservoir.     

Disengagement of the gas phase in bubble columns

The technique of gas disengagement is popularly used to assess the bubble size distribution in bubble columns. The technique involves the dynamic measurement of dispersion height when the gas supply is stopped. In this paper a mathematical model has been proposed for the process of dynamic gas disengagement. It has been shown that the initial faster disengagement is due to the presence of internal liquid circulation and not due to the presence of very large bubbles. Further, slower disengagement has been attributed to the transition from heterogeneous dispersion to homogeneous dispersion. The new model also explains the effects of superficial gas velocity, column diameter, column height and liquid phase physical properties on the gas disengagement.     

End effects in rarefied gas outflow into vacuum through a long tube

The problem of steady outflow of rarefied gas from a reservoir into vacuum through a long cylindrical tube of circular cross-section at a constant temperature is considered on the basis of the kinetic S-model. The kinetic equation is solved numerically using a conservative second-order method. The basic calculated quantity is the gas flow rate through the channel; the flowfields are also studied. The solutions obtained are compared with the known results.     

气藏压裂水平井非稳态流动分析

应用Green函数和Newman积原理,建立了压裂水平井与气藏耦合的非稳态流动模型,并给出求解方法。所建立的模型可应用多种约束条件,考虑了井筒压降和加速度的影响,并适用于各向异性气藏。实例计算表明：水平井在早期非稳态阶段的流量比拟稳态阶段大得多。早期非稳态流动阶段,各条裂缝之间未发生干扰,流量的大小与裂缝条数成正比例关...     

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT

The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.     

Similarity in stationary motions of gas and two-phase medium with incompressible component

In this paper we suggest the transformation between the equations for a perfect gas and the equations describing in one-velocity approach the two-phase medium with any volume occupied by the incompressible phase. It is proved that the motion of a two-phase medium in the transformed coordinate system is similar with certain accuracy to that of a perfect gas. It means that the solutions obtained for perfect gas can be used to solve wave problems for media with incompressible component. There is no necessity directly to solve the problem for medium with incompressible component, and it is only sufficient to transform the known solution of the similar problem for a homogeneous medium. Thus, the solutions of many hydrodynamic problems for multi-component media with incompressible phase can be obtained without solving the original set of equations. The scope for the suggested transformation is demonstrated by reference to the strong explosion in a two-phase medium.     

Galilean-invariant axisymmetric self-similar submodel of gas dynamics without swirling

This paper deals with one insufficiently studied submodel of invariant solutions of rank 1 of the equations of gas dynamics. It is shown that, in cylindrical coordinates, the submodel without swirling reduces to a system of two ordinary differential equations. For the equation of state with additional invariance, a self-similar system is obtained. A pattern of phase trajectories is constructed, and particle motion is studied using asymptotic methods. The obtained solutions describe unsteady flows over axisymmetric bodies with possible strong discontinuities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 46–52, March–April, 2009.     

Exact solutions for evolutionary submodels of gas dynamics

A new class of exact solutions with functional arbitrariness describing the motion of a polytropic gas is constructed on the basis of invariant submodels of rank two of the evolutionary type. In the solutions obtained, the velocity is a linear function of some spatial coordinates. These solutions describe continuous gas dispersion and the motion with density collapse at a finite time.     

Effect of decomposition of a gas hydrate on the gas recovery from a reservoir containing hydrate and gas in the free state

The problem of gas recovery in the process of methane hydrate dissociation in a reservoir saturated with gas-hydrate mixture is considered. The mathematical model of hydrate decomposition into gas and water is generalized to include the negative temperature interval and takes ice formation into account. The solution of the problem is represented in the self-similar approximation. It is shown that there exists a transition hydrate decomposition regime in which water and ice are formed simultaneously. A comparative analysis of the recovery is carried out on the basis of relations derived for the masses of recovered gas in different hydrate dissociation regimes. It is shown that an anomalous increase in the recovered gas volumes is observed in the transition hydrate dissociation regime.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 132–142. Original Russian Text Copyright © 2005 by Tsypkin.     

Burnett simulations of gas flow in microchannels

The Burnett equations with slip boundary conditions are used to model the gas flow in microchannels in transition flow regime. As the Navier-Stokes equations are not appropriate to model the gas flow in this regime, the higher-order Burnett equations are adopted in the present study. In earlier studies, convergent solutions of the Burnett equations of microPoiseuille flow could only be obtained when Knudsen number is less than 0.2. By using a relaxation method on the boundary values, convergent solutions of the Burnett equations can be obtained even when Knudsen number reaches 0.4. The solutions of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The pressure distributions and velocity profiles are then discussed in detail.