Arch. Min. Sci., Vol. 53 (2008), No 1, p. 97–114

Arch. Min. Sci., Vol. 53 (2008), No 1, p. 97–114

Arch. Min. Sci., Vol. 53 (2008), No 1, p. 97–114
97
STANISŁAW PRUSEK*, ELIGIUSZ JĘDRZEJEC*
ADJUSTMENT OF THE BUDRYK-KNOTHE THEORY TO FORECASTING
DEFORMATIONS OF GATEROADS
ADAPTACJA TEORII BUDRYKA-KNOTHEGO DO PROGNOZOWANIA DEFORMACJI
CHODNIKÓW PRZYŚCIANOWYCH
This paper presents a attempt to use the Budryk-Knothe theory for forecasting deformations of
gateroads subjected to direct influence of the front abutment pressure. The work was carried out on the
basis of underground measurements of vertical and horizontal convergence as well as floor heave in gate
roads. Based on a detailed analysis of these measurements’ results and statistical calculations there were
derived equations that allow determining magnitudes of deformations occurring in gateroads depending
on their location in relation to the extraction front that influences them.
Keywords: excavation, mining, gateroad, convergence, forecast
W artykule przedstawiono próbę wykorzystania teorii Budryka-Knothego do prognozowania deformacji chodników narażonych na bezpośrednie wpływy ciśnień eksploatacyjnych. Podstawą prac były
rezultaty pomiarów dołowych zaciskania pionowego, poziomego, jak również wypiętrzania skał spągowych
w chodnikach przyścianowych. W wyniku wstępnych analiz stwierdzono znaczne podobieństwo pomiędzy
deformacjami zmierzonymi w chodnikach w warunkach dołowych, a przebiegami ruchów górotworu, które
można otrzymać z teorii Budryka-Knothego. Z tego powodu podjęto próbę wykorzystania tej teorii do
predykcji deformacji chodników przyścianowych z uwagi na oddziaływanie czynnego frontu ścianowego,
czyniąc przy tym następujące założenia:
osiadanie skał stropowych w chodnikach opisuje się przy pewnym parametrze rozproszenia
wpływów charakterystycznym dla rejonu stropu chodnika,
wypiętrzenie skał spągowych opisane jest takim samym wzorem jak osiadanie stropu, lecz z innym
współczynnikiem eksploatacyjnym i innym parametrem rozproszenia wpływów, charakterystycznym dla obszaru spągu chodnika,
zaciskanie poziome wyznaczane jest jako iloczyn szerokości chodnika i poziomego odkształcenia
panującego w górotworze w rejonie pomiaru, które można opisać przy pomocy teorii BudrykaKnothego, stosując odpowiednie parametry charakterystyczne dla tego obszaru chodnika.
Po przyjęciu powyższych założeń opracowano funkcje aproksymujące wyniki pomiarów, wyznaczając
nieznane ich parametry w oparciu o rezultaty badań dołowych. Wytypowano grupę parametrów
charakteryzujących poszczególne badane wyrobiska przyścianowe, to jest długość frontu ściany, głębokość
*
CENTRAL MINING INSTITUTE, PL. GWARKÓW, 40-160 KATOWICE, POLAND
98
eksploatacji, wytrzymałość na ściskanie skał stropowych, spągowych, i węgla oraz typ i wymiary
obudowy chodnikowej. Następnie przeprowadzono analizę regresji poszukując zależności parametrów
aproksymacji z parametrami charakteryzującymi badane wyrobiska, w postaci zależności liniowych.
Uzyskane równania regresji liniowej wbudowano w model predykcji zaciskania pionowego i poziomego
oraz wypiętrzania spągu .
Słowa kluczowe: eksploatacja, górnictwo, chodnik, konwergencja, prognoza
1. Introduction
Gateroads play a critical role in the process of extracting hard coal seams by means of
the longwall system. The state of the workings is instrumental in rhythmical extraction
run, takes impact on the district’s ventilation, crew working conditions, transportation of
materials and gotten haulage. Therefore investigations and research work are continually
carried on in order to ensure appropriate size and geometry of the gate roads. The scientific
work is aimed, among others, at detailed getting to know the phenomena occurring in the
mine roads being under direct influence of the front abutment pressure. The most often
investigative practise is performance of measurements of changes in height and breadth
of the roads. Underground measurement results were for many researches a basis on
which they built up their analyses from which were retrieved for example methods that
allow to forecast squeeze of gateroads (e.g. Biliński, 1968; Marczak, 1996). Appropriate forecast information on the magnitude of deformations of the gateroads that may
occur during a longwall advance is a very important element at the stage of planning
extraction that influences proper choice of supports for these workings and organisation
of activities indispensable for ensuring their functionality and safety.
This paper presents a trial to use the Budryk-Knothe theory for forecasting gate
road deformations. The theory has been commonly used in assessing the influence of
active extraction work on the magnitude of surface and rock-mass deformations of for
many years. The work on the use of the theory in forecasting gate road deformations
was carried out in the framework of the statutory activity of the Central Mining Institute (GIG) (Prusek, 2005, 2006) and was based on results of underground deformation
measurements performed at ten and several gateroads in diverse geological and mining
conditions. Based on a detailed analysis of these measurements’ results and statistical
calculations there were derived equations that allow determining magnitudes of vertical
and horizontal convergence as well as floor heave that may occur in gate roads.
2. Underground investigations on gate road deformation
and its characteristics
In the work on adapting the Budryk-Knothe theory for the purposes of forecasting gate
road deformations results of measurements performed in twelve gateroads were used.
99
The results were gained by cyclical carrying out the following measurements: vertical
convergence, horizontal convergence, and floor heave. The measurements were performed
according to a proprietary investigation methodology worked out by the Central Mining Institute (Biliński, 1968, 1996), which consisted in setting up measurement stations
in the gateroads before an extraction front. At each such measurement station all the
aforementioned measurements were cyclically carried out along with the measurement
of distance between the extraction front and the measurement station.
The roads at which the measurements were performed were located in diverse geological conditions and represented various mining-technical solutions including different
support sizes (Prusek, 2005, 2006). Basic technical specifications that characterise the
gateroads are presented in Table 2.1.
TABLE 2.1
Basic technical specifications that characterise the gateroads in which
the investigations were performed
TABLICA 2.1
Dane charakteryzujące chodniki przyścianowe, w których prowadzono badania dołowe
ich deformacji
No.
1
2
3
4
5
6
7
8
9
10
11
12
Designation of
colliery and gate
road under
investigation
K1 – 4
K2 – 424
K2 – 426
K3 – 2502
K4 – 321
K5 – 762
K6 – 2E
K2 – 316
K2 – 422
K7 – 221
K7 – 3
K1 – 5
L, m H, m g, m
Rcst, Rcsp, Rcw,
MPa MPa MPa
sch,
m
hch, Size of support
m
set
345
250
250
180
250
210
205
250
250
220
200
230
32.2
21.6
26.1
10.0
36.6
11.4
20.4
16.0
21.6
28.0
31.0
32.2
5.5
4.7
4.7
5
4.7
5
5
5
4.7
5.5
5
5.5
3.8
3.3
3.3
3.5
3.3
3.5
3.5
3.5
3.3
3.8
3.5
3.8
990
500
500
490
695
390
550
340
500
270
710
970
2.4
2.0
2.0
3.0
1.5
3.0
3.0
2.6
2.0
3.3
2.0
2.2
39.5
20.0
31.4
32.2
15.7
23.5
16.0
16.0
20.0
14.0
24.0
39.5
14.7
14.8
17.4
32.2
14.9
22.6
16.0
15.0
14.8
14.0
15.7
14.7
ŁP10/V25/A
ŁP8/V25/A
ŁP8/V25/A
ŁP9/V25/A
ŁP8/V25/A
ŁP9/V25/A
ŁP9/V29/A
ŁP9/V25/A
ŁP8/V25/A
ŁP10/V29/A
ŁP9/V25/A
ŁP10/V29/A
Legend: K1-K7 – symbols of collieries and numbers of gateroads in which the underground
investigations were performed, L – longwall length, H – extraction depth, g – seam thickness,
Rcst – compressive strength of roof rocks, Rcsp – compressive strength of floor rocks, Rcw – compressive strength of coal, sch – road breadth, hch – road height.
Oznaczenia w tablicy: K1-K7 – kopalnie, w których prowadzono badania dołowe wraz z numerami chodników przyścianowych, L – długość ściany, H – głębokość eksploatacji, g – grubość
wybranego pokładu, Rcst – wytrzymałość skał stropowych na ściskanie, Rcsp – wytrzymałość
skał spągowych na ściskanie, Rcw – wytrzymałość węgla na ściskanie, sch – szerokość chodnika,
hch – wysokość chodnika.
100
The magnitudes vertical and horizontal convergences observed in the gateroads
were diverse which surely resulted from different extraction parameters, specificity of
geological conditions, extraction parameters and support sort applied in the roads under
investigation. The vertical convergence in the roads ranged from 326 mm to 1651 mm,
whereas horizontal convergence was included in a range between 100 mm and 742 mm.
As to the vertical convergence a dominant share in its magnitude in majority of cases had
floor heave with only a relatively small share of roof settling (Prusek, 2005, 2006).
3. Proposal of approximation model
Figure 3.1 shows a system of co-ordinates XYZ whose plane XY is bound up with
the coal seam in which a rectangular wall with a length of L has been assumed. An extraction front of longwalls advances in parallel to one of the rectangle sides beginning
Y
A2
A3
A(x,y,z)
-L/2
A
A4
A1
A diagram showing
location of points at road
cross section
X
-L/2
Xp
X
Fig. 3.1. Schematic presentation of spatial situation of longwall extraction in the system of coordinates
XY. Initial location and instantaneous location of the extraction front are designated with (Xp) and (X)
respectively. Over the longwall there are shown the road and arbitrary point A(x,y,z), at which w the
rock-mass deformation parameters are to be determined. Aside one may see a diagram showing location
of measurement points at road cross-section A1, A2, A3, A4 used to determine road convergence and floor
heave (A1) as well as point A serving for forecasting horizontal convergence of a gate road
Rys. 3.1. Schematyczne położenie eksploatacji ścianowej w układzie współrzędnych XY. Zaznaczono
początkowe (Xp) oraz chwilowe (X) położenie frontu eksploatacyjnego. Nad ścianą pokazano chodnik
oraz dowolny punkt A(x,y,z), w którym określa się wskaźniki deformacji górotworu. Obok schemat
położenia punktów pomiarowych A1, A2, A3, A4 umieszczonych w przekroju poprzecznym chodnika
i służących do pomiaru zaciskania chodnika i wypiętrzenia spągu (A1) a także punktu A służącego
do prognozy poziomego zaciskania chodnika
101
from its initial location Xp. The mined seam thickness amounts to g and the extraction
coefficient is equal to a. Axis Z of the co-ordinate system is directed to the ground surface, and plane Z = 0 divides the seam into halves.
The Budryk-Knothe theory (Knothe, 1953; Litwiniszyn, 1953) determines settling of
an arbitrary chosen rock-mass point w(x,y,z) with co-ordinates x, y, z located in a zone
z > g/2 above an extracted seam, without dealing with the zone below the seam’s roof.
In practical applications when influences of several extracted seams superimpose it is,
in forecasting rock-mass deformation, oft assumed that the influences below the seam’s
roof are null. It is justified because of the inconsiderable vertical range of the influence
zone under the seam’s roof in comparison with the influence range above the seam (up
to the surface). Moreover the theory defines lowering of a rock mass point depending
on roof settling magnitude which constitutes the boundary condition for solving some
partial differential equation of parabolic type. At the place of location of this boundary
condition, i.e. at the seam roof, the value of parameter r (radius of influence spread)
(compare below) is, out of definition, equal to zero.
In the considered case of gate road deformations the measurement points were located
at the roof level and below. Therefore from the formal point of view the Budryk-Knothe
theory may not be applied here directly. Yet the form of functions existing in the description of displacements in the Budryk-Knothe theory is very similar to the floor heaves
and horizontal convergences gained from the underground measurements. It suggests the
idea of using the functions that describe sag and horizontal deformation in the BudrykKnothe theory as functions that approximate the shape of roof settling, floor heave and
horizontal convergence of a gate road.
In the gateroads there were performed measurements of floor heave wsp at point A1,
vertical convergence Z of the considered road between points A1 and A2 and horizontal
convergence M of the considered road between points A3 and A4.
The formulas describing floor heave and roof settling may be applied to forecast
vertical convergence. Because horizontal convergence is measured using points located
at the same level z, it is possible to use the following approximation to describe it
M = - sch e yy ( A)
where
sch — distance between points A3 and A4 (road width)
εyy(A) — horizontal deformation perpendicularly to the longitudinal road axis determined at point A.
The following approximation formulas were proposed to describe floor heave wsp, roof
sag wst, vertical convergence Z and horizontal convergence M in gateroads depending
on longwall extraction front location X:
wsp = C1 F1 (x1 ( x, r1 , p1 ), x2 ( x, r1 , p1 ))
(3.1)
102
wst = C2 F1 (x1 ( x, r2 , p2 ), x 2 ( x, r2 , p2 ))
(3.2)
Z = C1 F1 (x1 ( x, r1 , p1 ), x 2 ( x, r1 , p1 )) + C 2 F1 (x1 ( x, r2 , p2 ), x2 ( x, r2 , p2 ))
(3.3)
M = C F1 (x1 ( x, r , p), x 2 ( x, r , p))
(3.4)
where:
C1, C2, C — coefficients independent from location x of point A in the road and
location X of the longwall front but dependent on location y of point A
as per the formulas
C1 = a1 gF1 (h1 ,h 2 )
(3.5)
C 2 = a2 gF1 (h1 ,h 2 )
(3.6)
C=Q
g sch
F2 (h1 ,h 2 )
r
(3.7)
r1, r2, r — parameters that determine the horizontal spread of extraction impact
at the level of the road floor (point A1), road roof (point A2) and point
A, respectively
p1, p2, p — parameters that determine the extraction boundary on the front side
(while p is positive – seeming retreat of the extraction front position
by a value of p),
Q — coefficient of road’s horizontal convergence at A-point level,
a1, a2, a — parameters (analogous to the extraction coefficients) for road floor
heave (point A1), road roof sag (point A2) and horizontal convergence
at A-point level, respectively
g — extracted thickness of seam,
sch — road breadth,
Xp — initial location of longwall front,
pch — boundary on the road side (further assumed pch = –sch), where
ξ1, ξ2, η1, η2, F1, F2 — are in formulae as below:
x1 ( x, r, p ) = p
Xp - p - x
r
1
L+ y
h1 ( y, r, pch ) = - p 2
r
x2 ( x, r, p ) = p
X - p-x
r
1
L - pch - y
h 2 ( y, r, pch ) = p 2
r
p — so-called extraction boundary on the front side,
pch — extraction boundary on the road side,
(3.8)
(3.9)
103
F1 (u ,v) =
1
[erf(v) - erf(u )]
2
F2 (u , v) = v exp(-v 2 ) - u exp(-u 2 )
erf (u ) =
2
u
e
p ò
-t2
dt
(3.10)
(3.11)
(3.12)
0
with the natural extension of the definition of function erf over negative numbers in
form
erf(-u ) = - erf(u)
(3.12a)
This description has been based on the formulae determining the coefficients of rock
mass deformations caused by a longwall extraction front (Jędrzejec, 1999).
3.1. Methodology of adjusting parameters on the basis
of measurements
The method of smallest squares was applied to each of the aforementioned coefficients that consisted in such a selection of the parameters’ set that sum S of squares of
deviations of measurement results from approximation values would be minimum (Smin)
which gives for each of the three levels a system of nonlinear equations of parameters
C, r and p.
A measure of quality of the coefficient adjustment is
s=
S min
N -3
(3.13)
where N is a number of measurements used for determining the parameters.
Because vertical convergence of a mine road is a sum of floor heave and roof sag
(3.3) and parameters C1, r1 and p1 of approximation (3.1) the floor heave is determined
from a separate their adjustment to observations of wsp, thus parameters C2, r2 and p2
for the approximation of roof sag may be determined in two ways:
• From adjustment to observations wst = Z – wsp,
• From adjustment to observations Z while assuming parameters C1, r1, p1 as
known.
Determining of the parameters was carried out by means of a computer programme
specially developed for this purpose.
104
3.2. Results of calculations and analysis of them
The calculation results are presented in Table 3.1. As parameters that describe the
road roof sag (C2, r2 and p2) gained from the adjustment to observations wst = Z – wsp
and from the adjustment to observations of Z (with already adjusted parameters C1,
r1, p1) are not essentially different so arithmetic mean values of them was assumed as
representative and placed in Table 3.2. A list of parameters C1, C2, C, r1, r2, r, p1, p2
determined in this way is presented in Table 3.2. Some exemplary curves of adjustment
are shown in Figure 3.2.
Fig. 3.2. Curves of adjustment for floor heave (left) and for vertical convergence
Rys. 3.2. Wykresy dopasowania dla wypiętrzeń spągu (z lewej) i zaciskania pionowego
TABLE 3.1
List of results of adjustment of the model’s parameters to individual
measurement series
TABLICA 3.1.
Zestawienie wyników dopasowania parametrów modelu
do poszczególnych serii pomiarowych
No.
Measurement
series
1
2
3
4
5
6
7
K1 – 4
floor heave
roof sag
vertical convergence
horizontal convergence
316
1200
1200
364
102
35
34
63
4
–8
–8
–2
7
28
29
11
1
Sort of deformation
C, mm
r, m
p, m
σ, mm
105
1
2
2
K2 – 424
3
K2 – C-426
4
K3 – 2502
5
K4 – 321
6
K5 – 762
7
K6 – 2E
8
K2 – 316
9
K2 – 422
10
K7 – 221
11
K7 – 3
12
K1 – 5
3
4
5
6
7
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
floor heave
roof sag
vertical convergence
horizontal convergence
1084
529
530
397
928
591
622
928
306
292
293
117
448
300
352
475
90
229
229
95
212
91
90
196
351
124
123
284
1109
322
354
437
185
468
461
208
760
376
381
663
372
666
666
382
145
26
19
106
97
26
15
97
44
78
78
54
26
51
133
97
62
38
37
43
117
109
105
97
91
10
10
135
91
67
80
79
55
14
14
63
48
37
34
32
263
41
41
69
36
6
5
2
10
0
1
10
6
–5
–3
11
8
19
48
24
27
28
27
13
34
38
37
38
42
12
8
47
13
17
34
16
40
4
–3
42
9
–10
–11
0
80
15
15
9
27
42
42
19
47
27
68
47
28
14
37
9
34
5
12
35
4
7
9
4
4
3
5
9
19
18
21
14
23
8
36
11
7
29
44
6
31
14
33
10
7
11
27
14
106
TABLE 3.2
List of the approximation model parameters C1, C2, C, r1, r2, r, p1, p2, p
TABLICA 3.2
Zestawienie parametrów C1, C2, C, r1, r2, r, p1, p2, p modelu aproksymacyjnego
No.
1
2
3
4
5
6
7
8
9
10
11
12
Measurement
series
K1 – 4
K2 – 424
K2 – 426
K3 – 2502
K4 – 321
K5 – 762
K6 – 2E
K2 – 316
K2 – 422
K7 – 221
K7 – 3
K1 – 5
C1
mm
316
1084
928
306
448
90
212
351
1109
185
760
372
C2
mm
1200
529
606
292
326
229
91
123
338
465
378
666
C
mm
364
397
928
117
475
95
196
284
437
208
663
382
r1
m
102
145
97
44
26
62
117
91
91
55
48
263
r2
m
34
23
20
78
92
38
107
10
74
14
36
41
r
m
63
106
97
54
97
43
97
135
79
63
32
69
p1
m
4
36
10
6
8
27
34
42
13
40
9
80
p2
m
–8
5
1
–4
33
28
38
10
25
1
–10
15
p
m
–2
2
10
11
24
13
38
47
16
42
0
9
Assuming the extraction boundary on the road side pch = –sch, where sch is a road
breadth, there were determined parameters a1, a2 that serve as extraction coefficients
for floor heave and roof sag
a1 =
C1
gF1 (h1 , h 2 )
(3.14)
a2 =
C2
gF1 (h1, h 2 )
(3.15)
where formulas (3.4) and (3.5) were applied with co-ordinates of measurement points
L s
y = + ch .
2
2
As for horizontal convergence – from some considerations follows that
M=Q
g
sch F1 (x1 , x 2 ) F2 (h1 ,h 2 ) = CF1 (x 1 , x2 )
r
(3.16)
where Q is a dimensionless coefficient which is awaited to be approximately constant
in given geological conditions. Knowing r and C, that have already been determined, it
is possible to derive the following formula of Q from equation (3.16):
Q=
rC
g sch F2 (h1 , h 2 )
(3.17)
107
Further, applying some analogies to the Budryk-Knothe theory the values of the
following proportions were also investigated
t1 =
H
r1
t2 =
H
r2
t=
H
r
(3.18)
k1 =
p1
H
k2 =
p2
H
k=
p
H
(3.19)
Values of calculated coefficients a1, a2, Q, t1, t2, t, k1, k2, k are presented in Table 3.3.
TABLE 3.3
Values of a1, a2, Q, t1, t2, t, k1, k2, k calculated based on the adjusted parameters
C1, C2, C, r1, r2, r, p1, p2, p
TABLICA 3.3
Wartości obliczonych wielkości a1, a2, Q, t1, t2, t, k1, k2, k na podstawie
dopasowanych parametrów C1, C2, C, r1, r2, r, p1, p2, p
Road
designation
1 K1 – 4
2 K2 – 424
3 K2 – 426
4 K3 – 2502
5 K4 – 321
6 K5 – 762
7 K6 – 2E
8 K2 – 316
9 K2 – 422
10 K7 – 221
11 K7 – 3
12 K1 – 5
No.
a1
0.25
1.05
0.89
0.18
0.51
0.06
0.14
0.26
1.05
0.10
0.69
0.34
a2
Q
t1
0.86 22.93 9.73
0.44 113.77 3.44
0.49 222.05 5.17
0.18 5.24 11.11
0.41 153.05 26.79
0.13 2.67 6.28
0.06 27.99 4.70
0.06 88.54 3.72
0.32 70.50 5.47
0.20 9.41 4.87
0.33 15.58 14.80
0.53 30.79 3.69
t2
t
28.89 15.60
22.19 4.72
24.52 5.17
6.27 9.01
7.58 7.15
10.38 9.05
5.12 5.65
33.95 2.52
6.78 6.30
19.23 4.28
19.80 22.22
23.81 14.10
k1
k2
k
0.0039 –0.0080 –0.0016
0.0712 0.0106 0.0034
0.0199 0.0011 0.0199
0.0132 –0.0077 0.0233
0.0117 0.0478 0.0344
0.0687 0.0710 0.0327
0.0616 0.0683 0.0684
0.1240 0.0294 0.1374
0.0260 0.0504 0.0312
0.1467 0.0024 0.1573
0.0120 –0.0144 –0.0002
0.0825 0.0153 0.0090
Extracted seam thickness, road height and breadth are so slightly diverse that these
do statistically not depart from the constants (diversity coefficients are inconsiderable).
Therefore investigations to find out dependence of the determined parameters on these
data were abandoned.
Coefficients of correlation between the determined parameters and L, H, Rcst, Rcsp,
Rcw, ŁP (support set type) were calculated and results have been placed in Table 3.4.
The large values of the following correlation coefficients are conspicuous:
• Coefficients of correlation between parameters a1 and Q and road support
set type
• Coefficients of correlation between parameters a2 and t and depth H.
108
TABLE 3.4
Coefficients of correlation between each of parameters a1, a2, Q, t1, t2, t, k1, k2, k and
data L, H, Rcst, Rcsp, Rcw, ŁP
TABLICA 3.4
Współczynniki korelacji pomiędzy każdym z parametrów a1, a2, Q, t1, t2, t, k1, k2, k
a danymi L, H, Rcst, Rcsp, Rcw, ŁP
L
H
Rcst
Rcsp
Rcw
ŁP
a1
a2
Q
t1
t2
0.1862
0.7655 0.2811 0.0175 0.5305
0.0482 0.7739 –0.0819 0.2964 0.1812
0.2253 0.6574
0.2568 0.4482 0.2509
–0.0361 0.6815 –0.1093 –0.1041 0.2964
–0.3152 –0.3129 –0.2757 0.0768 –0.4283
–0.7257 0.1353 –0.7166 –0.2412 0.3293
t
0.0468
0.7075
0.4029
0.5553
0.0153
0.3881
k1
–0.1922
–0.5173
–0.1982
–0.4353
–0.2785
0.3830
k2
–0.1606
–0.2641
–0.2862
–0.4906
–0.0847
–0.2941
k
–0.1732
–0.6640
–0.1948
–0.6331
–0.1829
0.2589
3 . 3 . P r e d i c t i o n o f t h e a p p r o x i m a t i o n m o d e l ’s p a r a m e t e r s
From an analysis of the data in Table 3.4 follows that it is possible to find the following relations:
1. a1(ŁP)
2. a2(L,H,Rcst,Rcsp)
3. Q(ŁP)
4. t1(Rcst)
5. t2(L,Rcw)
6. t(H,Rcst,Rcsp)
7. k1(H,Rcsp)
8. k2(Rcsp)
9. k(H,Rcsp)
An analysis of multidimensional linier regression, in which dependence coefficients
are determined by means of the minimum square method, was applied to determine the
selected dependencies in linier form. The following results have been received:
ì0.87
ï
a1 = í0.26
ï0.23
î
for £P 8
for £P9
for £P10
a 2 = –0.80 + 0.00245 L + 0.000077 H + 0.01153 Rcstr + 0.01343 Rcsp
ì139.8
ï
Q = í 28.2
ï 21.0
î
for £P 8
for £P9
for £P10
(3.20)
(3.21)
(3.22)
109
t1 = –0.188 + 0.3554 Rcstr
(3.23)
t 2 = 0.832 + 0.0992 L – 0.4018 Rcw
(3.24)
t = –1.790 + 0.01655 H – 0.01829 Rcstr + 0.06263 Rcsp
(3.25)
k 1 = 0.121 – 0.000087 H – 0.00073 Rc sp
(3.26)
k 2 = 0.062 – 0.00162 Rcsp
(3.27)
k = 0.146 – 0.0001 H – 0.00188 Rcsp
(3.28)
Values of parameters a1, a2, Q, t1, t2, t, k1, k2, k calculated with the help of linear
regression formulas (3.20)…( 3.28) for the real data from Table 2.1 are presented in
Table 3.5.
TABLE 3.5
Prediction of parameters a1, a2, Q, t1, t2, t, k1, k2 and k by means of formulae (3.20)…(3.28)
for the real data from Table 1
TABLICA 3.5
Predykcja parametrów a1, a2, Q, t1, t2, t, k1, k2, k przy pomocy wzorów (3.20)…(3.28)
dla danych rzeczywistych z tablicy 1
Road
designation
1 K1 – 4
2 K2 – 424
3 K2 – 426
4 K3 – 2502
5 K4 – 321
6 K5 – 762
7 K6 – 2E
8 K2 – 316
9 K2 – 422
10 K7 – 221
11 K7 – 3
12 K1 – 5
No.
a1
a2
Q
t1
t2
t
0.23
0.87
0.87
0.26
0.87
0.26
0.26
0.26
0.87
0.23
0.26
0.23
1.02
0.37
0.57
0.23
0.50
0.19
0.19
0.24
0.37
0.27
0.42
0.74
21.00
139.80
139.80
28.20
139.80
28.20
28.20
28.20
139.80
21.00
28.20
21.00
11.26
7.49
9.09
3.37
12.82
3.86
7.06
5.50
7.49
9.76
10.83
11.26
36.51
27.10
27.36
21.88
27.11
23.91
22.76
27.12
27.10
24.04
22.23
25.11
16.48
7.34
7.97
8.15
10.03
5.93
7.94
4.55
7.34
3.04
10.90
16.15
k1
k2
k
0.0060 –0.0020 –0.0273
0.0629 0.0296 0.0584
0.0546 0.0111 0.0370
0.0549 0.0098 0.0365
0.0491 0.0366 0.0470
0.0699 0.0239 0.0628
0.0615 0.0361 0.0609
0.0797 0.0361 0.0819
0.0629 0.0296 0.0584
0.0873 0.0393 0.0927
0.0417 0.0231 0.0299
0.0078 –0.0020 –0.0253
It worthwhile noticing that the parameters calculated by means of the formulae show
good agreement with those gained intermediately from measurements, in spite of the
not numerous data series – only 12 cases.
Coefficients a1 and a2 have similar values as the extraction coefficients in the BudrykKnothe model. Values of parameters t that conceptually correspond to parameters tgβ
110
are of similar value order – yet, one should keep in mind here that the description refers
to the direct seam vicinity. The order of values of parameters k is also the same like that
used in practical applications of the Budryk-Knothe model.
4. An example of forecasting gateroad deformations by means of
the approximation model
Lineal regression formulae (3.20)…(3.28) allow to calculate parameters a1, a2, Q,
t1, t2, t, k1, k2 and k when data characterising extraction L, H, Rcst, Rcsp, Rcw, and road
support set type ŁP are known. One should keep in mind that these make sense only in
the data range from which these were derived and any trial to go beyond the extrapolation
range may (but does not have to) lead to a loss of their physical sense. Next, one should
calculate parameters r1, r2, r, p1, p2, p using formulas (3.18) and (3.19) and parameters
C1, C2, C using formulas (3.14), (3.15)...(3.17). Finally the forecasted floor heave, roof
sag , vertical convergence and horizontal convergence for an arbitrarily selected longwall
front location X may be calculated from formulas (3.1)…(3.4).
In order to make the calculations easier A new computer programme Konwergencja
has been developed (Figure 4.1) being a form of specialised calculator that calculates
the model parameters for assumed data: L, H, Rcst, Rcsp, Rcw and ŁP.
The computer programme also plots curves of forecasted indices (Figure 4.2)
5. Summary
The Budryk-Knothe theory is used in many cases for determining ground surface
deformations caused by mining operations. This publication presents results of work
performed to check possibilities of adapting the theory for the purposes of forecasting
rock mass movements around the gateroads that are subjected to a number of mine
roadways located in diverse geologic conditions were analysed. The measurements of
deformations such as vertical convergence, horizontal convergence, floor heave and
roof sag depending on the longwall front location were measured. Based on performed
calculations some approximation formulae have been proposed to forecast of gate road
deformations. Comparison of approximation curves drawn up based on calculations with
results of the underground measurements showed a good agreement.
It is indispensable to verify the gained dependences based on results of new underground measurements in gate roads. Moreover it would be necessary to try and adjust
approximation dependences other than the linear ones which might lead to improved
accuracy of such forecasts.
111
a)
Legend: Zaciskania chodnika – Road convergences; Plik – File; Pokaż raport – Show report; Wykresy
– Curves; Pomoc – Help; Ściana – Longwall; wybieg – advancement; Chodnik – Road; odrzwia – support
set; szerokość – breadth; skał stropowych – of roof rocks; skał spągowych – of floor rocks; pokładu – of
seam; Oblicz - Calculate
b)
Legend: Zaciskania chodnika – Road convergences; Plik – File; Pokaż raport – Show report; Wykresy
– Curves; Pomoc – Help; Dane – Data; Wyniki – Results; Maksymalne zaciskania – Maximum convergences; pionowe – vertical; poziome – horizontal.
Fig. 4.1. Graphical Interface of the programme Konwergencja a) – Data input, b) – Results
Rys. 4.1. Graficzny interfejs programu. a) – wprowadzanie danych, b) – wyniki
112
a) Maximum vertical convergence = 1410 mm
Legend: Wykresy – Curves; Rysuj wykres zaciskań pionowych – Draw a vertical convergence curve
b) Maximum floor heave = 946 mm
Legend: Wykresy – Curves; Rysuj wykres wypiętrzeń spągu – Draw a floor heave curve
113
c) Maximum horizontal convergence = 1287 mm
Legend: Wykresy – Curves; Rysuj wykres zaciskań poziomych – Draw a horizontal convergence curve
Fig. 4.2. Exemplary course of convergence curves:
a) vertical convergence, b) floor heave, c) horizontal convergence
Rys. 4.2. Przykładowy przebieg krzywych zaciskania:
a) zaciskanie pionowe, b) wypiętrzanie spągu, c) zaciskanie poziome
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Received: 06 August 2007