Gas emission into the excavated space during the mining of coal deposits

We consider the problem of nonsteady filtration of a gas in a bounded domain with an unknown moving boundary. We obtain a closed-form solution both for the case when the moving boundary is smaller than the boundary of the filtration region and in the case when the two boundaries coincide. We study the regularities of variation of gas emission, pressure distribution, and advance of the free boundary as functions of time and the filtration properties of the coal vein. Four figures. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 81–85, 1991.     

Dynamics of coal dust flame acceleration: A feedback control model for unsteady flow

Several modeling concepts borrowed from control theory are employed to develop an algebraic and ordinary differential equations model for the dynamics of unsteady coal dust flame acceleration in a constant area duct closed at one end, e.g., in a coal mine tunnel. We are particularly concerned with modeling the feedback mechanisms which cause a coal dust flame to accelerate, leading to detonation. Previous experimental studies have been conducted on both coal dust flame propagation and on individual coal particle combustion. Based on the results, a physical model is proposed in which coal dust flame acceleration is entirely controlled, in a feedback fashion, by volatiles emission and their reaction. A control system model is developed that employs five well-stirred reactor subsystems with three feedback interaction mechanisms. The model consists of a leading shock wave, followed by a variable length volatiles emission region ahead of the flame, a fixed length burning region immediately behind the flame front, and a variable length exhaust region extending back to the closed end of the duct. The feedback mechanisms incorporated into the model include heat transfer and pressurization from the burning region to the volatiles emission region, and pressurization from the volatiles emission region to the turbulent mixing region behind the shock wave. Each well-stirred reactor is described by a system of algebraic and ordinary differential equations for the rate of change of conditions inside the reactor. Numerical simulation results reveal that, despite far-reaching simplifications (ordinary instead of partial differential equations, ideal gases insteady of two-phase flow, separation of volatiles emission and combustion, neglection of char burning), the model exhibits the fundamental dynamic properties of the flame propagation process. The model agrees with qualitative photographic experimental results and is applicable to both the case where the flame accelerates to detonation and to the case where the combustion process dies out.     

Influence of nonisothermal effects on gas production in northern regions

The influence of mathematical model parameters on the dynamics of pressure and temperature fields at nonisothermal gas filtration is investigated in a numerical experiment. A nonlinear system of partial differential equations obtained from the energy and mass conservation laws and the Darcy law are used to describe the process, and physical and caloric equations of state are used as closing relations. The boundary conditions correspond to a given pressure drop at the bottomhole. It is shown that the influence of the temperature field on such integral characteristics as cumulative gas production is most pronounced at moderate pressure drops. The size of the zone of possible hydrate formation in a gas reservoir is determined in a particular example.     

On the Existence of a Generalized Solution of the Saturated-Unsaturated Filtration Problem

We study the existence of a generalized solution of an initial–boundary value problem describing the process of unsteady filtration of a liquid in a bounded region of an n-dimensional space. We consider the case in which the Kirchhoff transformation used to determine the generalized solution takes the real axis into a semiaxis bounded below. An auxiliary problem is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem under study. The solvability of the auxiliary problem is established by using the method of semidiscretization in time and the Galerkin method.     

Estimating diffusion-capacity parameters of a coal bed using the gas pressure measured in a hole and the solution of an inverse problem

A geomechanical model is proposed for gas emission from a coal bed with block structure. Some coefficient inverse problem is formulated for finding the initial gas content as well as the diffusion and mass transfer coefficients from the measured pressure in a well. The solvability of this problem is tested, and the additional data on the gas-kinetic characteristics of the coal bed are shown to be necessary for the inverse problem to have a unique solution.     

Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner

In this paper, we study the Riemann problem of the two-dimensional (2D) pseudo-steady supersonic flow with Van der Waals gas around a sharp corner expanding into vacuum. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by a Goursat problem or a mixed characteristic boundary value and slip boundary value problem for the 2D self-similar Euler equations. We establish the hyperbolicity and a priori C¹ estimates of the solution through the methods of characteristic decompositions and invariant regions. Moreover, we construct the pentagon invariant region in order to obtain the global solution. In addition, based on the generality of the Van der Waals gas, we construct the subinvariant regions and get the hyperbolicity of the solution according to the continuity of the subinvariant region. At last, the global existence of solution to the gas expansion problem is obtained constructively.     

Double Aztec diamonds and the tacnode process

Discrete and continuous non-intersecting random processes have given rise to critical “infinite-dimensional diffusions”, like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary.     

On the solvability of the problem of saturated-unsaturated filtration consolidation

We prove the existence of a solution of the initial-boundary value problem modeling the process of liquid filtration in a viscoelastic medium in the case of semipermeability on part of the boundary. To determine the generalized solution, we use the Kirchhoff transform. We consider the case (most often used in applications) in which the range of the Kirchhoff function is only part of the real line. To prove the existence theorem, we use the method of the semidiscretization with respect to the variable t and the Galerkin method.     

煤层注水非线性渗流方程的解析解及应用

本文运用流体力学,多孔介质流体动力学,渗流理论等理论知识,结合实验室和现场试验,从理论上对煤层注水预湿煤体机理进行了研究.分析了煤层注水过程,建立起了煤层注水的数学模型;对煤层注水的边界条件进行了描述.由于描述煤层注水的方程组为非线性的,为简化它们,利用了因次分析理论,引入了注水压力,渗透速度,煤水份增加量等无因次量.之后讨论了其解析和近似解.另外:结合实际煤层注水的科研项目,说明了该理论指导煤层注水及设计的作用和重要性.     

Level–wise Approximation of a Markov Process Associated to the Boundary of an Infinite Tree

We study an approximation of a Markov process associated to the boundary of an infinite rooted tree. This approximation is constructed by projecting the infinitesimal generator of the original process (defined on the boundary) onto the spaces associated to the filtration spanned by the successive levels of the rooted tree. S. Martínez and J. San Martín acknowledge the support by Nucleus Millenium Information and Randomness P04-069-F. D. Remenik current address: Center for Applied Mathematics, Cornell University, 657 Rhodes Hall, Ithaca, NY 14853. The author acknowledges the partial support by Nucleus Millenium Information and Randomness P01-005 for his work in his undergraduate thesis at Universidad de Chile.     

Investment evaluation based on the commercial scope the production of natural gas

The annual production planning of a natural gas trading and transporting company is modelled as a linear system of (in)equalities. The model is used to quantify the increase of robustness with respect to commercial uncertainty, resulting from investments in production capacities. A novel concept is thecommercial scope, describing the set of future commercial scenarios that can be handled effectively. It is shown how relevant parts of the boundary of this set can be constructed using induced constraints. A numerical example is presented.     

A Lattice-Based Smoother for Regions With Irregular Boundaries and Holes

We consider the problem of estimating a smooth function over a spatial region that is delineated by an irregular boundary and potentially contains holes within the boundary. Methods commonly used for spatial function estimation are well-known to suffer from bias along such boundaries. The estimator we propose is a kernel regression estimator, where the kernel is an approximation to a two-dimensional diffusion process contained within the region of interest. The diffusion process is approximated by the distribution of length-k random walks originating from each observation location and constrained to stay within the domain boundaries. We propose using a cross-validation criterion to find the optimal walk length k, which controls the smoothness of the resulting estimate. Simulations show that the method outperforms the soap film smoother of Wood, Bravington, and Hedley in many realistic scenarios, when data are noisy and borders are highly irregular. We illustrate the practical use of the estimator using measurements of soil manganese concentration around Port Moller, Alaska. Supplementary materials for this article are available online.     

Effect of gas seepage on stress-field redistribution in the vicinity of a hole

The effect of gas seepage on the stress-strain state of a coal seam is investigated. The problem of the degassing of coal seams with long holes is discussed. It is established that normal tensile stress components develop with the movement of gas in the vicinity of a hole. For certain ultimate tensile strengths of the coal, this reduces to failure of the walls of the hole, and to the development of dynamic phenomena.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 115–119, 1987.     

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.     

The Spin-Coating Process: Analysis of the Free Boundary Value Problem

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.     

Numerical modeling of the influence of heat exchange of reservoir beds with enclosing rocks on gas production from a single well

In the computational experiment, the influence of heat exchange through the top and bottom of the gas-bearing reservoir on the dynamics of temperature and pressure fields in the process of real gas production from a single well is investigated. The experiment is carried out with a modified mathematical model of nonisothermal gas filtration, obtained from the energy and mass conservation laws and the Darcy law. The physical and caloric equations of state together with the Newton-Richman law of heat exchange of a gas reservoir with surrounding enclosing rocks are used as closing relations. It is shown that the influence of the heat exchange with environment on the temperature field of a gas-bearing reservoir is localized in a narrow zone near its top and bottom, although the size of this zone increases with time.     

Existence of solutions for a finite nonlinearly hyperelastic rod

A theoretical investigation of a mathematical model for the capillary-tissue fluid exchange, including the characteristics and influence of the boundaries and media through which the fluid flows, has been studied. Filtration from the cylindrical capillary into the concentrically surrounding tissue-space and flow from a capillary into the tissue across the thin membrane are analyzed in detail. It has been observed that the filtration efficiency of the functional unit decreases as the peripherallayer viscosity increases, and that contrary to the results of Apelblat, Katziv-Kutchalsky and Silborberg (Biorheology2 (1974), 1–49), the slip velocity plays dominant role on filtration efficiency. It is also noted that he filtration efficiency decreases as the slip velocity at the porous boundary increases.     

On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.     

On the instability of resonances in parallel-plane waveguides

It has been observed¹³ that the propagation of acoustic waves in the region Ω₀= ?² × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown⁹ in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω₀. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω₀ ?B, where B is a smooth bounded domain with B??Ω₀. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x₁, x₂, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω₀ will be discussed in a subsequent paper.