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9ПРОСТАЯ МОДЕЛЬ ГИДРОРАЗРЫВА УГОЛЬНОГО ПЛАСТА
Муллагалиева Л.
PhD докторант кафедры «Разработка месторождений полезных ископаемых» Баймухаметов С. доктор технических наук, профессор кафедры «Разработка месторождений полезных ископаемых»
Портнов В.
доктор технических наук, профессор кафедры «Геология и разведка месторождений полезных ископаемых»
Юров В.
кандидат физ.-мат. наук, доцент Карагандинский технический университет Казахстан, Караганда
SIMPLE MODEL OF HYDRAULIC FACING OF COAL BEDROOM
Mullagaliyeva L.
PhD doctoral student of the Department «Development of Mineral Deposits» Baimukhametov S. Doctor of Technical Science, Professor of the Department of «Development of Mineral Deposits»
Portnov V. Doctor of Technical Sciences, Professor of the Department of «Geology and Exploration of Mineral Deposits»
Yurov V.
Candidate of phys.-mat. sciences, associate professor Karaganda Technical University Kazakhstan, Karaganda
Аннотация
Рассмотрим угольный пласт на большой глубине как изотропную монолитную среду. Как и в реальных твердых телах в такой среде всегда есть дефекты структуры, радиус которых начинает изменяться при наложении внешнего давления. При достижении некоторого критического размера начинается перекрытие радиусов соседних дефектов и возникает трещина. Такую простую модель угольного пласта, содержащего сферические дефекты, полагаем в основу нашего теоретического рассмотрения гидроразрыва угольного пласта. Рассматриваем задачу роста дефекта в терминах уравнения теплопроводности и затем температуру заменяем на давление. Затем рассматриваем слияние дефектов с точки зрения статистической физики. Окончательно получаем приемлемую модель для расчета параметров гидроразрыва угольного пласта, которая совпадает с экспериментом.
Abstract
Consider a coal seam at great depth as an isotropic monolithic medium. As in real solids, such a medium always contains structural defects, the radius of which begins to change when external pressure is applied. When a certain critical size is reached, the overlap of the radii of neighboring defects begins and a crack appears. We assume that such a simple model of a coal seam containing spherical defects is the basis of our theoretical consideration of hydraulic fracturing of a coal seam. We consider the problem of defect growth in terms of the heat conduction equation and then replace the temperature with pressure. Then we consider the merging of defects from the point of view of statistical physics. Finally, we obtain an acceptable model for calculating the parameters of hydraulic fracturing of a coal seam, which coincides with the experiment.
Ключевые слова: угольный пласт, дефект, температура, давление, трещина, структура.
Keywords: coal seam, defect, temperature, pressure, crack, structure.
Introduction
A significant number of works have been devoted to degassing a coal seam by hydraulic fracturing [1-5]. The most common method in the coal industry to ac-
tively stimulate a seam is to increase its gas permeability by working or working over. Let's take a quick look at these methods.
Torpedoing of a coal seam is carried out by placing explosive charges in drilled holes. The magnitude
of these charges and their structural arrangement are chosen in such a way that only cracks of a certain size could form in the coal seam. These fractures cause an increase in the gas permeability of the formation. Due to the complexity of its implementation, this method is not currently used.
Pneumatic disengagement is carried out by supplying compressed air to the coal seam under high pressure. This also leads to the formation of a system of cracks in the coal seam, along which methane is extracted. This method is only applicable to coal seams drilled from the surface due to the low air density.
A high-power explosion in underground conditions - for example, a nuclear camouflage explosion in a well drilled from the surface or an old mine working - significantly changes the properties and state of the formation. These changes can be used later when drilling degassing wells to increase their production rate. The radius of influence of such an explosion on the properties of the reservoir can be up to 150-200 m, which determines the advance of its impact.
Hydraulic stimulation of a coal seam with water or an aqueous solution containing a certain amount of surfactants is the most common fracturing method.
Simple coal frac model
Consider a coal seam at great depth as an isotropic monolithic medium. As in real solids, such a medium always contains structural defects, the radius of which begins to change when external pressure is applied. When a certain critical size is reached, the overlap of the radii of neighboring defects begins and a crack appears.
We will use such a simple model of a coal seam containing, for the sake of simplicity, spherical defects as the basis of our theoretical consideration of hydraulic fracturing of a coal seam (hydraulic fracturing). So, let a spherical defect grow in a coal seam. The pressure distribution around the defect can be approximately described by a stationary equation similar to the heat conduction equation. We will consider the problem of defect growth in terms of the heat conduction equation and then replace the temperature with pressure in accordance with the Mendeleev-Cliperon equation. We then look at the merging of defects from the point of view of statistical physics.
So, the temperature field around a spherical defect can be approximately described by the stationary heat conduction equation:
dl+=o, (i)
dr2 r dr
where r is a variable radius in a spherical coordinate system.
The general solution to equation (1) has the form:
T(r ) = A + B, r
(2)
where A and B are arbitrary constants. On the surface of the defect, the equation describing the kinetics of the defect growth process simultaneously takes place:
dp dt
: K[Tk - T(p )1
(3)
where p is the radius of the spherical defect; K is the constant of the growth rate of the defect and the heat balance:
n dP /ST
QoY- = -X dt I Sr
(4)
where Qo is latent heat; y - specific gravity; 1 - coefficient of thermal conductivity. Substituting (2) in (3) and (4), we get:
£ = K(Tk - A - B) ; Qo y £ = x B (5)
dt p dt
p
If we take into account that limT(r) = To as c ^ ® and introduce the notation T(p) = ^(t), then we get:
T(r) = To + [<p(t) - To ]P.
(6)
In this case, equations (5) will be rewritten as fol-
lows:
d = K[Tk -p(t)] ; QoY dP = ^P(t) - To ]P. p dt r
(7)
From the system of equations (7) it is necessary to determine the functions ^(t) and p(t). We exclude the function ^(t) and obtain:
QoYAp2 + ldp_(T -T0) = 0, (8a)
2X dt K dt
, dp
or
d_ dt
P2 + 2—P-(Tk -To)*t
= 0.
(86)
(a - thermal diffusivity). Taking into account that in our simplified consideration p(0) = 0, we find:
p2 + 2 — p-2-(TK -T)*t = 0. (9) QK Qv K
Hence:
p(t) = -_!_ ± (_1_) + 2 —(Tk - To) • t (10)
QK v QK Q 0
Since p(t)>0, then the plus sign should be taken in front of the root. In this way:
^ x a
P(t) =-
QK
For small t:
1 + 2QK2(Tk - To) • t -1
. (11)
1 + 2QK2(Tk - To) « 1 + QK2(TK - To) • t(12)
p(t) « K(Tk - To) • t
(13)
Passing now to the pressure drop AP by the hydraulic fracturing method, we get:
p(t) = K0 • AP • t (14)
So, at short exposure times, the radius grows linearly with the pressure drop. If t is large, then:
1 + 2QK2 AP • t.. iE^;
or
^ AP
p(t) /2a Apt
(15)
r=p
a
a
a
a
a
Thus, at low t, the rate of the process is determined by the kinetics of defect growth. With increasing t, the role of the dissipation of mechanical energy increases and, finally, becomes decisive.
Let's move on to the next stage of our task. Let the density of defects in a coal seam be equal to n, then the probability W(p) that the nearest defect will be at a distance p (i.e., defects will merge) from the selected defect can be easily obtained from classical statistical physics, and it is equal to:
W(p) = 4rcnp2 exp[- 4n2p3 /3] (16) The probability of finding N particles in a defect of radius p is, obviously:
wn(p)=n wt(p)
(17)
On the other hand, we define the probability (16) as the ratio of the energy of one particle to the total energy of the system. Thus, we have:
= (4nn)N pNexp[- 4nn2p3/3] (18)
3/2nkT
Taking the logarithm of both sides of (18), performing simple transformations and discarding small terms, we obtain: N AP 4
N • — = 4nN • n2 • p3 ™ n P 3
AP 4 2 3
— = - nn2g3 P 3
A 0
4
(19)
For a unit volume y^ = _ n L and taking into account that n = ^, and N • g = L - crack length, we
get:
AP-L
or crack length
L = L„i
0
(20)
here Lo is the initial crack length. The initial crack length is determined by formula (14). The constant K has the order of unity, and the time t ~ 10-7 s and is determined by the speed of sound in the coal seam. At the maximum pressure AP = 600 MPa at fracturing of the coal seam, we have (Po = 1 MPa): AP
— = 10^0.6 « 7,,
P,
1
L„ « 200•¡O6 —10-7 = 10, 0 2
then L ~ 70 m - this practically coincides with the experimental value.
Thus, we have obtained an acceptable model for calculating the parameters of hydraulic fracturing of a coal seam.
Conclusion
The main results obtained in this article are as follows:
- hydraulic action of a coal seam with water or an aqueous solution containing a certain amount of surfactants is the most common method of hydraulic fracturing;
- the pressure distribution around the defect can be approximately described by a stationary equation similar to the heat conduction equation. On the surface of the defect, an equation describing the kinetics of the defect growth process is simultaneously valid. At short times, the rate of the process is determined by the kinetics of defect growth. With increasing time, the role of the dissipation of mechanical energy increases and, finally, becomes decisive. It is shown that at short exposure times, the radius grows linearly with the pressure drop;
- let the density of defects in a coal seam be equal to n, then the probability W (p) that the nearest defect will be at a distance p (i.e., the merging of defects) from the selected defect can be easily obtained from classical statistical physics;
- finally, we got an acceptable model for calculating the parameters of hydraulic fracturing of a coal seam.
References
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2. Zhao Z.M., Wang G. The research of gas drainage technology in daning coal mine // Applied Mechanics and Materials, 2014, vol. 580-583. - P. 2558-2563.
3. Malashkina V.A., Determination of the operating modes of degassing installations of coal mines with sections of underground gas pipelines made of composite materials // Mining information and analytical bulletin, 2018, CB 19. - P. 112-116.
4. Krings T., Gerilowski K., Buchwitz M., Hartmann J., Sachs T., Erzinger J., Burrows J., Bovensmann H. Quanti8cation of methane emission rates from coal mine ventilation shafts using airborne remote sensing data // Atmospheric Measurement Techniques, 2013, Vol. 6. - P. 151-166.
5. Malashkina V.A. Directions for increasing the efficiency of underground degassing to improve the working conditions of coal miners // Mining information and analytical bulletin, 2018, No. 7. - P. 69-75.
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