Solution of the differential equations representing physical processes is greatly complicated by the following factors: (1) each equation represents a physical process only approximately, the degree of accuracy being greater or smaller; (2) physical processes are usually represented by complex nonlinear differential equations in partial derivarives, for which it is very difficult to find accurate solutions. It is therefore important to have approximate methods for solving differential equations and to deal with the associated problems of the agreement of the approximate solutions with the accurate solutions.This article is devoted to the determination of approximate solutions of equations representing gas movement during degassing of coal seams.Below we deal with the corresponding problems of the determination of the degassing time at a known location of the degassing holes and find the pressure distribution in the seam after an interval tg after degassing has begun. As the initial model representing gas movement in a coal seam, we will use the Krichevskii equation [1]. Particular attention is devoted to determining the dependence of the degassing time on the seam porosity and permeabiliry, the viscosity and temperature of the gas, sorptton, the initial pressure, and other parameters included in the I

In working coal deposits, preliminary degassing of the seam is used to reduce the gas content and gas pressure in the coal. One of the most popular methods is to drain part of the seam by boreholes drilled in the solid coalalong the seam or through the rock strata from a stripping working [1].In theoretical papers on nonsteady movement of gas in coal seams, gas filtration has been considered in the one-dimensional approximation (see, for example, [2][3][4]). In practice, the main mass of coal is dug from massesof coal previously exposed on four sides. One can consider that the coal seam is bounded by parallel planes and that the rocks in which it is enclosed are imperbeable to gas [5]. Then filtration of gas in the exposed coal mass can be regarded as plane-parallel [6]. Two-dimensional movement of gas in coal seams was discussed by . In a linearized formulation, she discussed the problem of nonsteady filtration of gas in an infinite region shaped like a right angle. She found exact solutions to the equations of gas filtration for cases in which the region of motion is unchanging in time and in which one side of the right angle moves with constant velocity. The exact solutions in [6] belong to the class of self-modeling solutions.

Problems involving degassing of coal seams are, of course, very important. Most research on the processes involved has been based on equations proposed to represent isothermal gas filtration [i] in which Darcy's law holds.The gas permeability is regarded as a characteristic of the medium, independent of the filtration conditions, in particular, of the gas filtration parameters and the state of stress of the rock.Under real conditions all these characteristics, and to a great extent the mechanical load on the filtering coal seam, have a marked influence on the gas permeability.As a result, the coefficient of gas permeability obtained for filtration of a particular gas under some pressure in the absence of mechanical actions on the coal may have only a limited range of applicability.Thus, consideration of the state of stress of the rock leads us to reject the linear law of filtration.Nikolaevskii et al. [2] give a closed system of equations (based on experimental data on the gas permeability and the porosity as functions of pressure) in the filtration approximation in which account is taken of the deformations occurring in the porous collectors.Khodot [3] describes experiments to determine the porosity and gas permeability in relation to the fluid pressure in the pore of a coal seam.A class of exact solutions of the equations of motion of a ga s in a deformed coal seam was derived by us in [4].In this article, using as examples two typical problems concerning the gas flow in a borehole allowing for the state of stress of the seam, we will investigate the problems involved.Let us consider the steady isothermal filtration of gas in a degassing borehole. The equation for the motion of the fluid in a one-dimensional coal mass, involving desorption phenomena, iswhere m = m(p) is the porosity, a and b are Langmuir's sorptio n constants, R is the universal gas constant, T is the temperature, p is the gas pressure, ~ = l(p) is the coefficient of gas permeability; u = 0 in the plane case, and ~ = 1 in the axisymmetric case. Here ~in is some initial permeability of the coal seam, Po is the gas pressure in the undisturbed massif, and ~ and 8 are empirical constants.Problem i. Consider a one-dimensional coal seam of length 2Z bounded at either end by impermeable rocks.To reduce the gas pressure in it we bore a hole in which aInstitute of Mining, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk.

A. T. Gorbachev SEAM UDC 622.42/46 : 518.4 The nonsteady process of gas emission from solid coal, complicated by gas sorption in the pores of the coal, is represented by a nonlinear differential equation in partial derivatives of the parabolic type ['1]:

In mining practice attempts are frequently made to degas a local area of a seam so as to reduce the gas content and pressure in the coaL For this purpose, drainage boreholes are usually drilled in the seam through the rock from the face of the entry working [1]. However, like many other methods, this is not always effective and in some cases may even prove to be of no benefit [2]. In the author's opinion, this is due to certain physical properties of the coal, which depend on rock pressure, and to the absence of reliable methods for calculating the parameters determining the drainage borehole system, making due allowance for these properties.Below I examine the problem of degassing a zone of a seam with initial permeability k H by a system of drainage holes, making due allowance for the physical properties of the coal which depend on rock pressure and affect the movement of the gas in the seam during degassing.When boreholes of radius r e are drilled through a coal seam, a zone of relief from rock pressure, with radius r H, is established in the vicinity of each hole. This is accompanied by an increase in the volume of the pore space and of coal permeability in the zone.We will assume that in the sector of the seam to be degassed the borehotes are drilled in a dense network, i.e., the lines connecting them form a system of equidistant and equidimensional triangles, covering the whole degassing area. We will also assume that the outer row of boreholes on the boundary of the area prevents the entry of seam gas into the area. The drainage zone of each borehole within the degassing area will then be bounded by a regular hexagon, the faces of which are equidistant from two neighboring boreholes. At the boundaries of the drainage zonesOp : O,(1) On where p Is the gas pressure in the pores of the seam, and n is the direction of the normalto the borehole drainage zone.Therefore, in each hole not located on the periphery of the degassing area, gas will only flow from its drainage zone.The flow of gas into the boreholes from the drainage zone hounded by a regular hexagon will (provided that Op/On = 0 at the boundary) differ little from the flow of gas into a boreholr from an equidimensional zone bounded Op [ = 0. The relation between ~ and l, the distance by a circle of radius ~, provided that at this boundary ~ Irbet~een the boreholes, must be determined by the equation:Thus to determine how the degassing time varies with the borehole radius we need only examine the flow of gas into an open borehole from the zone with radius ~. provided that at the boundary ar ,=7 r=;