The document discusses prediction of the height of the destressed zone above the mined panel roof in longwall coal mining. It summarizes existing views on estimating the height of the caving and fracturing zones induced by longwall mining based on literature. The height of the destressed zone, which includes the caving and fracturing zones, is estimated to range from 6.5 to 24 times the coal seam extraction height in the short term, and 11.5 to 46.5 times in the long term.

1. Prediction of the height of destressed zone above the mined panel roof in longwall
coal mining
Abbas Majdi a,
⁎, Ferri P. Hassani b
, Mehdi Yousef Nasiri c
a
School of Mining Engineering, College of Engineering, University of Tehran, Tehran, Iran
b
Department of Mining and Materials Engineering, Faculty of Engineering, McGill University, Montreal, Canada
c
Khavar Tunnel Consulting Engineering Inc., Tehran, Iran
a b s t r a c ta r t i c l e i n f o
Article history:
Received 27 January 2012
Received in revised form 7 April 2012
Accepted 7 April 2012
Available online 18 April 2012
Keywords:
Longwall mining
Height of destressed zone
Mathematical model
Caving zone
Fracturing zone
Longwall mining is one of the most widely used underground mining methods most suitable in relatively flat-
lying, thick, and uniform coalbeds. Due to extraction of the coal seam, the panel roof strata above the mined
zone will be destressed and then the roof loads will be redistributed and transferred to the front abutment
and neighboring solid sections with higher load bearing capacity where the adjacent access tunnels and
the corresponding barrier pillars are located. The height of destressed zone (HDZ), in this paper is taken as
equivalent to the combined height of the caving and fracturing zones above the mined panel roof induced
due to longwall mining. The height of destressed zone plays a vital role in accurate determination of the
amount of loads being transferred towards front abutment and panel sides. The paper describes the mecha-
nism of development of the height of this zone. Finally, five new simple, yet conclusive, mathematical ap-
proaches to estimate the height of destressed zone are presented. The results of the methods proposed are
compared with each other and with the comparable methods. The methods proposed are further compared
and verified with in-situ measurements reported in the literatures. The comparative results confirm the
agreement that exist among the methods and those with the in-situ measurements as well. Finally the
methods have shown that, in short term, the height of destressed zone ranges from 6.5 to 24 times the extrac-
tion coal seam height; while, in long term, the height of destressed zone ranges from 11.5 to 46.5 times the
extraction coal seam height. Therefore, beyond this height the overburden pressure will be transferred to-
wards the front abutment, the adjacent access tunnels, the intervening barrier pillars as well as the panel
rib-sides.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Longwall mining is one of the most widely used methods in un-
derground coal extraction. One of the most important advantages of
this method is the automated form of underground coal mining char-
acterized by high efficiency. The main objective of coal mining is to
economically extract coal and safely remove them from the ground.
Hence, the mining efficiency solely depends on the coal seams overall
conditions. In this method, due to the extraction of coal seams within
a considerable panel width, after advancing the hydraulic jacks or
powered roof supports, the immediate roof of the mined panel is
unsupported and hence is allowed to collapse and cave in some dis-
tance behind the hydraulic jacks or in the goaf area. The downward
movement of the roof strata then gradually extends upwards and
will cause the disturbed roof strata to become destressed. Thus, the
overburden pressure above the destressed zone will be redirected
towards the front abutment and the two adjacent neighboring solid
sections where the gate roads (access tunnels serving the coal face),
the intervening barrier pillar and the adjacent un-mined solid sec-
tions are located. The upward extension of destressed zone (HDZ) de-
pends on a variety of parameters such as: depth, thickness of the
extracted coal seams, panel width, the number and the relative thick-
ness and the strength of the panel roof strata and the corresponding
coefficient of expansion.
Nowadays, the prime concerns of many coal mining researchers
are to find the appropriate approaches to appraise the panel roof stra-
ta behavior during and after panel extraction. Professional mining en-
gineers who are dealing with the longwall mining design need to
effectively determine the height of destressed zone above the panel
roof that induced due to longwall mining. The height of destressed
zone (HDZ), in this paper, is considered to be equivalent to the com-
bined height of the caving and the fracturing zones that exist above
the mined panel roof. To suitably evaluate the amount of transferra-
ble loads to the adjacent tunnels as well as the intervening barrier pil-
lars, the height of destressed zone must be estimated accordingly
which is the main objective of this paper.
International Journal of Coal Geology 98 (2012) 62–72
⁎ Corresponding author. Tel.: +98 514 398 2215.
E-mail addresses: amajdi@ut.ac.ir (A. Majdi), ferri.hassani@mcgill.ca (F.P. Hassani).
0166-5162/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.coal.2012.04.005
Contents lists available at SciVerse ScienceDirect
International Journal of Coal Geology
journal homepage: www.elsevier.com/locate/ijcoalgeo

2. 2. Existing views of mined panel roof caving and fracturing appraisal
Longwall mining is a highly productive underground coal mining
technique whose basic principles have been traced to the latter part
of the 17th century to Shropshire and other counties in England
(Anon, 1995). When a longwall panel is extracted then the immediate
roof strata are allowed to move downward. Due to the roof strata's
downward movements the original natural in-situ stress regime and
the hydraulic conductivity will be changed. Hence the roof strata
will collapse and fall into the extracted panel space. Depending up
on the volume expansion of the fractured rocks, the movements
will gradually influence the rock layers above the immediate roof
strata. The roof strata will behave differently, depending upon many
factors including: strength, thickness and the number of roof rock
layers and the thickness of the overburden in all, in one hand, and
the extracted coal seam height, the panel width and the panel length,
on the other hand. The behavior of the panel roof strata and the pro-
cess of the gradual upward movement have been of prime concern
and hence investigated by many researchers to account for the ob-
served movement as well. The following outlines various literatures,
in particular, those by Chekan and Listak (1993); Chuen (1979);
Denkhausi (1964); Dinsdale (1935); Eavenson (1923); Hasenfus et
al. (1998); Kenny (1969); Luo (1997); National Coal Board (NCB)
(1975); Palchik (1989); Peng (1992); Peng and Chiang (1984);
Richard et al. (1990); Ropski and Lama (1973); Singh and Kendorski
(1981); Styler (1984); Wiggill (1963); Zhou (1991); and many others
whose work will be referred later in this paper, wherever they apply.
Eavenson (1923) believed that the inner-burden shear failure dur-
ing multiple seam extraction in longwall mining extends to the ground
surface. Dinsdale (1935) proved that the height of destressed zone
(dome) is directly proportional to the depth of cover and to the excava-
tion width and inversely proportional to the horizontal reaction. Wiggill
(1963) depicts the concept of a composite destressed zone (dome) and
trough theory, where the movement trough does not start from the
face but from higher up at the dome boundary. According to Denkhaus
(1964), in treating the problem a distinction should be made between
sufficiently cohesive rock and insufficiently cohesive rock. For a dome
with sufficient cohesion, the maximum height is equal to 50% of the
depth of cover above excavation. If the rock is insufficiently cohesive,
then the maximum height is 63% of the depth of cover above excavation.
Kenny (1969) discussed a method of quantifying the description of the
caving zone by means of simple observations and measurements along
with the relevance of the manner of caving to roof control. Ropski and
Lama (1973) used the terms of primary and secondary regions of caving
while working coal seams by longwall mining. The latter authors showed
that regions of primary and secondary caving extend to a height of 3–3·5
times the thickness of the coal seam extracted.
Chuen (1979); National Coal Board (NCB) (1975); and Peng and
Chiang (1984) proposed empirical approaches to estimate the height
of caving and fracturing zones induced by longwall coal mining based
on the large experiences gained from the British coal mines where the
method of longwall mining was originated as well as the USA. Singh
and Kendorski (1981)have found that there are three distinct zones
(caved zone, fracture zone and continuous deformation zone) of dis-
turbance in the overburden strata in response to longwall mining.
Singh and Kendorski (1981)further suggested that the height of cav-
ing is dependent on the extracted coal seam thickness as well as the
strength and stratigraphy of the roof strata, generally extending up-
wards 3 to 6 times the thickness of the mined coalbed. Karmis et al.
(1983) have indicated that the height of the caved zone can be 12
times the underlying coal seam thickness. According to Styler (1984)
measurements of inter-burden deformations above 6 longwall faces, 5
in the United Kingdom, and 1 in the U.S.A. showed that the caving height
above a longwall face is equal to 8 to 12 times the extraction seam height,
with a zone of fractured rock extending to approximately 50 times the ex-
traction height above the seam.
Peng and Chiang (1984) differentiated four zones in the overbur-
den when a wide enough longwall coal panel is mined; Caved Zone:
the immediate roof caves irregularly, filling the void. The strata lose
their continuity and bedding planes disappear. Fractured Zone: Locat-
ed over the caved zone, its main feature is the loss of continuity and
breaking or yielding of materials but some bedding planes may re-
main. Continuous Deformation Zone: In which strata bend downwards
without breaking. Only occasional tension cracks can be observed. Soil
Zone: The surface layer, whose behavior is very site-dependent. Accord-
ing to Peng and Chiang (1984) the first two zones can be considered
plastified, that is, their forming materials yield and hence they can be
simulated by means of an elasto-plastic constitutive model. Fawcett et
al. (1986) based on their personal communication with Farmer (1980)
quoted an alternative formula which is based on the panel width rather
than the extraction thickness, predicts greater fracture heights at typical
widths between 100 and 200 m.
Palchik (1989) has found that there are three distinct zones
(caved zone, fracture zone and continuous deformation zone) of dis-
turbance in the overburden strata in response to longwall mining.
According to Palchik (1989) the thickness of the fractured zone varies
greatly from 20 to 100 times the seam thickness. The zone above the
fractured zone is the continuous deformation zone where there are
no major fractures. The extent of the zones of rock movement over
longwall mining may vary significantly.
According to Richard et al. (1990) as the mine roof is allowed to
collapse, there is an area of caving and severe fracturing which occurs
directly above the mined coal seam. This area extends upwards 30–60
times the coal seam thickness, depending on the mechanical qualities
of overlying rock strata. Strata sag above the caved area. A “zone of
continuous deformation” may extend to approximately 50 ft of the
ground surface, depending on the geomechanical properties of over-
burden layers. This zone is characterized by intermittent fracturing, bed-
ding plane separation, and some sliding of beds across each other. A near
surface zone comprising the upper 50 ft of overburden may show in-
creased fracturing or permeability because of compression and tension.
Richard et al. (1990) further quoted an interesting study included subsi-
dence monitoring, time domain reflectometry, static water level obser-
vations, and hydraulic conductivity in test wells, pre-mining and post-
mining coring, and seismic surveys conducted in Virginia coal mines
which was reported by Hasenfus et al. (1998); four distinct zones were
identified in the panel roof rock strata due to coal seam extraction:
Zone 1, the gob zone, just above the mined panel and extending upwards
4 to 6 times the mined seam thickness, Zone 2, the highly fractured zone,
above zone 1 and extending to about 30 times the seam thickness is a
transitional, highly-fractured zone with massive block-type caving and
vertical fracturing. Zone 3, the composite beam, extending from above
the highly fractured zone to within 50 ft of the surface, exhibits little ver-
tical fracturing and occasional horizontal slippages between strata. Zone
4, the surface zone, comprises the uppermost 50 ft. Strata in this zone are
susceptible to fracturing and movements.
Zhou (1991) believes that the height of caving and fracturing
zones follow a geometric function of the height of mining based on
extensive field measurements and then modified the empirical equa-
tion that was given by Peng and Chiang (1984). According to Peng
(1992) the combined height of caved and fractured zones is in gener-
al 20 to 30 times the extraction height, being bigger for hard strata
and vice versa.
Booth and Spande (1992) suggest that due to longwall mining, dif-
ferent forms of deformation at various levels within the overburden
will induce hydraulic changes accordingly. The latter authors believe
that there are three different zones of deformation above longwall
panel: the first zone is intensely fractured and tends to dewater into
the mined panel itself and extends typically up to 20–60 times the
mined thickness; the second zone is the intermediate levels of the
overburden which subside more coherently and commonly remain
a confining layer, and the third zone is the shallow, near surface
63A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

3. aquifers which deform more freely and exhibit fractured-controlled
hydrologic responses to subsidence separate from the deeper drain-
age effects.
Chekan and Listak (1993) mentioned three distinct zones of dis-
turbance in the overburden strata in response to longwall mining;
the caving zone, which is the immediate roof before it caves, ranges
in thickness from 2 to 20 times the height of excavation. The caving
zone can actually be further divided into three sub-zones: the com-
plete caving zone is a zone that the strata fall onto the mine floor.
This zone is generally 3 to 6 times the mining height depending
upon the bulking factor. The partial caving zone is a zone which ex-
tends from the top of complete caving zone, that is, from 6 to 12
times the mining height. The upper limit of the caving zone occupies
a distance of 12 to 20 times the mining height. The fracturing zone
which is located above the caved zone whose thickness ranges from
20 to 50 times the mining height. The continuous deformation zone
which is sometimes called the sagging zone is located between the
fracturing zone and the ground surface. Indeed, according to Chekan
and Listak (1993), the combined height of the caved and fracturing
zones reaches to about 50 times of the extracted coal seams. Luo
(1997) mentioned that the greatest potential for shearing occurs in
coalbeds lying within 12 times the extracted coal seam height or in
the partial caving zone where the strata have significant degree of
bending, leading to intense fracturing or displacement.
A surface subsidence and extensometer monitoring research pro-
gram undertaken above longwall panels 4 and 5 at Clarence Colliery
in Australia was reported by Mills and O'Grady (1998). The latter au-
thors aimed to investigate the behavior of the overburden strata during
longwall extraction on two faces of different widths. The monitoring in-
dicated that a dome shaped zone of large downward movement ex-
tends up into the overburden strata to a height equal to about 1.0 to
1.1 times the panel width (Mills and O'Grady, 1998).
According to Jeffrey and Zhang (2001) hydraulic fracturing was used
at Moonee Colliery in Australia, to induce caving as part of the routine
operation of this longwall mine. The weak roof coal sequence typically
caves immediately behind the supports leaving the 30 to 35 m thick con-
glomerate section bridging the 100 m wide longwall panel (Jeffrey and
Zhang, 2001).
Apart from the above remarkable referred literatures, an interesting
microseismic monitoring study was carried out at the Gordonstone or
Kestrel Coal Mine, in Australia (Kelly et al., 1998; Kelly et al., 2002;
Luo et al., 1998) to investigate the ground caving processes, the pattern
of the caving and the extent of ground failure induced by underground
longwall mining. The failure extended to a height of about 120 m above
the extracted coal seam as was observed and commenced about 15 m
ahead of the face. Kelly et al. (2002) believe that despite the impressive
growth in Australia's longwall production, many longwall mines have
experienced major geotechnical problems. The later authors further ex-
press that these types of problems are not just confined to Australia and
are common in many countries, restricting the production potential of
modern longwall faces.
Palchik (2002a) during his in-situ investigation over abandoned
subsurface coal mines in Donetsk, Ukraine reported the height of
the caved zone in porous weak rock mass over shallow, abandoned
underground workings (up to 80 m) that were detected by drilling
and linked with laboratory measurements of the physical characteris-
tics of the rock mass overlying the underground openings. Palchik
(2002a) has found that the height of the caved zone could reach 4
to 11 times the thickness of the underlying coal seam where overbur-
den rocks are weak and porous. The fractured zones caused by long-
wall mining were further studied by Palchik (2003) using vertical
wells drilled from the ground surface down to active underground
workings in the overburden of the Torezko-Snezhnyanskaya area.
The latter author found that the maximum heights of the zone of inter-
connected fractures and separate horizontal fractures may reach 19–41
and 53–92 times the thickness of the coal seam, respectively.
According to Karacan et al. (2005) the permeability of the caved
zone in the longwall gob is not easily predictable since it mostly de-
pends on the type of rock units, the extent of their fragmentation
and packing in the void space, and caving height. Therefore, it is natu-
ral to assume that this permeability value may vary widely for different
locations and for different roof rocks when they cave. Karacan et al.
(2007) believe that longwall mining is a high-volume coal extraction
method which creates large-scale disturbances around the longwall
face and in the overlying strata. They believe that occurrence of such
an extensive area of mining induced stress relief and resultant rock
damage changes the gas flow-related properties in the overlying (and
in some cases the underlying) strata, particularly the permeability.
According to Karacan and Goodman (2009); Kendorski (1993);
and Singh and Kendorski (1981) evaluated the disturbance of rock
strata resulting from mining beneath surface water and waste im-
poundments. The latter authors in their analysis, describe a caved
zone that extends from the mining level to 3–6 times the seam thick-
ness, a fractured zone that extends from the mining level to 30–58
times the seam thickness, an aquiclude zone where there is no change
in permeability that extends from 30 times the seam thickness to
50 ft below ground surface, and a surface cracking zone that is 50 ft
thick.
RafiqulIslam et al. (2009) employed a two-dimensional boundary
element modeling to analyze the deformation and failure behavior of
rock elements for a multi-slice longwall mining in Bangladesh, with
special reference to the Barapukuria coal mine. The results show
that fracture propagation would be about 240 m upward for single-
slice (height 3 m) mining extraction (up to 80 times the extracted
coal seam). From the contours of mean stress magnitudes, it is observed
that the high range of fracture propagation increased upward for multi-
slice extraction of coal (RafiqulIslam et al., 2009). Singh and Singh
(2010) illustrated a numerical modeling based integrated approach
for predicting the progressive caving behavior of strata and optimum
capacity requirement of powered support for longwall working in a
given geo-mining and strata condition.
Zhang et al. (2011) reported that in order to deal with the appar-
ent conflict between water protection and high production in under-
ground mining, an aquifer protection mining technique was trialed
in panel number 51201 in the Shangwan colliery, the Shendong
Coalfield, China. Overburden failure, water level of the unconsolidat-
ed aquifer, surface cracks, and ground subsidence were monitored in
a series of boreholes using borescope, and also by observations in
nearby water well and by surface surveys. The observation indicated
that the height of the caved zone was in the range of 5 to 6 times the
mining height, while the height of the fractured zone is about 10 to
11 times the mining height (Zhang et al., 2011).
In regards to the above referred views some existing empirical cri-
teria that were used to estimate the height of caving/fracturing zone
are also reviewed and presented in Table 1. To estimate the height
of fractured zone above the mined panel induced due to longwall
mining, in this paper, five mathematical models have been proposed,
analyzed, interpreted, and compared with each other and with the re-
sults obtained from the present comprehensive literature review
which will be outlined in the following sections.
3. New mathematical modeling for determining the height of
destressed zone
There are several approaches to determine the height of des-
tressed zone above the mined panel roof in longwall mining; in-situ
measurement, laboratory physical simulation, numerical modeling,
and mathematical modeling. Although in-situ measurements provide
helpful and reliable results, due to the inherent complexity of deter-
mining the height of the destressed zone using this approach, in par-
ticular for deep coal mining, actual implementation of this procedure
might be considered cumbersome, lengthy, and costly. Laboratory
64 A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

4. physical simulation is not a popular approach owing to discordantly
limitations for model construction length of time and cost required
to execute the experiment. The last two remaining approaches, how-
ever, are the simplest, least costly and may yield very handy and reli-
able result as well.
In this paper, the authors focus on the last method, that is, the math-
ematical approach so that with regards to mechanism of panel roof frac-
ture initiation and their extents, two general models are proposed;
Geometry-independent roof fracture and Geometry-dependent roof
fracture both of which will be outlined henceforth.
3.1. Geometry-independent roof fracture model
In this model, it is assumed that the width of destressed zone is
equal to the extracted panel width. In other words, the tensile failure
occurs at two extreme ends of the panel perpendicular to the advanc-
ing direction. On the basis of these assumptions the following two
new sub-models are proposed.
3.1.1. Sub-model-1
In this sub-model, it is assumed that all panel roof strata have equal
thickness and the average volume expansion of broken caved in materials
and fractured rocks is a coefficient of the extracted seam height (Fig. 1a).
The schematic diagram of the two access tunnels along with the corre-
sponding barrier pillars indicated by “B.P.”, before mining the coal seam
out is also shown in Fig. 1a. Next, it is presumed that the coal seam with
a thickness of (hs = ho), has been removed from the ground and the
pack supports have been erected at the panel ends as illustrated in
Fig. 1b. If the number of roof rock strata is assigned as n, then n=0 indi-
cates the coal seam layer alone. Now, if the coal seam is extracted then the
first roof layer is broken down. Theoretically, if there is no volume
expansion, then the volume of broken layer must fill the extracted
space equally. However, it is a proven fact that every solid material after
fracture expands. In this sub-model, the volume expansion factor is
taken as a coefficient of the extracted seam height and represented by
“d”. Thus, the height of broken materials is equal to (ho+d) and left
over space above the broken materials is equal to (ho−d) as shown in
Fig. 2a. Fig. 2b shows the sequential failure of the roof rock strata after
the coal seam extraction by assuming that all the strata have equal thick-
ness and each stratum is as thick as the coal seam. The sum of the frac-
tured roof strata thickness indicates the height of destressed zone, Hc,
above the extracted panel and can be expressed as follows:
Hc ¼ n þ 1ð Þhs−1=2n n−1ð Þð Þd ð1Þ
In order to prove the convergence of Eq. (1), it can be proved that
the difference between any two consecutive terms in an arithmetic
sequence is in a descending order, that is;
anþ1−an ¼ n þ 2ð Þho−1=2n n þ 1ð Þd− n þ 1ð Þho−1=2n n−1ð Þd½ Š
¼ ho−nd ð2Þ
It must be born in mind that in Eq. (2), the terms hs and n are variables
so that with an increase of n, its overall value will be decreased and hence,
the Eq. (2) represents a convergence series. For determining the limiting
value of n, it can be expressed as follows:
n ¼ hs=d ð3Þ
Table 1
Fracture/caving height prediction empirical formulas extracted from the existing
literatures.
Fracture/caving
height
prediction
formulas
Overburden
rock
conditions
Rock property
constants
Remarks References
a b c
Hc,f =100hs/
[(a.hs +b)±c]
Hard rock 0.640 16.00 8.20 Modified
from Peng
and Chiang
(1984)
Yingxin
Zhou
(1991)
Medium hard 1.433 19.00 7.20
Soft rock 1.890 32.00 4.90
Weathered 2.134 63.00 3.90
Hf =a.w−b – 0.83 11.00 – – Fawcett et
al. (1986)
Hc =(hs −Ss)/
(b−1)
If the lowest
strata sagging
– – – – Peng and
Chiang
(1984)Hc =hs/b−1 If the strata
break and fall
without any
sagging
– – –
Hf =56(hs)1/2
General
formula
– – – 0.0≤hs ≤3.5 Singh and
Kendorski
(1981)
Hf =100hs/(a.hs
+b)
Weak
overburden
3100 5.00 – – Chuen
(1979)
Medium
overburden
1.6 3.60 –
Strong
overburden
1.2 2.00 –
Hf =105 NCB
minimum
cover
– – – hs ≤1.7 NCB, PI/
1968/
8 (revised
1971)
Hf =43hs +a 32.00 – – 1.7≤hs ≤4.0
Where in equation proposed by Peng and Chiang (1984), Hc is the caving height, b is
the bulking factor, Ss is the sagging of the lowest uncaved strata and Smax is the
maximum allowable sagging. If the strata break and fall without any sagging, then,
Ss =Smax =0. In the remaining equations given in Table 1, W is the panel width, Hc,f
or Hf is the fractured height or minimum cover as appropriate, hs is the extracted thick-
ness and all quantities are in meters.
a)
b)
Ground Surface
Ground Surface
Fig. 1. Schematic cross-sectional view of a longwall with the corresponding adjacent
access tunnels and barrier pillars: a) before coal seam extraction, b) after coal seam ex-
traction and pack supports erection for the Arithmetic model.
65A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

5. Hence, the limiting equation to be used for determining the height
of destressed zone can be given in the following form:
Hc ¼ hs hs þ 3dð Þ=2d ð4Þ
It is obvious that if expansion factors of the panel roof strata are
known then Eq. (3) can be used to determine the number of roof strata
that must be fractured until the extracted panel space is completely
filled. Thus, Eqs. (1) or (4) is considered as an arithmetic series which
can be used to estimate the combined height of fractured and caved
zone or simply the height of destressed zone (HDZ) above the extracted
panel.
3.1.2. Sub-model-2
In this model, it is assumed that an increase in volume of broken
panel roof rock strata is a function of the free space in mined panel
area. Hence, if the thickness of first roof stratum is equal to the extracted
seam height then the height of broken layer will be increased by a factor
of “α”, that is, the height of free space will be decreased accordingly
(Fig. 3a). Therefore, the leftover space will be equal to [ho(1−a)]. If
the second stratum above extracted panel caves in, then the free
space will be equal to [ho(1−a)2
] as illustrated in Fig. 3b. Summing
up the terms shown in Fig. 3b yields:
Hc ¼ nhs−ahs n−3ð Þ−ahs 1−að Þ n−5ð Þ−ahs 1−að Þ
2
n−7ð Þ−…
þ ahs 1−að Þ
n−1
ð5Þ
Eq. (5) represents a geometric series whose limiting value can be
computed as follows:
Hc ¼
hs 1 þ að Þ
a
ð6Þ
By employing Eq. (6), the height of destressed zone above the mined
panel roof can be estimated.
3.2. Geometry-dependent roof fracture
3.2.1. Sub-model-1
In this sub-model, it is assumed that the roof failure depends on the
geometry of the panel roof situation and follows a vertical parabolic
function as has been illustrated in Fig. 4a. Thus by considering the mid-
point of the panel roof as the origin of the coordinate system then
where the vertical axis intersects the maxima of the parabola represents
the height of fracture zone. Since the parabolic equation can be written
y ¼ ax
2
þ bx þ c ð7Þ
Then, by employing the boundary conditions, that is, (0, Hc), (Lw/2, 0),
(−Lw/2, 0) and solving Eq. (7) Hc can be determined which represents
the height of destressed zone as shown in Fig. 4b. Hence,
y ¼ 4Hc=L
2
w
x
2
þ Hc ð8Þ
a)
b)
Ground Surface
Ground Surface
Fig. 2. Schematic cross-sectional view of a longwall: a) after the first panel roof stratum
failure due to the coal seam extraction, b) due to a gradual increase of the height of
fracture zone above the extracted panel for the Arithmetic model.
a)
b)
Ground Surface
Ground Surface
Fig. 3. Schematic cross-sectional view of a longwall: a) after the first panel roof stratum
failure due to coal seam extraction, b) due to a gradual increase of the height of fracture
zone above the extracted panel for the Geometric model.
66 A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

6. Taking the expansion factor of the broken materials into consider-
ation and on this account cross-sectional area of the caved in rocks will
be “α” times the area of the parabolic shape shown in Fig. 4b. Thus, the
Eq. (8) can be further simplified and presented as follows:
Hc ¼ 1:5hs=α ð9Þ
By using Eq. (9) the height of destressed zone above the extracted
panel can be estimated.
3.2.2. Sub-model-2
In this model, it is assumed that the shape of destressed zone is
similar to half a vertical ellipse as depicted in Fig. 5a and b. Similar
to the parabolic model, it is presumed that the cross-sectional area
of the fractured materials is “α” times the cross-sectional area of the
caved zone. Thus the height of destressed zone can be estimated by
the following formula
Hc ¼ 1:273hs=α ð10Þ
3.2.3. Sub-model-3
In this sum-model, it is assumed that the shape of destressed zone
resembles to a triangle as depicted in Fig. 6a. Employing the boundary
conditions (Fig. 6b) similar to the elliptical model and assuming the
cross-sectional area of the fractured materials is “α” times the cross-
sectional area of the caved zone then the height of destressed zone
can be estimated by the following equation:
Hc ¼ 2hs=α ð11Þ
By employing the Eq. (11), the height of destressed zone above the
mined panel roof can easily be estimated.
4. Discussion and comparative analysis of the results
In this section, comparative analyses of the results obtained from
the five new approaches are given for estimation of the height of des-
tressed zone (HDZ). The results are further compared with the only
comparable formula given by Peng and Chiang (1984). In this paper,
variations of the height of destressed zone versus the extracted coal
seam height are illustrated in Figs. 7–14. Figs. 15 to 22 illustrate the
non-linear variations of the height of destressed zone with the mined
panel roof rock expansion factors.
Triangular, Parabolic, Elliptical, and Geometric models as well as
Peng and Chiang's model illustrate a linear relation between HDZ
and parameters such as; the expansion factor and the extracted coal
seam height. Hence, in these models, the ratio of (Hc/hs) is always
constant, no matter what the coal seam extraction seam height is.
While, the Arithmetic model represents a nonlinear relation between
HDZ and the parameters such as; the expansion factor and the extracted
coal seam height. Therefore, in the case of Arithmetic model, the ratio of
a)
O
Lw
Hc
hs
b)Ground Surface
Fig. 4. a) Schematic cross-sectional view of a longwall with the corresponding adjacent access tunnels, barrier pillars after coal seam extraction and pack supports erection with
parabolic roof failure concept, b) Theoretical Parabolic panel roof failure with a prescribed boundary conditions.
a)
hs
Lw = 2a
Hc=b
b)Ground Surface
Fig. 5. a) Schematic cross-sectional view of a longwall with the corresponding adjacent access tunnels, barrier pillars after coal seam extraction and pack supports erection with
elliptical roof failure concept, b) Theoretical elliptical panel roof failure with prescribed boundary conditions.
67A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

7. (Hc/hs) is not constant; rather it is a linear function of both the expan-
sion factor and the extracted coal seam height. In this paper, the average
expansion factor of the panel roof rock strata, in all, is taken in the range
of 5% to 80% for the extracted coal seam thickness ranging from 1.0 to
4.5 m.
As it can be seen from Figs. 7 to 14, the Arithmetic model is the
only one that reflects a non-linear relation between the height of des-
tressed zone and the extracted coal seam thickness. This model also
provides the maximum height of destressed zone for most of the
cases. One of the most important results of the proposed formulas is
that; the highest height of destressed zone is attained when the ex-
pansion factor is the least. It is obvious that the least expansion factor
is attainable with time as the goaf materials are compressed. If an av-
erage of 10% expansion factor for the panel roof rock strata is taken to
represent the short-term condition of the fractured rocks for all the
cases, then the following results are obtained; the Triangular model
yields Hc =20hs; the Parabolic model yields Hc =15hs; the Elliptical
model yields Hc =12.7hs; and the Geometric model yields Hc =11hs.
However, the Arithmetic model is extremely sensitive to the extraction
height and yields Hc =6.5hs when hs =1 m, and plus 2.5hs for each half
a meter of increase in the thickness of the extracted coal seam, that is, if
the value of hs =1.5 m, 2 m, 2.5 m, 3 m, 3.5 m, 4 m, and 4.5 m, then the
corresponding height of destressed zones are; 9, 11.5, 14, 16.5, 19, 21.5,
and 24 times the extracted coal seam thickness, respectively. Indeed,
according to the Arithmetic model, for the short-term condition, the
maximum height of destressed zone for a 4.5 m of coal seam to be
extracted is equal to 24hs. It is interesting to notice that Peng and
Chiang's formula also have been used in all the cases for comparison
purposes. As it can be seen from Figs. 7 to 14, Peng and Chiang's Model
yields the height of destressed zone equal to 10 times the extracted
coal seam thickness for the short-term condition. In other words, Peng
and Chiang's model can be considered as a special case of the Arithmetic
model proposed in this paper. Thus, the results of the five different math-
ematical models presented in this paper, have proven that the height of
destressed zone (HDZ) induced due to longwall mining is in the range of
6.5 to 24 times the extracted coal seam thickness for the short term panel
roof fractured rock condition. In fact, the Arithmetic results represent
both the lower and the upper limits for all the five models proposed in
this paper. Indeed, the other four models can be considered as special
cases of the Arithmetic model as well. The authors strongly believe that
the long-term height of the destressed zone could be much more than
those considered for the short term condition. Hence, in this paper, an
average of 5% expansion factor has been taken to represent the long-
term condition of the goaf materials and the fractured zone located
above the gob-side. On this basis, the long-term results of the models
are almost twice as much as the height that is induced in the short
term. That is, if an average of 5%expansion factor for the panel roof
rock strata is taken to represent the long-term condition of the fractured
a)
Hc
hs Lw
b)Ground Surface
Fig. 6. a) Schematic cross-sectional view of a longwall with the corresponding adjacent access tunnels, barrier pillars after coal seam extraction and pack supports erection with
Triangle roof failure concept, b) Theoretical Triangle panel roof failure with prescribed boundary conditions.
Fig. 7. Relationship between the height of destressed zone and the extracted coal seam
height for the panel roof rocks' average expansion factor of 10%.
Fig. 8. Relationship between the height of destressed zone and the extracted coal seam
height for the panel roof rocks' average expansion factor of 20%.
68 A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

8. rocks for all the cases, then the following results are obtained; the Trian-
gular model yields Hc =40hs; the Parabolic model yields Hc=30hs; the
Elliptical model yields Hc =25.5hs; and the Geometric model yields
Hc =21hs. Peng and Chiang's model, for long term condition, yields
Hc =20hs, versus the value ranging from 20 to 30 times the extracted
coal seam thickness, as shown in Table 2. However, the lower and the
upper limits for all five models can be represented by the Arithmetic
model which ranges from 11.5 to 46.5 times the thickness of the extrac-
tion seam height, respectively. Beyond this height the overburden pres-
sure will be transferred towards the front abutment, the adjacent access
tunnels, the intervening barrier pillars as well as the panel rib-sides.
Hence, no matter at what depth the longwall is located, the complete
overburden pressure will not be inserted on the broken rocks located
some distance behind the hydraulic jacks in the gob-side. In fact, the dif-
ference between the longwall depth and the critical height of destressed
zone must be considered for the calculations of the stress transfer to-
wards the front abutment and the adjacent rib-sides. The long-term re-
sults of the five new models presented in this paper are shown in
Table 2 and are further compared with those obtained from the literature
review. The authors gathered two series of information both of which are
provided in Table 2; one series is based on the empirical inferring, which
show the combined height of caving and fractured zone (height of des-
tressed zone) in the range of 2 to 105 times the extracted coal seam
thickness. Whereas, the second series is obtained from the in-situ mea-
surements which show the height of destressed zone is in the range of
4 to 92 times the extracted coal seam thickness. The lower and upper
limits of both series of information are significantly far from each other
which may show the inherent complexity of the actual geological situa-
tion of the mining sites. Though all the five models' results, for the long
term condition, are in the range of 11.5 to 46.5 times the extracted coal
seam thickness, it seems that the Arithmetic model embraces the lower
and upper limits which better represent the in-situ conditions compared
with the other four models and all other existing empirical approaches as
shown in Table 2. According to Table 2, the long-term results of the
models proposed in this paper are in a close agreement with the in-
situ measurements and with those proposed empirically by researchers
as well. Depending upon the depth of longwall mining and the panel
width or the length of longwall, the height of destressed zone, may
have a direct influence on the corresponding ground surface subsidence
which is out of the scope of this paper. Finally, the authors believe that
further universal in-situ measurements are required to firmly express
the integrity and applicability of the existing empirical approaches as
well as the models proposed in this paper.
5. Conclusions
Five new mathematical approaches were proposed to model the
height of destressed zone based on the assumptions of being independent
and/or dependent on the geometry of the mined panel roof rock strata.
Fig. 9. Relationship between the height of destressed zone and the extracted coal seam
height for the panel roof rocks' average expansion factor of 30%.
Fig. 10. Relationship between the height of destressed zone and the extracted coal
seam height for the panel roof rocks' average expansion factor of 40%.
Fig. 11. Relationship between the height of destressed zone and the extracted coal
seam height for the panel roof rocks' average expansion factor of 50%.
Fig. 12. Relationship between the height of destressed zone and the extracted coal
seam height for the panel roof rocks' average expansion factor of 60%.
69A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

9. The results were analyzed and compared with both in-situ measurements
and those proposed empirically by researchers, obtained from a compre-
hensive literature review. An average of 5% expansion factor has been
taken to represent the long-term condition of the goaf materials as well
as the roof fractured rocks. Beyond this height the overburden pressure
will be transferred towards the front abutment, the adjacent access tun-
nels, the intervening barrier pillars and the panel rib-sides. Hence, no
matter at what depth the longwall is located, the complete overburden
pressure will not be inserted on the broken rocks located some distance
Fig. 13. Relationship between the height of destressed zone and the extracted coal
seam height for the panel roof rocks' average expansion factor of 70%.
Fig. 14. Relationship between the height of destressed zone and the extracted coal
seam height for the panel roof rocks' average expansion factor of 80%.
Fig. 15. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 1 m.
Fig. 17. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 2 m.
Fig. 18. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 2.5 m.
Fig. 16. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 1.5 m.
70 A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

10. behind the hydraulic jacks in the goaf area. In fact, the difference be-
tween the longwall depth and the critical height of destressed zone
(HDZ) must be considered for the calculations of the stress transfer
towards front abutment and the adjacent rib-sides. On this basis, the
long-term results of the models are proven to be in a close agreement
with the in-situ measurements and with those proposed empirically
by researchers as shown in Table 2. Though the present models are in-
dependent of the strength properties of the mined roof rock strata,
they yield very promising and reliable results.
Fig. 19. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 3 m.
Fig. 20. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 3.5 m.
Fig. 21. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 4 m.
Fig. 22. Relationship between the height of destressed zone and the panel roof rocks'
average expansion factor for an extracted coal seam thickness of 4.5 m.
Table 2
The results of in-situ measurements and those proposed empirically by researchers in-
cluding the results of the five new mathematical models proposed in the present paper.
Height of
caving zone
(×hs)
Height of fracture
or destressed zone
(×hs)
Reference Method of
appraisal
3–3.5 – Ropski and Lama (1973) Empirical
– 50–105 NCB, PI/1968/8 (revised 1971) Empirical
– 5–12 Chuen (1979), for weak rock Empirical
– 13–31 Chuen (1979), for strong rock Empirical
3–6 26–56 Singh and Kendorski (1981) Empirical
12 – Karmis et al. (1983) Empirical
8–12 50 Styler (1984) In-situ
– 20–30 Peng and Chiang (1984) Empirical
– 71–105 Fawcett et al. (1986) Empirical
4–6 30 Hasenfus et al. (1998) In-situ
– 20–100, 19–92 Palchik (1989, 2003a) In-situ
– 30–60 Richard et al. (1990) Empirical
– 2–3 Zhou (1991), for soft rock Empirical
– 5–6 Zhou (1991), for hard rock Empirical
– 20–60 Booth and Spande (1992) Empirical
2–20 20–50 Chekan and Listak (1993) Empirical
– 12 Luo (1997) Empirical
– 1–1.1 WP,
WP =160 m,
200 m
Mills and O'Grady (1998) In-situ
– 30–35 m, hs =?,
(WP =100 m)
Jeffrey and Zhang (2001) In-situ
– 40, 120 m,
hs =3 m
Kelly et al. (2002) In-situ
4–11 – Palchik (2002a) In-situ
3–6 30–58 [Singh and Kendorski (1981) and
Kendorski (1993), from Karacan
and Goodman (2009)]
Empirical
– 80, hs =3 m RafiqulIslam et al. (2009) In-situ
5–6 10–11 Zhang et al. (2011) In-situ
– 40 Present paper (Triangular model) Theoretical
– 30 Present paper (Parabolic model) Theoretical
– 25.4 Present paper (Elliptical model) Theoretical
– 22 Present paper (Geometric model) Theoretical
– 13–48 Present paper (Arithmetic model) Theoretical
71A. Majdi et al. / International Journal of Coal Geology 98 (2012) 62–72

11. Acknowledgments
Acknowledgement is due to the School of Mining Engineering, Col-
lege of Engineering, University of Tehran, Tehran, Iran for providing the
sabbatical opportunity to the first author to complete this research
work at the Department of Mining and Materials Engineering, Faculty
of Engineering, McGill University, Montreal, Canada. The first author
also acknowledges that this work could not be completed without re-
search support provided by the Department of Mining and Materials En-
gineering, Faculty of Engineering, McGill University, Montreal, Canada.
The views expressed in this paper are those of the authors and not nec-
essarily of the institutes they work for.
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