ROCK

MECHANICS

AND

MINE PRESSURE

MEASUREMENT OF GAS PRESSURE IN COAL SEAMS S. A. Khristianovich and Yu. F. Kovalenko

In coals, in rocks associated with coal (sandstones), and in salt strata there is often gas at high pressure (methane, carbon dioxide). In mining these rocks catastrophic phenomena in the form of sudden bursts occur in which gas release from the coal (rock) is accompanied by breakage and transfer of considerable masses of coal and gas into the goal (tens or even thousands of tons of coal and sometimes many hundreds and thousands of cubic meters of gas). In order to anticipate sudden coal and gas outbursts and to understand processes causing them, it is necessary to know and determine correctly the initial condition of the gas in seams which are undisturbed by mining. There is extensive information in the literature on questions connected with measuring gas pressure in coal seams [1-4]. Gas is contained in coal seams in the free state in pore space and in solution (sorbed) within the coal itself. A t constant temperature the amount of sorbed gas depends on its pressure and with an increase in pressure it rapidly tends toward saturation. Gas content in coal is estimated as 15-20 kg/m 3, and it may reach 25-30 kg/m 3 (density of methane under normal conditions 0.72 kg/m3). A considerable part of this gas is in the sorbed condition. According to laboratory test data a typical sorption isotherm for gas (methane) by coal has the form [3] (Fig. i): M -- abp/(~ + ap),

(i)

where M is mass of gas in a unit volume of coal (the volume is calculated from the mass of coal taken for the test and its density) at pressure p; b is mass of gas in a unit volume of coal with p ~ | (typical value of b is 10-20 kg/mS); a is a constant having a dimension inverse to pressure [typical value of a - 1 (MPa)-I]. The saturation pressure for sorption p depends on temperature and it is normally within the limits 3-4 MPa. Concerning the pressure of free gas contained in coal pores, there is as yet no single opinion about its amount and the structure of the pore space itself. In addition, there are instructions in [5] determining the process for measuring pressure and mass of gas in an intact coal seam, and many such measurements are carried out. The method adopted for measuring gas pressure is based on the assumption that in coal sesm~ still intact from mining there is permeability. There is a suggestion that coal pores are always joined to each other} and although the permeability is very small it always differs from zero. Measurements [3] give a value for the permeability in coal of the order of I0"~-10 -3 D (i D = i0 - ~ cm2). This is very small compared with the permeability of gas producing strata (i0-I0 s D). A suggestion has been made that with the presence of initially stable filtration capacity coal seams, as for other rock strata exhibiting a stable filtration capacity, would be in communication with water-lmpregnated strata, so that gas pressure and strata would be hydrostatic. However, measured data by no means confirm this. If pores in an isolated coal seam communicate with each other, then the initial gas pressure should be the same throughout the seam. This also contradicts measured data. Therefore, it is more probable that in seams intact from mining, gas is contained in isolated pores.

Fig. i. t

I

Typical curve for saturation of sorption.

P Institute of Mechanics Problems, Academy of Sciences of the USSR, Moscow. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 3, pp. 3-24, MayJune, 1988. Original article submitted November 20, 1987.

0038-5581/88/2403-0181512.50

9 1989 Plenum PublishinR Corporation

181

p, MPa

b

0

Fig. 2

t5,90

fo90

H, m

Fig. 3

Fig. 2. Scheme for measuring gas pressure in coal seams: i) surement chamber; 2) control chamber; 3) working; 4) tube; 5) microvalve; 6) manometer; a) coal seam; b) rock. Fig. 3.

mea-

Measured data for gas pressure in coal seams.

Since coal exhibits flow, the equilibrium condition over geological time could be achieved with considerable mine pressure only with complete disappearance of tangential stresses, i.e., with uniform hydrostatic compression. Therefore, it is probable that the initial pressure of free gas contained in isolated pores is close to local mine pressure. There is also a suggestion that in burst-prone coal seems free gas may be entirely absent and all of the gas is in the sorbed condition. Coal seams containing only sorbed gas apparently exist, but the initial filtration capacity in these se-m~, as in clay or salt strata, should be entirely absent. I. Data for Gas Pressure Measurement in Coal Se~m~ and Discharge of Gas from Boreholes Gas-pressure measurement in coal se-m~ is currently accomplished by driling boreholes through the coal seam from the surface of the face in direction of both the strike of the seam and rise, as well as by drilling boreholes from the working through the thickness of the surrounding rocks along a normal to the seam and at different angles. Boreholes are used for measuring gas pressure in the seam and for measuring gas discharge. In order to measure gas pressure of the borehole it is sealed [3, 4]. In this way in the coal seem an unsealed part of the borehole ("measurement chamber") r~-mins, into which gas enters from the seam. Shown in Fig. 2 is a diagram of the measurement horehole [3]. A borehole thus equipped is a device making it possible to measure gas pressure which is established in the measurement chamber after closing the valve, and also gas discharge with an open valve varying with the passage of time. In order to measure gas pressure after drilling the borehole, it is sealed and the valve is closed. Pressure in the borehole gradually increases and then it stabilizes maintainin E its value for practically any length of time. During borehole drilling a small amount of dust coal is normally yielded. The volume of dust coal often exceeds the volume of the drilled hole by severalfold. If V d is the volume of dust coal yielded during drilling per unit thickness of plateau, and R0 is equivalent radius cavity formed, then V d - ~R0 2. The process of equipping the borehole sometimes requires several days. From the open borehole during this time a considerable mass of gas is released. However, as observations have shown, this has little effect on the value of gas pressure which is established after closin E the valve, end it is only affected by the time of establishlng this pressure. If after an interval the valve is opened again and after some time it is closed, almost the same pressure is established in the borehole as before openin E the valve. All of this, as it were, indicates that gas enters the borehole from an almost unbounded seam exhibiting a filtration capacity~ and the pressure which is established in the borehole is the gas pressure in this seam. Often some increase in pressure is observed after the next short-termopenlng of the valve.

182

Fig. 4. Diagrem of the test for studying the change in gas pressure in measurement boreholes on the approach of the working face.

TABLE i. Change in Gas Pressure in Measurement Boreholes on Approach_ of the Working Face 1 I I m [ P, a t . I

20 16 13

10 9 6 5

4

3

2 l,m [ P, a t .

l,

4 i P~t.[ l,

p,at.

2,2 I

I

l, re p,at,

l, m I p' a t .

!

24,6 24,71 20 I 28 ~,71 o,81

~S" P, a t

7

6

5

t5 I 3t lO I 4~ 7,51

o

28

40

27 27 28

t8

14~ 6,5

4~1 20 tt

26,4 26,6 28 33

7

24

9

20

4

2,2 [ 2,8 27

2

0,8

34

Note. s is the distance from the working face to the borehole; p is the gas pressure in the measurement boreholes. By using the Darcy law, the permeability [3, 4] is calculated from measured data for gas discharge from the borehole and the change of this discharge over a period of time. With an open valve gas discharge from the borehole often decreases with time and finally settles down to a very low level (several liters per hour or even per day). This discharge gradually (over months) also decreases a llttle, which would confirm the hypothesis about the presence of stable, although very low, initial permeability of the seam and inflow of gas to the borehole from an almost unbounded space. If this hypothesis is correct, then in order to obtain reliable data for gas pressure it is only necessary to be careful about good sealing of the borehole and to hold it for the necessary time in order for it to settle down completely. However, the collection of observed phenomena are not entirely explained if a hypothesis is adopted about the presence of a stable initial permeability for the coal seem. To begin with there is an enormous scatter of measured data for pressure in different se-m~, in a single seam, and even in closely placed boreholes. Currently there are numerous measured data for gas pressure over a whole coal deposit and through all burst-prone seams. Measured data for pressure p~ in boreholes published in [3, 4, 6, 7] are given i~Fig. 3, where the depth of borehole drilling is given along the abscissa axis, whence it can be seen that the measured results have, as it were, a random character. There are numerous data for gas-pressure measurement in measurement boreholes located initially at a considerable distance from workings. On the approach of workings reducing mine pressure in the vicinity of the boreholes and creating a drop in gas pressure in the direction of the working, gas pressure in the measurement boreholes changes. If a seam exhibited a stable filtration capacity, this drop would cause a gradual reduction in pressure in boreholes as the face is approached. However, data for observations give a completely different picture. Very often on approach of a working gas pressure in the measurement borehole starts to increase and often it increases severalfold [6]. With the passage of time pressure in the measurement borehole gradually starts to decrease. Wlth further approach of the workings, gas pressure increases again (often very sharply). In 1958 in the Makeev Scientlflc-Research Institute of Safe Working in the Mining Industry (MakNII) under the leadership of I. V. Bobrov and with participation of S. V. Kuznetsov 183

P

kgflcm 2

f $0

~

II ^t~,e, l

| ' ,

/

t"r

'1 I

I ~

i-

i

',

i~

i

'

I

i

,

,

.1i',

9

s

'

m

i

i.i>. .'

, N[

iI

I=,il=',l

I

':

":7'

';;..i. I

,

1%4;

"~ '

~" ~' ~i-"4-'I"

~ I~', ~I =lu~i,,.

o 5o ~ zoo 3oo ~o soo eoo 700 ~,h Fig. 5. Change in gas pressure in two measurement boreholes on approach of the working face. a considerable number of studies of the sudden outburst phenomenon and factors causing it were carried out in the "Derezovka" seam at levels 537-695 m in the "Krasnii Profintern" mine [1]. A study was made of changes in the state of the seam on approach of a mine working toward it. Areas of the seam were selected which were initially at a considerable distance from the working, and a study was made of the change in condition on approach of a preparatory working (Fig. 4). This drift was driven by a sequence of shock demolitions, some of which caused sudden outbursts. Before the start of the tests an experimental section was equipped with 56 measurement boreholes located in three rows in a checkerboard sequence. These boreholes were drilled in the "Derezovka" seam from a special drift driven through the "Zolotarka" seem located parallel to the "DerezovkaY' seam and at a distance of I0 m from it. Some of these boreholes were equipped with instruments making it also possible to measure approach of the roof and floor of the seam. At the start of the test the face of the drift along the "Derezovka" seam was 90 m distant from the test section. Pressure established in the measurement boreholes varied within the limits 16-31 arm. As the face approached the measurement borehole, the following picture of change in pressure was observed. Up to a distance of 10-15 m from the borehole no changes in pressure were observed [1], and with a further approach after the next explosion pressure started to change. A particularly marked change occurred after demolition caused by sudden outbursts. In certain boreholes pressure increased very markedly. In one of the boreholes a pressure exceeding 100 arm was recorded (a manometer with a scale up to i00 arm was used). Given in Table i are measured data for some boreholes. Shown in Fig. 5 is the change in pressure in two measurement boreholes with the passage of time as the face approached, blastings accompanied by sudden outbursts are marked. In these boreholes as the face approached gas pressure increased jumpwise, and it decreases slightly in the time intervals between blastings. Only with approach of the face to a very small distance from the borehole d i d pressure start to decrease sharply. In some boreholes pressure did not increase, and gas pressure in the borehole started to decrease smoothly with approach of the face. Measurement of the approach of the roof and the floor of the seam showed that this approach does not as a rule exceed 2-3 mm per 1-1.5 m thickness of seam. What could cause an increase in pressure in the measurement boreholes on approach of the face? It has been suggested [I] that the increase in pressure was a consequence of pressure of free gas in the pores of the seam as a result of a reduction in porosity under the effect of an increase in mine pressure with approach of the working and a reduction in vol=me of the measurement borehole. However, this suggestion is not well founded since even if it is assumed that approach of the roof and floor causes a corresponding reduction in porosity, this reduction would increase gas pressure overall by only several percent. A reduction in volume of the measurement chamber could not affect gas pressure since it equals the corresponding increase i n porosity within the vicinity of the borehole (coal particle compressibility is negligibly small).

184

There are numerous data for similar observations for a change in pressure in measurement boreholes on approach of a drift or longwall face. Attempts to degas a seam have led to positive results only in the case of a seam heavily loaded due to mine pressure. With relatively small loads an initially almost zero discharge from a degassed borehole on approach of the face for an unloaded seam increased sharply, reached a maximum, but as the face approached further to the body of the borehole the discharge dropped again almost to zero, although the total amount of gas released from the borehole was insignificant in relation to the amount of gas contained in the seam within the vicinity of the borehole. An attempt is made below to explain all of the phenomena observed on the basis of hypothesis for complete absence of initial filtration capacity in burst-prone coal seams and its development in the vicinity of workings and boreholes with relief of stresses due to mine pressure. During subsequent analysis it is assumed as a second hypothesis that initial gas pressure in pores P00 equals the compressive stress of local mine pressure. As will be seen below, with these two hypotheses it is possible to explain all of the phenomena observed, and this makes it probable that the initial condition of coal and gas in the undisturbed seam in fact confirms these hypotheses.

2. AnalTsis of the Stressed State and Breakage of Coal within the Vicinity of a Measurement Borehole During drilling of a borehole compressive stresses due to mine pressure in its vicinity are shed and the process of gas filtration commences. We consider the case of a measurement borehole drilled through a rock stratum along the normal to the seam through its whole thickness. The stressed state in the vicinity of this borehole will be radially symmetrical, and if approach of the seam roof and floor and adhesion forces at its surface are ignored, coal deformation can be assumed to be planar with reasonable accuracy. Consideration of stresses within the vicinity of the final part of a long borehole driven directly through the seam will be entirely similar. Hole deformation may also be assumed to be approximately planar, identical in planes orthogonal to the borehole axis, but the stressed state at a certain distance from the borehole axis will be symmetrical, and this without altering the phenomenon complicates the calculation somewhat. We direct axis 0z along the borehole axis (Fig. 6), polar coordinates r and 6 are placed in the plane of the seam. The stressed state at a distance from the borehole will be assumed to be a state of initial all-around compression with stress q. As is normally adopted, compressive stresses will be assumed to be negative. At some distance from the borehole axis, compressive stresses o r are still quite large, it is possible to assume that although the coal contains a system of defects as well as pores filled with gas, it retains its coherence and deforms elastically. If this material is on average uniform in all directions, then its deformation is governed by two elasticity moduli [8]. From considerations of synunetry it Is assumed that before drilling the borehole the state of the seam would be uniform, and it is possible to assume that a crack system occurring under the influence of expanding gas is located symmetrically in relation to the borehole axis. Deformation of this material may also be assumed to be continuous on average and it is assumed that within known limits the change in stresses for this material will follow an elasticity rule during unloading. The material will already be transversally isotropic, and its deformation will be governed not by two, but by five elasticity moduli [9]. Stress distribution within the vicinity of a borehole if the material is elastic, isotropic, or transversely isotropic, and it does not depend on the value of elasticity moduli [9], and has the form t

tl

*

o, = q - - ( q - - o,) (R,@-,

9

I

i

oe = q + (q - - ~ ) (R /@',

(2)

where or ' i s normal s t r e s s i n r - R' (R' i s boundary of e l a s t i c d e f o r m a t i o n ) . Presence of cracks o n l y a f f e c t s changes in e l a s t i c i t y moduli compared w i t h values o f moduli f o r m a t e r i a l in the absence of cracks [ 8 ] . The amount of change in moduli o n l y a f f e c t s the amount of d i s placement, and i t does not a l t e r the stressed s t a t e ( 2 ) . 185

Fig. 6

Fig. 7

/~-I _,-cA

--I :-v,, 1 i- s / I W ,.,,I;Yl '~

o,

Fig. 8 Fig. 6.

Coordinate system selected in the work.

Fig. 7. Schematic representation of a section of a coal seam with a borehole: left to right: core zone (coal in gas); zone of limited equilibrium; elastic filtration zone; elastic with a set of circumferential cracks; intact seem. Fig. 8. Schematic form of an oriented crack system: a) allaround hydrostatic compression; b) unloading along the axis. Shown in FiE. 7 is a section within the vicinity of a borehole with plane B = const. In the zone r > R there is no gas filtration in the direction toward the borehole. In region R < r < R there is a system of disk-shaped cracks in the surface r = const, and outside r = R 2 material deformation is negligibly small. Zone r < R is a zone of broken coal. Coal in the intact state can be represented as a material pierced in all directions by a system of planes located at surfaces with cohesion and resistance to breaking and shear significantly less than the average resistance for the material as a whole. These surfaces are called crack surfaces in the case when they are not pore boundaries, for example cleavage cracks. In order to distinguish them from natural cracks (pores) in whose surfaces cohesion and frictional forces equal zero, it is possible to call them defect surfaces and planes. With relief of coal compressive stresses, by expanding high-pressure gas breaks the coal thus creating a system of cracks. If coal failure were connected with a uniform all-around drop in compressive stresses, the gas would break coal mainly by a system of surface defects forming pieces of coal of different size pressed to each other. If compressive stresses are not relieved uniformly, the crack systems forming in this way are mainly directed by the stress tensor and the initial system of surface defects loses its governing importance. If one of the principal stresses is relleved, whereas the other two remain unchanged or decrease to a much lesser degree, then an initial pore of any configuration should finally degenerate to a disk-shaped crack surrounded by a whole set of finer cracks forming along defect planes in areas and directions where gas pressure exceeds the compressive stress by a value depending on the coal crack resistance factor [I0]. A schematic view of this crack system is shown in FiE. 8.

186

Fig. 9

A~

7.. ,~

o

Fig. I0 Fig. g.

Fig. ii

Diagram of filtration channels formed in a coal seam.

Fig. i0. Change in crack volume in relatoin to initial cross-sectional area with different values of parameter ~ = K T ~ . . broken line) ~ = I0 kg/cm312; solid line) y = 45 kg/cm 3/=. Fig. ii.

Diagram of the model test.

Stresses 0 2 and o a were unchanged (Fig. 8a, b), and stress 01 decreased. Shown in Fig. g is a chain of cracks in the vicinity of smaller cracks obtained by calculation. On approach toward the borehole stresses o r decrease according to (2), but o 8 grow; with a reduction in the value of o r to a certain limit o R development of the crack system and an increase in tangential stresses T = i 1 2 ( o 8 - o r) = (q - Or)(R'/r) 2 does not lead to more complete breakage of the coal. Coal filled with a system of cracks still retains its integrity and cohesion. As the set of oriented cracks formed in surfaces r - const surrounded by a set of finer cracks (see Fig. 9) develops, finally it creates a system of filtration channels mainly directed orthogonally to planes 8 = const. Coal permeability becomes different from zero, and with presence of a drop in pressure in a direction orthogonal to 8 = const gas filtration may occur. The filtration capacity occurring for coal is different in different directions. Permeability becomes anisotropic. For an anisotropic material it is also assumed that the generalized Darcy law is valid [11]. It follows from this that permeability is characterized by a second-order tensor. In the case in question, in the vicinity of a borehole directions of the principal axes for this tensor should coincide with directions of the principal axes for the stress tensor. We designate the principal values of tensor for permeability in terms of K r and K 8. If with relief of compressive stresses o r in the vicinity of a borehole in zone r 9 R, where r = R is the boundary of the broken coal zone, a set of cracks develops (see Fig. 7) since absolute values of o e in this way increase (2) and 1o81 9 lql, cracks do not form in the radial direction and K r = 0. Then in directions consisting of angle ~ with direction r = const the permeability factor will be K a = K 8 cos 2a. At the boundary r - R with some critical value of stresses o R coal breakage will occur. This breakage may be caused both by layer-by-layer separation of coal, and by an increase in tangential stresses I/2(o 8 - o r ) in a rock mass weakened by cracks.

187

In the vicinity of a disk-shaped crack located in surface r = const the value of o r may be assumed to be approxin~ately constant. This direction, as it were, corresponds to the direction at infinity for cracks in a limitless elastic space. Presence of gas in pores whose pressure exceeds furl creates tensile stresses at the perimeter of the crack, and as a result of this cause its growth. A crack [12-14] grows due to an increase in P0 + Or, where P0 9 0 is the gas pressure in the crack and o r < 0 is stress at a distance from the crack. In the limiting equilibrium condition p,+ o, ~ KJY2=-- ~, where u is crack radius; K T is crack resistance factor,

kg'cm "s/=.

With a reduction in absolute value of o r (2) increases crack radius u. As calculations based on accurate solution of the problem for a disk-shaped crack [14] show, the volume of a crack with an increase in u changes much more slowly than its area as a result of crack flattening. Therefore, gas pressure in a crack changes relatively weakly. Shown in Fig. i0 is the change in volume in relation to its cross-sectional area with different values of parameter y = KT/2V~'~. We designate P0 as gas pressure in cracks for surface r = R, P0 < P00If m 0 is initial porosity with gas pressure P00 and m0' is porosity with pressure P0, then moPoo

=

mofPo 9

With some critical value of stress oR at radius R there is coal breakage. If surface R = r were a free surface, it would cause a crushing wave. A sudden outburst would occur. Coal failure at a surface located in the rock mass does not cause a pressure wave only due to absence of a free space for large displacements. Coal breaks forming weakly connected elements (grains) with each other this process is accompanied by formation of a more isotropic structure. Overall porosity may be almost unchanged with a change in pore shape and formation of filtration channels in all directions. We consider the possible nature of the stressed state in region r < R (see Fig. 7) and equilibrium conditions at boundary r - R. With r 9 R the integrity of the coal mmss is disrupted by expanding gas. Coal in this zone consists of indlvidual elements (pieces of coal) held together pressed toward each other by mine pressure. T h e surface of each element is covered partly by areas over which this element is in contact bonded with neighboring grains and free surfaces washed with gas. Stresses due to an external load are transferred through contact areas. Our further consideration is based on the assumption that the medium formed of crushed coal is such that coal elements deform elastically, and their linear dimensions in all directions are of the same order (elements are not plates or rods). Contact areas between these elements are located on average symmetrically on the surface of each of them so .that gas pressure on part of the surface free of contacts with neighboring elements creates in each element only an all-around compressive stress. We also assume that the value of 6 representing the ratio of the portion of the surface occupied by contact areas to the whole surface area of the element is quite small. A material consisting of quite hard spheres of different size in which pores formed by the lattice of spheres of large dimension filled with smaller spheres may serve as a model for this material. Porosity of this material may be as low as one pleases and correspondingly 6 ~ I. Stress transmitted through contact areas between elements create in each element both tangential stresses and volumetric compression. First of all we turn to studying the stressed states occurring in the region of disturbed coal, and we consider some simple model problem. We present a cylindrical vessel with absolutely rigid walls filled with a crushed material with the properties indicated above and low porosity, but exhibiting permeability. Let this material be on average uniform in all directions. The transluscence w, i.e., the area occupied by a section of a pore in any quite large area, is equal to porosity m. Let the vessel be closed with a piston (Fig. ii) and loaded with weight G. The cross-sectional area of the piston S and force of the pressure per unit area of the piston o z - G/S. Let pores be filled initlally with gas with a pressure negligibly small compared with o z. The pressure of the piston creates in each elemerit a stressed state which consists of tangential stresses and all-around compression. Now let pores be filled with gas with pressure p. According to our assumption, this pressure creates in each grain a hydrostatic compressive stress, i.e., (i - 6)p.

188

We intersect the vessel (see Fig. Ii) by a plane parallel to the piston, and we consider equilibrium for the parts of each grain intersected by this plane and lying on one side of it. In the section parallel to the base of the piston from the equilibrium condition we have

a,- s,-[~ +(I- ~) (I- 6)]p- s,- [(I- 6)+ ~6]~, where s z is average normal stress transferred through contact areas between grains referred to the whole cross-sectional area [15, 16]. Gas pressure reduces stress s z by a value (i - 6)p + ~6p. In exactly the same way there is a reduction in normal stresses transferred from the walls of the vessels through contact areas between elements. Volumetric stress at the surface of each element transferred through contact areas decreases therefore to a value of (i - 6 + u6)p. In total the compressive stress for each element is almost unchanged (~6 ~ i). Thus, gas pressure hardly disturbs equilibrium, and by not creating pistonlike displacement (~6 ~ i) it reduces the load on the soil framework to a value p[(l - 6) + ~6] = (i - 6)p and with pressure p equal to or greater than s z it returns to zero. Tests carried out at the Institute of Mechanics Problems with compression for polystyrene spheres in a cylindrical vessel up to 100 MPa and with a change in pressure in pores to 25 MPa satisfactorily confirm these conclusions. Pistonlike displacements with a change in gas pressure were almost lower by a factor of two than the corresponding displacements with a change in external load. Let there be a cavity to whose boundaries both normal and tangential stresses are applied be filled with a material with the properties indicated above. Then let the pores be filled with gas with pressure p. Without hardly changing the equilibrium condition gas pressure reduces normal stresses transferred between contacts between elements of the material to a value of (i - 6)p + ~6p or approximately to (i - 6)p. 3.

Limiting Equilibrium State in the Vicinity of Borehol 9

Let the valve be closed and gas pressure in pores with r < R be the same everywhere and equal p~. We consider the equilibrium condition for the material at surface r = R and in the vicinity of the borehole. Let this surface isolate region r > R, where gas filtration is absent in the direction toward the borehole, from zone r < R of broken rock where permeability is already different from zero. At this surface gas pressure changes jumpwise from pressure P0 in cracks, r > R, to pressure p~ the borehole and pores, r < R. Let at certain instant nominal stress in this surface from the direction r > R equal IOr'l > IoRl and s r' is correspondingly normal stress transferred through contacts between coal grains in zone r < R. Let s R be the maximum stress in surface r = R, which in the given instant may withstand without breaking the coal within zone r < R. Let ISRl < loR[ and ISr'l < ISRl, then a r' = s r' - (i - 6)p, and this equilibrium be stable, i.e., it is not disturbed with small changes in p. We will reduce gas pressure p. In this way stress s r' will increase in value. The state of limiting equilibrium before breakage sets in with a certain gas pressure p~ when u r ' = o R and s r' becomes equal to the corresponding value of sR. Then oR=

sa--~,,

(3.1)

where ~ . = (I - 6)pm. If IsRl 2 loRl, then equilibrium were to be disrupted with any value of p. The maximum value corresponding to the u~ximum load which in the equilibrium condition is withstood by the coal in zone r < R is s R. This means that load in the surface r = R, r < R in the equilibrium condition cannot become greater than the related stress s R. If s r ' exceeds s R in value, then equilibrium of the coal is disturbed. There is coal breakage in surface r - R z (see Fig. 7) and displacement of this surface. This will occur until load s r' decreases so much that a new e q u i l i b r i u m s t a t e becomes possible. If in the original equilibrium condition there was o r = aR and ISr'l < IsRl, a reduction in gas pressure would not at first disrupt equilibrium. Only values of compressive stresses s r' increase. With a further reduction in pressure, limiting equilibrium sets in, a R = sR ~,.

189

On the surface r = RI stress o = -p, and s r I O. In region r < R I stress in the surface of coal elements equals the gas pressure (coal in gas). When a cavity initially free from coal r < R0 is totally filled with pieces of coal and a nucleus forms for uniform coal compression, then at boundary r = R I, Srl becomes different from zero. Let us assume that on closing the valve an equilibrium state was established and pressure in the borehole was p,. Let the valve be opened again and in this way some of the gas flows out of the borehole; then it is shut again, and an equilibrium state sets in with another radius R. If limiting stress sR does not depend on the absolute value of radius R, then with oR ffi const gas pressure in the borehole under equilibrium conditions will be constant. On opening the valve and outflow of a small amount of gas and, then closing the valve, an equilibrium condition is established with the same gas pressure in the borehole. On opening and closing the valve there is only a change in self-modeled (similar) stressed states with a different value of radius R depending on the amount of gas flowing out of the borehole. We show that with the prescribed conditions the limiting state of limiting stress s R and ~ , i s only governed by the condition in surface r = R. The supporting capacity of coal, as for other rocks, is mainly due to resistance to shear and separation. A critical value of shear resistance in a material, which on average is assumed to be uniform in all directions, in a given oriented plane, i.e., criterion for local failure, is normally in the form Tn ~ Zmax, where the value of ~max is the strength limit and 9 . . z ' - K c + Io.[ t g ~ ,

(3.2)

o n is normal stress in the plane; K c is cohesive coefficient; ~ is angle of friction. We call to mind analysis of the failure criterion [17]. If in the vicinity of any point of a material a plane is drawn passing through an axis parallel to axis 0z (see Fig. 7), then depending on the angle which this plane forms with the direction of principal stresses o r and o0, there will be change in values of on and Tn by a known rule. The maximum value of difference ITnl - lOnl t a n ~ is achieved in planes forming with axis o 8 = const angle (~/4 - ~ / 2 ) [17], and it equals

Ix-I--I~

~

1 I%-a,I 2

--

tg~l%+~ 2

(3.3)

This means that compressive stresses o r and o 0 are taken as negative, and we obtain a condition for the limiting stressed state at a given point of the material in the form O , - O , - --2K~r

~+(oe + o,)sin~.

(3.4)

With an increase in load limiting stresses occur at a greater number of points. With a further increase in load an instant sets in when the rock mass cannot withstand an increase in external load. This is the extreme equilibrium condition corresponding to the load, and the external stress is called the limiting stress. The limiting stressed state normally occurs in individual surfaces. In the analysis it is assmned as is normal that the limiting stressed state occurs simultaneously at all points of the rock mass. This gives an increased value for limiting loads on the rock mass since the equilibrium condition could be disrupted with smaller loads with the presence of only individual sllp surfaces. All the same, it is assumed that the limiting condition for stresses under limiting equilibrium conditions, when s r' ffi sR, sets in throughout the whole zone R x < r < R. At boundary r I R stresses o r are continuous, and volumetric compressive stresses accurate to 8w ~ i with a change in gas pressure also vary continuously (see 2). Consequently, stresses o 8 also change continuously. We designate s r and s 8 as stresses transferred through contacts between coal elements. Then

O" -- s ' - - SP'

190

O' ~

s ' - - ~'~'

(3.5)

where ~ We have

= (i - 6)p.

In the limiting equilibrium condition for the material

~'*

= (i

se - - s. = -2K~cos ~ + ( s , + s~)sin

-

6)p,.

(3.6)

or

(3.7)

Oe -- oR = -- 2Kc cos ~ + (us + os) sin ~ + 2~, sin and (2) oe +

o.

-- 2q.

We refer all stresses to q and label them with a line. -Kc/q. Then compressive stresses will be positive

We also designate ~ ,

Os + Oe -- 2 and

=~/q

and Kc =

(3.8)

ue -- us = 2Kc cos ~ + (ue + Ua) sin ~ -- ~ , sin ~,

whence

oR = 1 -- Ke cos ~ -- sin ~ (1 -- ~',),

oo = I + K--ccos ~ + sin ~ (I -- ~,)

(3.9)

~$) (i -- sin ~) -- A'c cos ~, se = (i --'~,) (i + sin ~) + Kc cos ~.

(3. i 0 )

and, c o n s e q u e n t l y , sa =

(i - -

From known values of ~ and with constant Kc and ~ we find ~--, and SR, and these values do not depend on the absolute value of R. The limiting stressed state according to the assumption is reached in all points of region RI < r < R. From the equilibrium equation _ ~ + "e--', = 0, Y

and this means that with r = RI, s r = 0, we obtain [18]

_s , = K c c t g ~

]

--i,

c=

l--sin~'

(3.11)

so that

[(I)

i3.n)

From known values of constants oR, Kc, and ~ we find the ratio R/R I. If with a large amount of gas flowing out of the borehole a uniformly compressed nucleus with stresses Srl formed in its vicinity, then we would have

sR = -- Kcct' ~ + ~rl + Kr c t ' ~) (~-~l)e"

(3.13)

Thus, values of SR, ~ . , R/RI and, consequently, pressure p~ measured by the borehole, and also stresses in zone r < R are only determined by values of oR , K c, and ~. With an increase in compressive stresses in a uniformly compressed nucleus there is a reduction in the ratio RI/R and at the limi~ with R I - R, Srl becomes equal to s R. With the absence of cohesion and crushing the equilibrium state is only possible with ~ . = IORl or with the presence of this nucleus. It may be suggested that equilibrium conditions are possible with which at boundary r = R, in which case gas filtration takes place toward the borehole, a limiting stressed state for the material does not arise simultaneously, and after coal failure in this region with R' < r < R (see Fi E . 7) an elastic deformation regime is maintained. The limiting state occurs with r - R', R' < R, with oR ' < o R and a 8' = 2 - OR'. Correspondln E values of sR' and ~' are determined at the boundary_r = R' from the same relationships (3.9) and (3.10) with substitution of oR by a value of o R' lower in magnitude. With oR ' < o R there is correspond191

ingly a reduction in the value of ~. with a closed valve.

and measured gas pressure established in the borehole

After opening the valve and closing it, with establishment of equilibrium a changeover to another equilibrium state is.possible with an increase in R' and the corresponding value of aR'. But oR ' cannot exceed oR . So that with R' s R, pressure in the borehole assumes the maximum possible value. Scatter of pressure in boreholes in one and the same seam, and particularly in closely positioned boreholes, is probably explained not only by scatter of values of material strength characteristics oR , K c, and @, but also by the occurrence of diferent equilibrium states. These states with the same material strength characteristics may form at the very beginning due to differences in the process of drilling and sealing a borehole. Low values of gas pressure in measurement boreholes do not depend on the value of P00 and they correspond to solid coal (low values of ~ and of course they are not entirely guaranteed due to occurrence in ~he seams of sudden outbursts with rapid exposure of the coal surface. Presence of confined water in coal or water penetrating into the seam during borehole drilling with counterpressure from the surface may exclude coal breakage in the zone of the borehole surface and ten~orarily create complete sealing in the borehole with a very low pressure. We consider the process of borehole operation. During drilling of a borehole there is a gradual change in the stressed state in its vicinity and flow of gas from the borehole. On closing the valve a certain initial 14miting equilibrium state is established with initial radius R and pressure p,. These values depend on time expended in drilling, the amount of coal dust taken out and gas flowing from the borehole. If the drilling process occurred instantaneously, then on the borehole surface a pressure drop would exist equal to P0 - Pa, where Pa is atmospheric pressure. This drop would generate a crushing wave. A burst would occur in the borehole cavity. With gradual relief of stresses as during proceeds continuously a change in stressed state occurs, and finally after closing the valve some limiting equilibrium state is established. When the valve is opened again gas pressure at surface r = R starts to drop. Expansion of the filtration zone occurs jumpwlse. W l t h a reduction in gas pressure at surface r s R and a corresponding reduction in pressure oR by ~r there is coal breaking by gas in some annular zone R < r < R + 8R. At the initial instant pressure of gas released from cracks is close to P0 in this zone. Stress o r at the newly surface therefore becomes greater in value than oR and expansion of the zone r - R ceases until this pressure falls

again.

Under the action of the pulse created at boundary r - e by stresses [sr'l > ISRl,

the limiting equilibrium state in region r < R is disturbed. At boundary r - R l there" is spalllng and movement. If there is an increase in the volume of the coal, i.e., an increase in porosity in zone R < r < R + dR is absent, a drop in gas pressure new surface r = R only occurs as a result of filtration. As gas flows out of the borehole, the radius R increases. By conditionally considering this process as continuous, it is possible to assume that on average a self-modeling regime is established, (i.e., a change in similar stressed states) with which stress distribution in the vicinity of the borehole does not depend on absolute values of radii R, R', R I, and R0, but it depends only on their constant ratio, and gas pressure p~ at the boundary r = R is constant. This state may exist until cavity r < Rl is filled uniformly with compressed coal. If a further increase in R becomes impossible, there is liberation of region r < R from gas. The absolute value of R with a closed valve, when r < R, is determined by the amount of gas flowing out of the borehole. We consider the followlng problem. Let the valve in the measurement borehole be closed and in the vicinity of the borehole let the coal be in a limiting equilibrium state and pressure in the borehole be equal to p~. In region r < R gas pressure p, is constant and it exceeds the saturation pressure for sorption Ps" Now let the valve be opened end remain open for a certain time. Then it is closed again and the limiting equilibrium state is established anew. The radius of the filtration zone increases and becomes equal to R", and the gas pressure in the vicinity of the borehole again increases to value p~. Let during operation of the borehole a mass of gas M be released from it and during filtration with an open valve and then during presure leveling after closing it gas of mass M a be released from the coal be desorption.

192

From the gas mass conservation condition it is possible to determine the dependence of the increase in filtration zone size on difference M - H a, and from an equation expressing constancy for the coal mass to estimate movement of coal particles in the region r < R. After sealing the borehole the mass of coal in the vicinity of the borehole cannot change. During drilling some coal dust is removed from the borehole and this must be taken into account in deriving the effective radius of the borehole R0. Let r - R 2 be the boundary of the zone of noticeable deformation in the vicinity of the borehole whose movement it is possible to ignore. We will ignore elastic changes in the volume of particles of coal with a change in its compressive stresses; coal mass conservation in the zone r < R 2 is equal in force to the volume occupied by it. This volume per unit o f seam thickness is Vy ffi ~ ( R 2 2 - R02)(l - m0). Let the zone of filtration be determined by the value of R in the first equilibrium state, and in the second by R". We call zone R < r < R" zone A. The volume of coal, and consequently the volume occupied by gas in zone r < R", remains constant in the first and second equilibrium states. All of the processes of gas expansion in coal pores may be assumed to be with restriction, and by ignoring the Joule effect gas temperature is constant and density is proportional to pressure. Let the average gas density in zone A in the first equilibrium state equal P0 corresponding to porosity m0' and pressure P0, and in the second let density equal p~ and correspondingly pressure p,. Since the volume and mass of gas in zone r < R2 does not change with a changeover from the first equilibrium state to the second, the mass of gas flowing out of the borehole leaving out the amount of gas released from the coal be desorption, should equal the difference in mass of gas contained in zone A in the first and second equilibrium states, i.e., M -

=

(R-' -

' m ) m0 (po -

p,) =

(A" -

R , ) m0p00

m;

( 3 14)

I t i s w e l l known t h a t t h e d i s c h a r g e o f gas from an open b o r e h o l e d e c r e a s e s v e r y r a p i d l y w i t h t h e p a s s a g e o f t i m e ( d a t a p r o v i d e d i n Sec. 4 ) . This i s t h e r e s u l t o f a r a p i d i n c r e a s e in filtration r e s i s t a n c e as a r e s u l t o f an i n c r e a s e in t h e zone o f a c t i v e f i l t r a t i o n with v e r y low v a l u e s o f f i l t r a t i o n coefficient. With t h e p a s s a g e o f t i m e d u r i n g p r o l o n g e d b o r e hole operation radius R almost ceases to increase. F r e e gas w i l l be c o m p l e t e l y removed from a considerable part of the filtration z o n e , and t h e r e i s o n l y g r a d u a l r e l e a s e o f t h e r~m~ini n g s l o w l y d e s o r b i n g p a r t o f t h e gas c o n t a i n e d i n t h e c o a l . Let gas pressure p , under limiting equilibrium conditions be less than the sorption saturation pressure Ps- If with an open valve the rapidly desorbing part of the gas will manage to escape and if this is taken into account, calculation of the state of the gas and dimensions of the filtration zone will be different from that discu~ssed with p~ 9 Ps" Only over a very long time interv~l after opening the valve could pressure in the vicinity of the borehole increase to a value exceeding p, as a result of desorption of slowly released gas. This could only occur in very solid coals, i.e., with quite low values of p~. What occurs in the vicinity of a measurement on approach of a working causing a reduction in mine pressure in the vicinity of the borehole? Shown in Fig. 12 is the stress distribution in a horizontal seam in planes orthogonal to the face surface and to the zone broken coal similar to zones in the vicinity of a borehole. Axis 0x is directed along the strike, axis 0y is orthogonal to the seam, and oxis 0z is in the plane of the seam. A change in the value of uX with distance from the free surface depends not only on the deformation characteristics of the coal (elasticity moduli, coefficients of friction and cohesion in the plastic deformation zone), but also on limiting values of coal adhesion with adjacent rocks [19, 20]. In zone x < &,, where &, is distance from the

Ill

-"';-..

"

~, i ".'...':-S-..--~" Pz

I L~

7,. l.t-

_

Fig. 12. Schematic picture of a section of a horizontal coal seam.

193

"

~

b

(,

'I

Pig. 13. Schematic explanation of test results for measuring gas pressure in boreholes on approach of a working face.

free surface to the boundary of the filtration region in which pores containing gas communicate with each other end filtration is possible along axis 0x. Let x " &~, be boundary of the filtration zone in the direction of axis Oz. Plane x = s can be assumed to coincide with the boundary of the zone of plastic deformation, end x = & ~ is the boundary of the region where permeability in direction 0z differs from zero in the zone of oriented cracks. Pressure P0 of free gas in pores (cracks) in zone x 9 &~ is close to P00, i.e., gas pressure in pores of the seam undisturbed by mining, and in zone x < s at surface x = &~ pressure p = p,, where p, is gas pressure which relates to the limiting equilibrium state of broken coal. With x = s o x = s x - ~, where s x are compressive normal stresses transferred through grains of soil. If s x relates to the limiting equilibrium constant in region x < s then ~, = (I - 6)p~. In plane x = s there is a jump in gas pressure P0 close to P00 to a value of p~, where p~ is determined by coal strength. How will gas pressure change in the measurement borehole as the face approaches it? While boundary x = s is at a significant distance from the face r < R (Fig. 13a) in the vicinity of the borehole, pressure in the borehole will not change. When the zone of oriented cracks &~ < x < & ~ intersects zone r < R (Fig. 13b) gas filtration from the seam commences into the region surrounding the borehole, and pressure in the borehole will increase. Wlth further advance of the face when the filtration zone x < &~ intersects region r < R (Fig. 13c) gas filtration commences from the region surrounding the borehole in the direction of the surface of the face and gas pressure in the borehole will fall. 4. Gas Filtration in the Vicinity of a Borehole. Outflow of Gas from an Open Borehole We write a filtration equation for gas in the vicinity of a borehole drilled transverse to the seem as a result of its desorption from the coal. Gas desorption may be represented as the effect of uniformly distributed sources with thickness ~ = ~(p, t) per unit volume of coal. The equation for mass conservation will be

-divo~ + x , We also assume that Darcy's law is valid

(4.1)

_

(4.2)

Opm/Ot --

where ~ is filtration velocity vector.

K!

v =----~-gradp,

For radial filtration in the vicinity

where Kf is permeability factor; ~ is gas viscosity. of a borehole we have consequently: E! i

O(pap~

P r Or ~o r ] r

ap

~m~'~X.

(4.3)

Sorption isotherms obtained in laboratory tests with prolonged pressure measurement are in good agreement with the Lansmuir rule (1). With rapidly occurring processes when it is possible to assume approximately that the amount of gas released from the coal follows directly from the corresponding change in pressure, only a small part of the sorbed gas is released. It is assumed in [21] that this amount is about 10-20% of the total volume of sorbed gas, i.e., that pert of the gas which desorbs at high pressure. In the high-pressure section the desorption curve can be approximated satisfactorily by a straight line, for example coincidins with the tangent to this curve at point p = Ps (see Fig. i). The angular coefficient of this tangent is

194

dM)

-~ = ( ~ "ab The corresponding sorption isotherm for rapidly released gas will be

dM

(4.4)

The total amount of rapidly released gas

Ms ( P - - i ) M - - M. ---- i--'~-3~' ~"

.

I f ap. = 4, t h e n t h e p r o p o r t i o n o f t h i s g a s i s 20%. E x p a n s i o n o f an i d e a l g a s w i t h r e s t r i c t i o n will be isothermal, p ffiRETO, where Rg is gas constant and T is absolute temperature. O(Ms-- M) #t

Then

ab 8p i -.1-aps Ot

and Eq. (4.3) takes the form

gt t

of ~

Op

"~ 7 " Or---~rp or ] = m B "gF,

(4.5)

where

abRr?~

B = 1 + m(i +=p,)'"

(4.6)

For methane, Rg = 519 J/(kg.K), with T ffi 300 K, m ffi 2.5%, a = i (MPa) -I, Ps = 4 MPa, b ffi 15 kg in i m s of goal and B ffi 5. We introduce dimensionless variables: y = r/r., ~ -- p/p..,

x -- t/tH,

where to is time scale for filtration.

For example with m ffi 2%, R0 = 7 cm, p ffi 0.1 MPa.sec, Pu0 ffi 20 HPa, Kf ffi 10 -s pD, to = 50 sec. In new variables (4.5) will have the form t

o / apS~

7.~LyWj

B ap~

= T.~

(4.8)

We consider the problem of outflow of gas from an open borehole and at first we iet P~ < Ps" Some time x. after opening the borehole and forming a filtration region y < y0 a stressed state arises and filtration flow forms, subsequently almost independent of the value of initial radius R0. With r < RI in the nucleus of crushed coal formed in the vicinity of the borehole gas pressure becomes equal to atmospheric pressure. With the passage of time the filtration zone y < Y expands and in this way there is a change in similar stressed state and filtration regimes depending only on the amount of gas flowing out of the borehole. Gas flow is self-modeling and it corresponds to the discharge in a borehole of infinitely small radius. This flow conforms with solution (4.8) depending on a single variable ~ ffi B(~), where

--YlY~ S i n c e a ~ / S y = B'/~I~-~ and aS/Sx ffi - ( S ' ~ ) / ( 2 x ) , we o b t a i n f r o m ( 4 . 8 )

~'+z

(4.9)

w h e r e ~' = d $ / d C , t h e n by d e s i g n a t i n g

N~+

§

•0.

~2 = z ,

(4.1o)

Th.is equation is convenient for solving by successive approximation, substituting in the first assumption B(~)/2B for some function ~. For rapidly desorbing gas it was assumed above that for B the value is constant. Following L. S. Leibenzon it is possible also to suggest that B = ~ (this approximation is acceptable with large values of ~) or to assume that B = ~,(C/~,)n, n < I, where ~, and ~, are values of C and B in the filtration front. Assuming that ~ ffi ~ , Eq. (4.10) will be

195

z" + ,, (.4~ ++)=o

(~.iz)

or, with ~ = ~(~/r

," +

z'.(.4"~,-- + ~-) = o,

(a.12)

where A and A' are constants. If, for example, B = 4 and

~, = 0.25, then A ~ 8; if B = i, then A = 2.

We demonstrate that solution of Eq. (4.10) meets the conditions for an expanding boundary of the filtration region y = Y. At this boundary gas pressure is constant ~ = ~ , where ~ is pressure which is established in a measurement borehole after closing the valve (see Sec. 3). At boundary y ffiY with its displacement conditions should be fulfilled for gas mass conservation flowing into this boundary in time dt in region y < Y equal to -2~Rp~vdt, and it should equal the amount of gas compressed to pressure P0 contained in cracks in volume 27. Rm0'P0dR ahead of the filtration front. By using (~.2) and bearing in mind that m0'p 0 = moPoo and P~/Poo " P , / P o o , we obtain

~ffi~.~.~ di P.o mo~ ~r o r , changing o v e r t o d i m e n s i o n a l v a r i a b l e s , ~ ~dy-_ ~,~.

Let ~ _ <

8~/Oy ffi t ! ' ~

(~. 13)

~s" For our solution at boundary y = Y, hearing in mind that y = ~ we obtain

, ' . . ~,~. ffi ~ .

(~,. 14 )

Now we use a solution for approx~_mation Eq. (4.11).

By integrating we obtain

z' -- c e-C~sg', T

(/~. 15)

where c is a constant. Discharge through surface r - const at instant t is 2~pvr. this discharge to the value

t,

uo= = ~'~p~ = _ ~ '

q

and

We refer

'

ffi - ~

= - ~-<~'>:;

(4.16)

uo

On the filtration front,

q, ffi-rH2.

(4.17)

q ~ qoe(-~us)~',

(4.18)

From (4.15)

where q0 is a value of q with ~ * 0 (discharge of the outlet). to = - ( ~ / 2 )

By integrating (4.15) with conditions: tain

,<~'~i:.

with ~ - ~

So that (4.19)

at the filtration front z = ~.2, we ob-

~'~e--(Mslus

P: - ~ = - q . Y - 7 -

~" (4.20)

196

fl =.P/Po

0,4 F

....

-"0!,-,-rI ' ~

~,-p,l~o

I

f

l

Y+Z~Y

0

D

Fig. 14

Fig. 14. hole.

' 4 0'

'

&o

r'

Fig. 15

Schematic distribution of gas pressure close to a bore-

Fig. 15. Change in time of dimensionless discharge of gas from a borehole.

But

[22] '*'-(*/')=' _.v---du

=

~

[Ei

(--u,) -

Ei (u)l,

where u = (AI2)c = and -Ei(-u) = 0.5772 + inu + u - u2/4 - ... With u 9 0.4 with an accuracy up to 5% it is possible to limit ourselves to the first three terms of the expansion, and with u < 0.i with the same accuracy to the first two terms. In region r < RI, where initial coal grains are not pressed to each other, as well as in the zone of uniformly compressed coal formed in the vicinity of the borehole axis filtration resistance is small and gas pressure at surface r = RI is close to atmospheric. For an open borehole the value of RI/R will not be completely determined by (3.13) or (3.11) since coal in this case is loaded additionally with a gas pressure gradient. The corresponding stressed state of limiting equilibrium will also be determined by the same initial values of s R and ~ from relationships (3.8) and (3.10) and with sR1 = 0 it will change with an increase in R retaining similarity. With formation of a compressed nucleus of coal with an increase in Srl this state of similarity is disturbed and the ratio n = R/Rz will decrease gradually. With Srl = 0 a picture and filtration flow. If CI set of equations (4.19) and proximately with ~a ~ ~, we

is established of a change in similar states of both stresses is the value of C at boundary r = RI, then ~,/CI = n, and from (4.20) the corresponding values of ~ and q0 are determined. Aphave for determining q0 and ~, the equations i

i

8(A12)~s 9

With an open borehole coal in the zone of limiting equilibrium is additionally loaded by a volumetric force proportional to the pressure gradient so that the equilibrium equation takes the form dr -- ~ ~P

+'"T .'e

--m~-

or, in similarity variables,

With low porosity the effect of the pressure gradient is small and it only appears in the vicinity of a borehole. 5. Calculation of the Process in a Computer for Establishing Gas Pressure in a Borehole After drilling and sealing a borehole a filtration zone forms which gradually expands as gas flows out of the borehole. An increase in the filtration zone will proceed while the

197

borehole remains open. Let the borehole be closed at a certain instant of time t = tF. We designate in terms of Y dimensionless radius of the filtration zone relating to the instant of closing the borehole. After closing the borehole gas pressure in the filtration zone and in the measurement chamber starts to level out, and as a result of this p - p~ is established. In this way there is an additional increase in the size of the filtration zone. Calculation of the filtration process with a computer is carried out as follows. It is assumed that after drilling a borehole an initial filtration zone forms around it with radius y - y0 and initial gas pressure in this zone is p = P0- As gas flows from the filtration zone y < y0 pressure in it starts to drop. When gas pressure at the boundary y = y0 drops to a value p = p~ coal breaking occurs in zone y0 < y < y0 + AY. At this instant pressure distribution in zone (y0 + AY) is shown in Fig. 14. In time At pressure in the new boundary of the filtration zone again becomes equal to p~, again there is spalling, etc. Calculation of the filtration process in each stage is carried out by the finite difference method [22]. The value of AY is determined by the structure of the material, but as calculations show, with a quite fine breakdown network over the radius the results are almost independent of the value of AY. We introduce data of the calculation for a specific case. We consider a seem 1 m thick lying at a depth of 600 m which corresponds to lql = 15 MPa. A borehole is drilled transverse to the seam with radius R00 = 2.3 cm and during drilling an additional volume of coal (V d) compared with the volume of the drill (V0) was removed. Then the borehole was sealed, and pressure p~ was established in the measurement chamber. As an example it is assumed that p~ = 60 arm, the valve is opened for time tF.and then closed again, and in the measurement chamber pressure p~ is established in time t ~ after closing the valve. For the time while the valve is open the size of the filtration zone increased to a value of Y at instant t = t F. After closing the valve the radius of the filtration zone continues to increase, reaching at the instant of establishing the pressure a value of Y~. Given below are values of Y = R/Ro, Y~ = R~/Ro, and t ~, where R0 is actual radius of the borehole taking account of the coal dust removed (R02/Ro02 - Vd/V0) for three values of t. It is assumed that Vd/V 0 = (Ro/R00) z = 5, Kf = I0 -s mD, and the gas is methane. The time scale with the chosen parameters t = 44 sec. Establishment of gas pressure in the measurement chamber t, h y y * t *, h

i 6 9 2

5 il iS,5 8,5

iO 16 17,5 12

Shown in Fig. 15 is the change in time of the dimensionless discharge of gas from a boreh o l e Q/Qo (Qo : R o P o / t o ) 9 LITERATURE CITED I. 2. 3. 4. 5. 5. 7. 8.

198

I. V. Bobrov and R. M. Krichevskil, Combating Sudden Coal and Gas Bursts [in Russian], Tekhnlka, Kiev (1964). I. L. ~ttinger and N. V. Shul'man, Distribution of Methane in the Pores of Coal Resources [in Russian], Nauka, Moscow (1975). S. V. Kuznetsov and R. N. Krigman, Natural Permeabillty of Coal Seams and Methods for Determining It [in Russian], Nauka, Moscow (1978). A. T. Airuni, Theory and Practice for Combating Mine Gases at Considerable Depths [in Russian], Nedra, Moscow (1981). Instruction for Safe Introduction of Mining into Seams Inclined Toward Sudden Coal, Rock, and Gas Bursts [in Russian], Nedra, Moscow (1977). I. V. Bobrov, Methods for Safe Driving of Prpearatory Workings in Burst-Prone Se-m~ [in Russian], Gosgortekhizdat, Moscow (1961). I . V . Bobrov, Driving of Preparatory Workings into S e a m s Prone to Coal and Gas Bursts [in Russian], Makeevka-Donbass, MakNII (1959). A. S. Vavakin and R. L. Salganik,"Effective elasticity characteristics of bodies isolated by cracks, planes, and rigid heterogeneities," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2 (1978).

9. I0. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21.

S. G. Lekhnitskii, Elasticity Theory for an Anisotropic Body [in Russian], Nauka, Moscow (1977). Yu. F. Kovalenko, "Elementary act of a sudden burst phenomenon. 5urst in a borehole," Preprint No. 145, IPM Akad. Nauk SSSR, Moscow (1980). G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Movement of Liquids and Gases in Natural Seams [in Russian], Nedra, Moscow (1984). I. N. Sneddon, The Fourier Transform [Russian translation], IL, Moscow (1955). G. I. Barenblatt, "Mathematical theory of uniform cracks forming during brittle failure," Zh. Prikl. Mekh. Tekh. Fiz., No. 4 (1961). Yu. F. Kovalenko, "Effective characteristics of bodies with isolated gas-filled cracks. Failure wave," Preprint No. 155, IPMAkad. Nauk SSSR, Moscow (1980). Yu. P. Zheltov and S. A. Khristianovich, "Hydraulic breakage of an oil stratum," Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 5 (1955). K. Terzaghi, Soil Mechanics Theory [in Russian], Gostroiizdat, Moscow (1961). V. V. Sokolovskii, Loose Material Statics [in Russian], GITTL, Moscow (1960). N. S. Bulychev, Mechanics of Underground Structures [in Russian], Nedra, Moscow (1982). S. V. Kuznetsov, "Effect of tangential stresses on the contact surface of a seam and rock on the stressed state of the rock mass," Fiz.-Tekh. Problm. Razrab. Polez. Iskop., No. 4 [1970). A. N. Tumanov, "Stress-strain state close to a longwall face," Dep. in VINITI 16.08.82, No. 5635. S. V. Kuznetsov, R. M. Krivitskaya, and N. Yu. Lavrenkova, "Kinetics of gas release from recovered coal with a different degree of breaking," in: Questions of Ventilation, Cooling Air, Combating Dust and Monitoring the Mine Atmosphere in Mines [in Russian], Donbass-Moscow (1983).

ELASTOPLASTIC PROBLEM OF AN EXTENDED CYLINDRICAL WORKING D. I. Imamutdinov and A. I. Chanyshev

As mine workings go deeper, so the mine pressure increases. At sufficient depth of mineral extraction, regions of plastic deformation form around the mine workings. Depending on the mechanical properties of the rock mass and on the ratio between the gravitational and tectonic forces, the plastic region exists partially or completely over the working; Solution and analysis of the corresponding elastoplastic probl~m~ is of some urgency in mining technology. The present work is devoted to the solution of the elastoplastic problem of an extended cylindrical working. It is assumed that a state of plane deformation is realized in its vicinity, the plastic state of the rock mass is determined by the Coulomb~ohr condition, and that the relation of [1] holds. The cases of complete and incomplete coverage of the working periphery by the elastoplastic boundary are considered. A step-by-step method of solution is proposed for the elastoplastic problem. This method assumes division of the loading path into small steps (stages), linearization of the initial system of equations, and subsequent summation of the basic (known) and additional states. From a math~me~tical viewpoint, the given problem is equivalent to the problem of bilateral compression of a plane with a circular hole. The elastoplastic problem of bilateral tension (compression) of a plane with a circular hole was solved in [2-15]. In [2-13], the case of a plane-deformed state was investigated, and in [14, 15] the case of a plane-stressed state. It is assumed that the stresses in the plastic region are related by the Tresk [2-12], Sokolovskil [8], and Annin [13] plasticlty conditions. These stresses correspond to a generalSpecial Design Office of Geophysics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 3, pp. 24-32, May-june, 1988. Original article submitted October 26, 1987.

0038-5581188/2403-0199512.50 9 1989 Plenum Publishing Corporation

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