CC BY
8Journal of Siberian Federal University. Engineering & Technologies 1 (2013 6) 75-82
УДК 622.831.325.3
Numerical Studies of Coal Bed Fracturing for Effective Methane Drainage
Andrey V. Patutin*, Petr A. Martynyuk and Sergey V. Serdyukov
Institute of Mining, Siberian Branch, Russian Academy of Science 54 Krasnyi, Novosibirsk, 630091 Russia
Received 11.02.2013, received in revised form 18.02.2013, accepted 25.02.2013
An effective methane drainage from the coal seam is essential for safety mining operations. This paper describes the algorithm used for calculating trajectory of crack formed by the hydraulic fracturing carried out in parallel wells. The results of numerical analysis show that direction of crack propagation is affected by the pressure applied and surrounding rock's stress state. The maximum possible distance between hydraulically fractured wells at which they linked was estimated. The obtained results help to design hydraulic fracture treatments in coal measure rocks in order to solve the problem of underground methane production or intensification of beds degassing.
Keywords: numerical modeling, methane drainage, hydraulic fracturing control, coal bed methane.
Introduction
One of the main challenges of underground coal mining is methane-bearing formations and sudden coal and gas outbursts related to them. The probability of catastrophic events of gas-dynamic processes increases with the depth of the coal seam. It is necessary to note that problems caused by high methane concentrations not only affect the safety of mining operations, but also the economic and environmental aspects of coal production.
Coal bed methane (CBM) extraction is complicated by low coal permeability, methane sorption (trapping) on the surface, water saturation, and various geological formations with natural or artificial fracture systems. There is a particular complexity in gas production during preliminary degassing. At this stage the coal bed is not affected by development, the channels for gas flow are not formed and the overall degassing operations are not efficient for methane removal from the coal measure. Due to these factors the risk of gas outbursts is still very high and the labor productivity is poor.
One of the main methods of CBM production stimulation is hydraulic fracturing which is used in wells drilled both from the surface and from the underground.
© Siberian Federal University. All rights reserved
* Corresponding author E-mail address: [email protected]
Statement of the problem
Simultaneous propagation of several fractures and, as a result, formation of main crack that will link several degassing wells isttne of ohe main conditkan for effitieno gas recovery. To achieve the goal it is necessary ro solve the peoblrm oa wells positioninf which is strongly affected by stress conditiof of coal measure;.
There is a problem of determining the relative position of simultaneous fracturing wells under a given stress state of the coal measure rocks at ihe desif n stage oh ehe degas sing system.
To assess the control efficiency of a synchaonous propagation of tracks we have developed special software that implements incremental algorithm for aalculating quasi-sratic trajectories of cracks.
Mathematical model and algorithm description
Solution to the problem based o n the mathematic al model which suggesf s pro pagation of N smooth curvilinear cracks from the boundary of the circulan hole with eadius i. There is a unifoem field of compressive stresses acrieg ir infinity witf intensify e<> and f0 (|e0| =e |f0|). Let us consider the form of each cut Ls in local coordtnate system xkOsys is known and determine by parametric equation:
ts = as (ir£) = xs (SO++ns (<=); ^n s e Le
Relationship between coordfnates (tit points in the maip and local systems is given by:
z = x + iy = ^sextr»fiass) + z0 = (xk + %)expO'«s) + 4 + iyk
where as angle betwe en Ox h OsXs axes, and z0 — the origin of coordinates of local system in the main.
We seek s solurion to fhe preblemof elasticity theory, when normal ind tangential sfresses ari known on the contours of the cuts:
Wl + tTj = pa SSs); ^Li; k = \N
The plus siggn reSeas to thei upper edge ok the cut, and the minus sign to the bottom. There is a compressive stress field et infinity and stfesset are deteemined at the confour of ahe cut (the pressure in the fracture is equal to c0 +nd its contour is stretched fn the veattc nl direction lay hty = nir).
When integral expressions of complex potentials of stresses for a plane with a circular hole [1] are used rhe probiem eeduces to finding solutions jfs p), s = 1, N of She eomplex system consisting ol N singular integaal equations:
C ts J, ___
s = 1 _
whrrr g 's(^) = afkStk)o)({lk=') and g'k(tk( - derivative of displacement shift
[ +iV()+ -)Us +ivk)-]=i4)1 E ) g^t), tseLs
where us, vs horizontal and vertical edge displacements of the cut in the local coordinate system, v - the Poisson's ratio, £ - modulus of elaoticity of tfe medium. Kesnels of the irrtegfal equatisnr ate:
-76 -d
Rkn tne=Re ffk n-a+eef ookn --+ +t+P = • r_ ta, +ee
Sef a a += s+ eo ,rna+e'a a a, a++• ea, na
Te = Tk (aa = e'a* 2 —) + z(e; T„=T„ (+) = e!i0n a a (LV) + ;
= V 0O7) -2ta„
He ~ '*e
m'n (jf)
cknza, +t=
A = l- T( Tk,
tzf ■= Tf -1,
a! = Tf - fn
k
k
- + —= + -
A k
1 + —
a2 Taal Taal2 z
C+ r 2TnA an k A(3AfTA-T a
r^ -a, ze •——+z=tA-= + v f n__' z= t_ -nk akT„ ^Tf T„
sfy--,+) = -
a2
f ^ 32kTf k2 T^^-kf T2~TnTf
a22 TkT- ak2 T-
n V f n
T3
Pn--)y-Pn(+)-\p+ -P-2Pe
fe2Y)
pTf -
+ Pn
P + +°0
+ P-
2T„e
-2iy
3e
-2iY
■- A
00
2 Pee
-ln:
T -/
V . nTn Tn +. T
+ Pn
k Tn - Tn Trnk-2 _ ln T^ri_+ 2-TJ T -J)
in
Tn-i 7fT20k + T2)
where p+ = (no + q0)/ 2; p- = Off -q0k12.. As cr-ckr ^«Jgt^uii to grow from tlie kounda+ of the hole, the kernels have: the jhc^d^f^rrtAh" kTR-l,-) = nknt-k,-) = 0. Note that if the number of cracks is even, the central symmetey ef tine problem can be used, and it is possible to reduce he order of the equatiess syikem by half.
The srlhloon for kctnels (tit inSegbal enuafieni is sought in the ^turtc ut (h) = ^^ Ob) j ^^^; ¡mod using tJne; Gauss quadratee foi*nn:^l£is>, the solution of integral equations system is reduced to solving a system ot linear algebraic equations:
N IN
]T [ (a,, n — n)+s^ (a,, n —k (.a, )]a=„n! n);
(=1 i=1
j = 1, N; m = 1, n -1
where n is tiie order oS npproximntihn of solution arred = cos ^P—=, S = 1, n '
2 n
nm = cor-, m = 1,n-1 nre zeros oO t^e Chebyrhnv potynomialr e^OO = chi(n7nrcchi0)s
n
rln(_7nrCCOr 77)
Ub^-i (n =-1 „— of first and second kinds respectively. Mathematical conditions that will
etr^it^^ finiteness o:f (^^^]f)lacenr^nt; iEtlr ^l^e ieft end of the cut located on the border of the hole and will close the ryrteib of equations are (pk (-1) = 0; k = 1,N.
— eu -
The valtaet of the function <pk (f) at the ends of cuts are defined by tha fol lowing equaaions: ) n< 71(21 e^
V1 <
i=1
<e—< = ---yo{-iype-t)ct-
tr e_t
4e
n(2i -1) 4e '
whe<ep— —k) is the sfolutnion (31s system on equutiont at nodal poiott. Stress intenstiy factors at ahe tip of k-crack:
Kik-iK2p=e)l\^(l\ ^
Pk (1) -k (1)
The limit equilibrium condition for any crack can be written as:
K,
1 3 -
4 v 2
Kk not3(^)
K1k + 3VK1k + SK2k
MC
Vnr
where K=, Ku are stress intensity f2ctors for the tip of k crack, K1C is critical stress intensity factor,
K\k - —Kile ~+=>K2k
and nngle S»k = 2arct---determines the direction of further crack growth, which
4K
2k
coincides with the plann on which the principal component of the tnnsile ttrers seachesiitr maximum value, sncf ¡heat stresces sire zsre. Ttus iss alitec known at as irr:^terion [2].
Cracf e^mng diiiijsi^isc^ns^^^ [v] = v- " -f ¡^lonj? i-tess itath can tee determined ns:
M=-
4((-v2)
e
Ik Re
I®i(Eo)|f Vk (E)
E
T Vk
2(Eo)E^V(-E
wehere tk is tno ^(nfte^^ni oe k nrack, anin tunctioe vr (Cl can be cnlculaSed through then kcown values in neCal pniees -^ as CoSlpws:
Vk<
(JV = -Y(-D^V- 22)
n E—(
t.
E-Ej
2=( " E Thus, we can finC tne crack or crackrs where limit equilibrium sondition is tatisfied. Then the stepw[se algorithm for building quasiisteitic arbjeceories of fracture is ured. It has iieen
tesied snd d^::>c;»-^i^ctd in [3-5-.
To estimute Ae ceiltLciiiiicjit of simultaneous fracture propagation caused by longitudinal hydraulic fracturing special software bared on quasseslntic approaimation war devefoped; The pnogram allows to calculane ttie cle¡IifSIsclliLIl^ on several Sactorp
a) the dirlance between cunters te;^ enitial feastnees d; lb ) initio fracturer length S;
tt) stress statu oS coal bed, perticuiarly mii^^mum ehiic» minimum compressive stress components P1 and q, respectively and thtiesi :r relation qdp;
— ur —
<y.
gam
OP
Fig. 1. The initial position of cracks in the; compressive field
Fig. 2. Calculation termination conditions
d) the angle gam between coalescence plane of adjacent cracks and maximum stress direction (Fig- 1).
It was assumed that there are five cracks in isotropic elastic plane with half length equal to one located on straight line along X coordinate; p0 and q0 were compressive stress components at infinity (p0>q0) and the angie between maximum snre ss p0 and anitial cracks direction is gam. Tae fracnure growth is provided with pressure appUied to the initial cracks (e00><p0, where e00>0).
According to the software algorithm, the count stopped when growing fractures approached each other closer than axcrit or one of fracture wings came out from the aycrit interval (Fig. 2). These critical values were obtained during auxiliary studies covered the process of fractures interaction and coalescence on relatively small ranges.
The limitation for axcrit can be explained due to certain features of used software: the algorithm is not designed for modeling of fractures intersection. That is why the value of axcrit was chosen small enough 0o coesiper sepanate crecks as ajointed snotem. The reason for aycrt, limit is due to the fact that cracks do not merge during the fracturing when they are out of ±aycrit interval.
The purpose of numerical studies was to determine the maximum distance between fracturing wells dmas at which separate cracks meage into a single system, and to esrimete the impact of various factors on this value.
Numerical experiments and results
Numericalexpeyimentr we re carried out as follows: n) qdp valuewns chosen equal to 0.5 or 0.8;
b) angle gam was chosen equal to 10, 30, 45, 60 or 80 degrees;
c) t°e watio oe praseure ie the crack to the maximum compaessive strees e00 varied m the range ftom 1.2 to 10 . 0;
Fig. 3. Dependence of d„„(eoo) for different angles: a) qdp = 0.8 and b) qdp = 0.5
d) axcrit = 0.05; aycrit = 1
During experime nts the desired value of dmax was dete rmined.
Fig. 3 shows the calculated dependence of the maximum distance dmat on the e00 when qdp value equal to 0.8 and 0.5 respectively.
Presented graphs show that with increasing pressure in the cracks there is a growth in the distance dmax. According to the dqta obtained this growth is faster when the snple gam getting closer to the direction oP o ne of the principal stresses. The total shependence of amax(e00) i s given by dsax ^f4xlnfe00)+s3 with high degree of approximation and correlation coefficient R2> 0.99.
When the relation qdp is decreased, the maximum distance between wells dmax becomes smaller. Another feature is that in the nod-uniersi^ compreasion field this spacing; is less than ia the hhdrrstatic (or umform) condition. With deceessing angle gam from 80° to some ghmcrlr the value mf dmax is reduced, and a further decrease leads to gam value growth. Fig. 4 shows the dependence of the gamcri, on qdp, based on the numerical experiments results. The value of gamcrit increases linearly from 36° at qdp? equals 0.5 to 44° with qdp equals 0.9; and approximate -value for the critical angte is 45° when the stress state of coal measure rocks can be described as hydrostatic.
Overall, these results are consistent with the generally accepted linear fracture mechanics, which confirms the correctness of the algorithm and software program.
Fig. 5 shows a plot of dmax dependent on e00 at different qdp. The angle between the direction of the maximum principal stress and the fracture line is either equal to 1°, or close to 90°.
In this case, when the direction of cracks propagation is close to the direction of one of the principal stresses, the formation of a combined fracture system is performed at the greatest distances between fracturing wells. The maximum value of dmax is observed during the cracks propagation in the less energetically favourable direction (gam equal to 89°) which is corresponds to a minimum principal stress.
The results show a strong influence of the hydraulic fracturing pressure on the crack propagation stability in a given direction. At small pressures, which can be modeled by low pumping rates of fluid,
46
bû 01
■n 42
Ê 3S
<T) M
M 30
R3 = 19qdp + 26,6 0,9977 ,
0.4
e,i
ow
0,7
d| so
o.s
0,9
Fig. 4. The gamcri, value for different qdp
k
* e.im-r
• lyim-ftr y
r
/
ï sa
■ jîsm-K* ____j
_ *
9
a)
^ prevnjfe ratio
b)
prifïïury rjliu
Fig. 5. Dependence of dmfX(e00) far principal stress fire ctions: a) qdp = 0.8 and b) qpp = 0.5
the value of dmax is approximately three times smaller than at high pressures and that is the evidence in favour of the pulsed nature of the fracturing. In order to achieve a high rate of fluid flow it is necessary to use more efficient pumps or special high-chpahty hydropneumatic accumulators automaticalry switching on at time of fracturc formation.
Summary
1. The aagorithm for calculating the trafeetory of the synchronous locgitudinal hydraulic fnacturing carrier ouS in parallef wells and research software based on thia algorithm wece developed.
2. The influence of coal rrcks stress statr to the maximum possihle dirtanco Oetween wells providing fractures linkage was determined. Also it is emphasized the desirability of pulsed fracturing with high rates of fluid flow in to the growing crack.
3. The obtained results help to design hydraulic fracture treatments if coal measure rocks in otder tr solve the proffem of undergaound methane profacfion or intensification oU beds degassing.
!
Acknowledgements
This study was supported by the Ministry of education and science of Russian Federation and partially funded by grants from Russian foundation for basic research (projects №11-05-00390 and 12-05-31358)
References
[1] Savruk M.P. Two dimensional problems of elasticity for bodies with cracks. K.: Naukova Dumka, 1981. P. 323.
[2] Panasyuk V.V. The limiting equilibrium of brittle bodies with cracks. K.: Naukova Dumka, 1968. P. 246.
[3] Alekseeva T.E., Martynyuk P.A. // Journal of Mining Science. 1991. Vol. 27. N 2. P. 15.
[4] Martynyuk P.A., Sher E.N. // Journal of Mining Science. 1996. Vol. 32. N 6, P. 19.
[5] Martynyuk P.A. // Journal of Mining Science. 2002. Vol. 38. N 4. P. 53.
Численные исследования процесса гидроразрыва угольного пласта для эффективной дегазации метана
А. В. Патутин, П. А. Мартынюк, С. В. Сердюков
Институт горного дела СО РАН Россия 630091, Новосибирск, Красный пр. 54
При проведении горных работ необходима эффективная дегазация угольных пластов. В статье представлен алгоритм, использующийся для расчета траектории трещины, которая формируется в результате проведения операции гидроразрыва в нескольких параллельных скважинах. Результаты численного анализа показывают, что направление распространения трещины зависит от приложенного к ней давления и напряженного состояния массива. Оценено максимальное расстояние между скважинами гидроразрыва, при котором происходит их сбойка. Полученные результаты могут быть использованы при проектировании гидроразрыва в угольных пластах с целью добычи метана либо для интенсификации дегазации углепородного массива.
Ключевые слова: численное моделирование, дегазация метана, управляемый гидроразрыв, метан угольных пластов.
