Global Trends in Coal Mine Horizontal Stress Measurements 

Christopher Mark, Principal Research Engineer 

NIOSH-Pittsburgh Research Laboratory 

Pittsburgh, PA
 

Murali Gadde, Senior Engineer 

Peabody Energy
 

St. Louis MO 

ABSTRACT 

 Knowledge of in situ stresses is fundamental to many studies 

in earth sciences, and coal mine ground control is no exception. 

During the past 20 years, it has become clear that horizontal stress 

is a critical factor affecting roof stability in underground coal 

mines. The theory of plate tectonics and the World Stress Map 

(WSM) project has been extremely helpful in explaining the 

sources and the orientations of the horizontal stresses observed 

underground. Recently, WSM geophysicists studying deep-seated 

stresses have developed a model of how stress magnitudes vary 

with depth in the crust. They have devoted relatively little 

attention to near-surface stresses, however. This paper explores 

the relationships between deep-seated and shallow in situ stresses 

in several of the world’s coalfields, using a data base of more than 

350 stress measurements from underground coal mines. The 

analysis indicates that distinct regional trends exist, corresponding 

roughly to the regional stress fields identified by the WSM. The 

paper presents equations for estimating stress magnitudes that 

were developed by treating depth and elastic modulus as 

independent variables in regression analysis. The magnitude of 

the horizontal stress increases with depth, at rates that range from 

0.8 to 2.0 times the vertical stress, just as the WSM “critically 

stressed crust” model predicts. Overall, it seems that the stress 

regimes encountered in underground coal mines are closely linked 

to those that exist deep in the earth’s crust. 

ACKNOWLEDGEMENTS 

The authors would like to express their sincere appreciation to 

Dr. Winton Gale for making the Strata Control Technologies (SCT) 

stress measurement data base available for this analysis. The SCT 

data base is undoubtedly the most extensive of its kind in the 

world, and its inclusion has added immensely to the value of this 

paper. We would also like to thank Rock Mechanics Technology 

(RMT) for making a large portion of their very impressive worldwide data base available to other researchers via their website. 

Thanks are also due to Russel Frith, Hamid Maleki, Dennis 

Dolinar, and Leo Gilbride for their contributions to the data base. 

INTRODUCTION 

As early as the 1940’s, researchers in British coal mines 

postulated that large horizontal stresses were responsible for much 

of the roof damage experienced underground. Philips, (1946) 

observed that “At depths greater than 700 ft (215 m)...lateral 

compressive forces cause fracturing along the laminations of the 

roof beds...the lateral compressive forces increase at a greater rate 

than the vertical compressive force, and ultimately both forces 

may be equal.” Once rock mechanics researchers began to 

measure in situ stresses, it became clear that in many cases the 

horizontal stress actually exceeded the vertical, often by factors of 

three or more (Dahl and Parsons, 1972; Hoek and Brown, 1980). 

A number of theories were proposed to explain the presence of 

horizontal stress. Two of the earliest were the “Poisson’s effect” 

and the “lithostatic stress state.” Both of these theories presumed 

a static earth, in which the horizontal stresses were generated in 

response to the vertical overburden load. The “Poisson’s effect” 

model predicted that the horizontal stresses should be about 1/3 of 

the vertical, while the lithostatic model predicted that the three 

principal stresses would be approximately equal (McGarr, 1988). 

These static earth theories could not explain two key 

characteristics of horizontal stress: 

•	 Why the horizontal stress often exceeds the vertical (SV) 

in magnitude, even at depth, and; 

•	 Why horizontal stresses are typically highly anisotropic, 

with the major one (SHmax) significantly larger than the 

minor (Shmin). 

In addition, the Poisson’s effect model suffers from severe 

theoretical errors, because it implicitly assumes that soft sediment 

(or magma) lithifies in the absence of gravity, and then gravity is 

instantaneously switched on once the rock has reached an elastic 

state (McGarr, 1988). As Hoek (2007) writes, the Poisson’s effect 

model was “widely used in the early days of rock mechanics, but 

proved to be inaccurate and is seldom used today.”1 

1 The most elegant of the static earth theories is the “spherical shell” model 

proposed by Sheorey (1994), based in part on earlier work by McCutchen 

(1982). Sheorey’s model considers stresses arising from the geothermal 

gradient in addition to the Poisson’s effect, and it predicts that the 

horizontal-to-vertical stress ratio may be very large near the surface but 

declining with depth. Like all static earth theories, however, Sheorey’s 

model can explain neither the near universal anisotropy of the horizontal 

stress, nor the prevalence of horizontal stresses that are significantly 

greater than the vertical deep into the crust. 
 Fortunately, during the 1970’s earth scientists were 

constructing a revolutionary new theory with the breadth and 

depth to explain all the major tectonic processes observed on the 

earth. Plate tectonics describes a dynamic earth, in which the 

crust of the earth consists of a number of continental plates that are 

sliding across the softer rock in the mantle below. Where the 

plates contact each other, their different directions of relative 

movement create large forces that are transmitted across the plate 

interiors. The scientists associated with the World Stress Map 

(WSM) have used a variety of indicators, including earthquake 

focal mechanisms, wellbore breakouts, and hydraulic fracturing 

stress measurements, to identify the lithospheric stresses that result 

from these plate movements. They found that the state of stress is 

remarkably uniform over vast regions of plate interiors, and that it 

is due to present-day forces, and not due to residual stresses from 

past tectonic activity (Zoback and Zoback, 2007). Today there is 

no argument within the geophyisics community about the general 

validity of the trends identified by the WSM. 

 The WSM has found that at any given location, the stress 

direction within the crust is typically consistent from the “upper 1-

3 miles (2-5 km), where essentially all of the wellbore breakout 

and hydraulic fracturing data come from, down through the lower 

3-12 miles (5-20 km), where the majority of crustal earthquakes 

occur” (Zoback and Zoback, 2002). However, the WSM has 

specifically excluded “near surface” measurements from its 

database, because they have sometimes found that there are 

“marked changes in stress orientations and relative magnitudes 

with depth in the upper few hundred meters, possibly related to 

effects of nearby topography or a high degree of surface 

fracturing” (Zoback, 1992). Later, Zoback and Zoback (2002) 

stated that “only in situ stress measurements made at depths 

greater than 100 m are indicative of the tectonic stress field.” 

N W E Western US 

Coalfield 

Stress Measurements 

S N

Illi W E nois Basin
 

US Coalfield
 

Stress Measurements
 

S N W E

Eastern USS
 

Coalfield
 

Stress Measurements
 

Figure 1. World Stress Map of the United States, compared with stress orientations determined from coalfield stress 

measurements. The solid arrows show WSM stress direction and the dotted lines delineate stress provinces (Map after 

Zoback and Zoback, 1989; Coalfield stress measurements from Mark and Mucho, 1994). 

Since most underground coal mining takes place within several 

hundred meters of the surface, it is legitimate to ask how relevant 

the WSM is to underground coal mining. On the one hand, we do 

know that topographic features can significantly affect the stresses 

we observe underground (Molinda et al., 1992; Hasenfus and Su, 

2006). On the other hand, since the near-surface is part of the 

crust, we would certainly expect some relationship to the deepseated stress patterns. The question is, how closely are the two 

related? Do we observe the same general trends that the WSM has 

identified in the deep crust, or are the topographic and other nearsurface effects so powerful that they completely mask any 

relationship? 

 The first part of the answer to this question was provided in 

the early 1990’s when researchers compiled a data base of stress 

measurements from U.S. coal mines. The WSM had identified 

the eastern portion of North America as a stable mid-plate region 

with a consistent ENE horizontal stress orientation (Zoback and 

Zoback, 1989). Sure enough, analysis indicated that 75% of the 

coal mine stress measurement orientations fell within the NE 

quadrant (Mark, 1991; see figure 1). This finding was 

particularly meaningful because the data base included 

measurements made all over the eastern U.S., by a variety of 

researchers using a number of different techniques. The observed 

trends were highly significant statistically even though no attempt 

was made to minimize the effects of bad data by applying a 

“quality ranking” to the individual measurements. Such quality 

rankings are normally considered essential for discerning 
 

 

 

 

 

 

 

 

 

underlying trends in stress measurement data (Zoback, 1992; 

Stacey and Wesseloo, 1998). 

The WSM also defined the stress regime within eastern North 

America as either strike/slip (where the magnitude of the vertical 

stress falls between the two horizontal stresses) or reverse faulting 

(where the vertical stress is smaller than both the principal 

horizontal stresses). Here, the US stress measurement data was in 

even better agreement, with the maximum horizontal stress 

exceeding the vertical 97% of the time. 

United Kingdom Coalfield 

Stress Orientations N W E S

 

Figure 2. World Stress Map of northern Europe, compared 

with stress orientations determined from UK coalfield stress 

measurements (Map after Muller et al. 1992; Stress 

measurements after Cartwright, 1997). 

THE WSM AND STRESS ORIENTATIONS IN 

COALFIELDS AROUND THE WORLD 

Stress measurement data bases have now been constructed for 

a number of the other of the world’s coalfields. It makes sense to 

ask how well the predictions of the WSM compare in those cases. 

Western U.S.: According to the WSM, the coalfields of Utah, 

Colorado, Wyoming, and New Mexico fall within regions of 

“extension” or normal faulting, where the vertical stress is 

predicted to be greater than either horizontal stress (see figure 1). 

In contrast to the eastern “midplate” region, the western regions 

are seismically active. Stress directions also vary throughout the 

region. 

Mark (1991) analyzed stress measurements from 17 Western 

U.S. mines, and found no significant regional trends in orientation. 

The maximum horizontal stress was significantly lower than in the 

east, and was approximately equal to the vertical stress in most 

cases. 

Germany and the UK: The WSM defines the stress regime in 

western Europe, including the coalfields of Germany and the UK, 

as a stable mid-plate area subject to uniform NNW maximum 

horizontal stress. As in eastern North America, the stress is 

controlled by plate driving forces acting on the plate boundaries 

(Muller et al., 1992). Western Europe is considered to be a strike 

slip environment, with the vertical stress as the intermediate 

principal stress. 

A series of 11 hydrofracturing stress measurements were 

conducted in four German coal mines between 1989 and 1991 

(Muller, 1991). These measurements confirmed that the greatest 

horizontal stress was oriented NNW. The maximum horizontal 

stress was reported to exceed the vertical stress down to depths of 

4,000 ft (1,200 m). 

Cartwright (1997) describes the results from 26 successful 

overcores conducted by Rock Mechanics Technology (RMT) at 16 

mine sites in “virgin or near virgin conditions.” The depths of 

cover ranged from 1,000 to 3,300 ft (300-1,000 m). In every 

overcore the maximum horizontal stress was located in the NW 

quadrant, and the vast majority were oriented within a few degrees 

of NNW (figure 2). The magnitude of the stress was 

approximately equal to the vertical stress, but there was 

considerable spread. 

Bowen Basin, Queensland, Australia: Australia is considered 

somewhat unique by the WSM because the stress orientation 

varies considerably between different regions of the continent, 

reflecting a variety of plate boundary forces rather than the 

direction of absolute plate motion (Hillis et al., 1999). In the 

Bowen Basin coalfields of the central Queensland, the major 

horizontal stress is consistently oriented NNE, and the vertical 

stress is either the minor or the intermediate principal stress. The 

region is not seismically active, and evaluation of the available 

stress measurements found that few were indicative of faults on 

the verge of movement. 

Nemcik et al. (2005) presented the results of 235 

measurements of pre-mining stress made by SCT in Australian 

underground mines. About a third of these measurements were 

conducted in the Bowen Basin, all at depths of less than 1,000 ft 

(300 m). Nemcik at al. (2005) reported that “the direction of the 

major lateral stress was in most cases confined to the N to NE 

quadrant” (figure 3). The magnitudes of these stresses almost 

always exceeded the vertical, sometimes by factors of 3 or more. 

Bowen Basin 

Coalfield 

Stress Measurements N W E

S

Figure 3. World Stress Map of Australia, compared with 

stress orientations determined from Bowen Basin coalfield 

stress measurements (Map after Hillis et al. 1999; Stress 

measurements after Nemcik et al., 2005). 

Sydney Basin, NSW, Australia: The Sydney Basin appears to be 

the exception that proves the rule. Early studies of horizontal 

stress in underground mines there found that stress directions 

could vary widely, even from one section of a mine to another 

(Gale et al., 1984; Gale, 1986). No consistent regional trend 

could be observed. As it turns out, the WSM found the same thing 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Hillis et al., 1999). The Sydney Basin is one of Australia’s most 

seismically active, and stress orientations vary widely throughout 

it. It seems that plate margin effects superimposed on the regional 

stress direction have resulted in a relatively low anisotropy 

between the major and minor principal horizontal stresses. As a 

result, mild perturbations caused by local effects, such as density 

contrasts, faults, or major geologic structures may cause local 

stress rotations (Hillis et al., 1999). Moreover, as many as 40% of 

the stress measurements are indicative of faults at incipient failure. 

Nemcik et al. (2005) confirmed that even with a very large data 

base, consisting of approximately 170 stress measurements from 

mines all over NSW, no consistent trends in stress direction 

emerged. 

Summary: Hillis (1999) concludes that “the apparent consistency 

between in situ stress measurements and seismicity of the Bowen 

and Sydney Basins suggests that relatively shallow (1,000-3,300 ft 

(300-1,000 m)) data ... may be representative of the stress at 

greater, seismogenic depth.” Our quick tour around other 

coalfields of the world leads to the same conclusion, at least with 

regards to stress orientation and relative magnitude. In every case, 

the WSM model provided a reasonably accurate prediction of (and 

explanation for!) the typical stress regime that is observed in 

underground coal mines. The next question is whether we can 

also predict the magnitude of the stress using the WSM model. 

THE CRITICALLY STRESSED CRUST2	 

2 The discussion in this section is based almost entirely on the summary 

provided by Zoback and Zoback (2002). 

The “dynamic earth” plate tectonics model implies that lateral 

forces are constantly being applied to the brittle upper crust. 

These forces would continue to build unless there was some 

mechanism for their release. That mechanism is failure of the 

crust itself, through faulting. Decades of research along a number 

of lines of evidence have resulted in what is, in the end, a simple 

but profound model of the magnitude of the stresses that the crust 

can carry (Zoback and Zoback, 2002; Zoback and Zoback 2007). 

The model begins with Anderson’s (1951) classification 

scheme for relative stress magnitudes in the earth: 

•	 Normal faulting regions, where Sv>SHmax>Shmin (1) 

•	 Strike slip faulting regions, where SHmax>Sv>Shmin (2) 

•	 Reverse faulting regions, where SHmax>Shmin>Sv (3) 

Research has shown that the strength of faults can be 

adequately described by the Coulomb criterion: 

T = Co + uSn	 (4) 

where T = the shear strength of the fault plane, Co = the fault 

plane’s cohesion, u= the friction coefficient, and Sn = the confining 

stress applied perpendicular to the fault plane. 

Using two-dimensional Mohr-Coulomb analysis, the shear 

stress at failure of an optimally-oriented fault is a function of the 

difference between the minor (S3) and the major principal (S1) 

stresses (Jaeger and Cook, 1979): 

(S1-Po)/(S3-Po) = ((u2

 + 1)1/2 +u)2 (5) 

where Po is the pore pressure, and (S-Po) is the effective stress. 

Studies have shown that, at depth, the cohesion is much smaller 

than the frictional component of the fault strength, the friction 

coefficient u is typically 0.6-1.0, and the pore pressure is 

hydrostatic on active faults (Townend and Zoback, 2000). 

Assuming u=0.6, the following approximate relationships can be 

derived: 

•	 SHmax = 2.3 Sv in reverse faulting regions; (6a) 

•	 SHmax = 1.6 Sv in strike-slip faulting regions (assuming 

SV = (SHmax+ Shmin)/2,) and; (6b) 

•	 SHmax < Sv and Shmin = 0.6 Sv in extension faulting (6c) 

regions. 

Stress measurements have now been conducted in several deep 

boreholes to depths of almost 5 miles (8 km). Figure 4 shows that 

the measurements confirm the general stress gradients derived 

above. In particular, in the mid-plate compressive stress regions 

where these measurements were made, horizontal stresses well in 

excess of the vertical persist far down into the crust, and the 

horizontal stress gradient (k) is fairly consistent with a value 

ranging from approximately 1.3 to 2. 

Maximum Horizontal Stress (MPa) 

0 50 100 150 200 250 300 350 400 

0.0 

1.0 

2.0 

Depth (km) 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

2.2 * Vertical 

Stress 

1.6 * Vertical 

Stress 

Vertical 

Stress 

Figure 4. Stress measurements from boreholes deep in the 

earth’s crust (data from Townend and Zoback, 2000). 

Zoback and Zoback (2002) also state that they have found “no 

evidence” that “residual stresses” from past tectonic events play 

any role in today’s stressfields. They speculate that if such 

stresses exist at all, they can only be important “in the upper few 

meters or tens of meters of the crust where the tectonic stresses are 

small.” 

If plate tectonics are responsible for virtually all the stresses 

measured at depth, and if the critically stressed crust model allows 

us to predict those stresses, then it is reasonable to expect that 

there is some relationship between the deep crustal stresses and 

those measured in coal mines. After all, the “near surface” is part 

of the crust! In fact, if we don’t find a relationship, then we have a 

significant problem. For instance, if we conclude that the 

horizontal stress increases less rapidly than the vertical stress to 

depths of one or two thousand feet, but we know it exceeds the 

vertical stress at greater depths, the implication is that there is a 

major discontinuity in the stressfield somewhere. Let us then see 

what the actual measurements tell us. 
 

 OVERCORING STRESS MEASUREMENTS 

IN COAL MINES 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Overcoring has been the most common technique for 

measuring stress in underground coal mines. In the US, most 

measurements have been made using the Bureau of Mines biaxial 

“borehole deformation gage” (Bickel, 1993). Internationally, the 

triaxial ANZI or CSIRO HI cells have been by far the most 

common (Mills, 1997; Nemcik et al., 2005; Cartwright, 1997). 

Most recently, downhole wireline stress measurement devices 

have been developed (Conover et al., 2004). 

One important feature of overcoring stress measurements is 

that interpretation of the data requires the determination of the 

rock’s elastic modulus (E). The modulus is not required for the 

other types of stress measurement contained in the WSM data base. 

Studies in the layered sedimentary geology of coal measure rock 

have found that the measured stresses in a single hole vary in 

proportion to the modulus of the rock (Aggson and Mouyard, 

1988). 

Cartwright (1997) pointed out that the relationships between 

horizontal stress and depth, like those of Hoek and Brown (1980), 

have typically displayed high scatter, particularly near the surface. 

Within his data base of UK stress measurements, there was a 

better correlation between stress and modulus than between stress 

and depth. He proposed that the two factors might be combined 

into a single equation: 

SH = B0 + B1 [(v/(1+v)) (Depth)] + B2 (Modulus) (7) 

where B0 is a constant with units of psi (MPa), B1 is a constant 

with units of psi per ft of depth (MPa/m), v is Poisson’s ratio, and 

B2 is a dimensionless constant called the “tectonic strain factor” or 

TSF. Regression analysis provided the following values for the 

constants, with an r-squared of 0.94: 

•	 B0 = -580 psi (-4.0 MPa); 

•	 B1 = 0.4 psi/ft (0.009 MPa/m), and; 

•	 B2 = 0.78*10-3 

Cartwright’s analysis indicated that for his data set, the 

modulus was more important than the depth for predicting the 

maximum horizontal stress. 

Dolinar (2003) studied stress measurements from 37 eastern 

U.S. underground mines, including several stone mines. His 

analysis employed a version of equation (7), with B0=0 and B1 

fixed at 1.1 psi/ft (0.025 MPa/m). He found that the remaining 

regression coefficient, the TSF (B2), varied between 400 and 900 

for the different geographic regions studied, with the highest TSF 

values in two small areas in central Appalachia. 

Nemcik et al. (2005) also calculated the TSF for the large SCT 

data base of 235 measurements from Queensland and NSW mines. 

In their analyses, they also used equation (7), setting B0=0 and 

B1=1.1 psi/ft (0.025 MPa/m). In contrast to both Dolinar and 

Cartwright, however, Nemcik found that: 

•	 There was a strong correlation between depth and the 

SHmax in both NSW and Qld, 

•	 The TSF also increased significantly with depth, averaging 

0.4 when the depth was less than 330 ft (100 m) deep, but 

more than 1.3 for mines at depths exceeding 1600 ft (500 

m), and; 

•	 At any given depth, a wide range of TSF values were 

measured. 

Nemcik et al.’s work indicates that the TSF can vary significantly 

within a single region, and that the TSF cannot always explain a 

large proportion of the variation in SHmax. 

 INTERNATIONAL OVERCORING MEASUREMENTS 

DATA SET 

If the magnitude of the near-surface stresses measured in coal 

mines were closely related to the deep-seated stresses measured by 

the WSM, we might expect to find that: 

•	 The depth is at least as important as the modulus in 

predicting the horizontal stress, though both factors 

together should be better still; 

•	 The depth gradient should be somewhere between 1.0-1.6 

times the vertical stress for coalfields located in stable, aseismic mid-plate areas, like those in the eastern U.S., the 

UK, Germany, or central Queensland; 

•	 The depth gradient should be higher in a seismically 

active compressive regime like the one found in the 

Sydney Basin, and it should be lower an active extension 

regime like the one found in the western U.S. coalfields. 

To test these hypotheses, a data set of 565 stress measurements 

was compiled. The heart of the data set is 373 measurements 

from underground coal mines. The breakdown of these by region 

is shown in figure 5. Approximately two-thirds of the coal data 

were from Australia, and were provided by SCT. Preliminary 

statistical analyses indicated that the four eastern U.S. coalfields 

could be combined into a single “eastern U.S. coal” grouping, and 

that the UK and German data could be combined into a “northern 

European coal” grouping. 

India (5) 

South Africa (22) Australia - 

Northern Europe Queensland (64) 

(52) 

Western US (20) 

Eastern US (40) 

Australia - New 

South Wales (170) 

Figure 5. Locations of the stress measurements included in the 

coal data set. 

In addition, about 200 non-coal measurements that were 

readily available in the literature and at the WSM website 

(Reinecker et al., 2005) were collected. The purpose of the noncoal data was to provide an independent check on the general 

regional trends observed within the coal data. The non-coal data 

set includes stress measurements from the same general regions as 

the coal data set, though it does not include any measurements 

from Australia. The non-coal data also provides an opportunity to 

compare stress trends within bedded coal measure strata to those 

in other geologic settings. However, it is recognized that while the 

coal data set is easily the most comprehensive of its kind ever 

compiled, the non-coal database is quite small compared to how 

many measurements could be available. 
 

 

 

Preliminary analyses were conducted to see where data sets 

could be combined. It was determined that the measurements 

from Ireland, the UK, northern Europe, and Scandinavia could be 

combined into a single “All Europe” grouping. Similarly, data 

from the eastern U.S., Canada, and the western U.S. were 

combined into a “North America” grouping. It was surprising that 

while there were distinct differences between the western and 

eastern US coal data sets, the trends within the U.S. non-coal data 

did not seem to vary much by region. Figure 6 shows the number 

of data points within each regional grouping. 

India (16) 

South Africa (14) 

Northern Europe 

(47) 

North America (115) 

Figure 6. Locations of the stress measurements included in 

the non-coal data set. 

Figures 7 and 8 show the range of depth and modulus within 

each regional group in the coal data set. The greatest depths are 

encountered in the western European mines (UK and Germany), 

while the highest modulus rocks are in the eastern U.S. 

Queensland has the lowest modulus rock. Note that the small size 

of the data set from India (n=5) and the relatively small range of 

depth in the South African data set mean that the results from 

these two regions would be expected to be less reliable than those 

from the other regions. 

 

Depth (ft )

0 1,000 2,000 3,000 4,000 5,000 

Eastern Western UK & South India NSW Qld United States Germany Africa Australia 

Figure 7. Range of depths of the coal stress measurements, by
 

region. The middle line represents the median value, and the
 

upper and lower hinges of the box represent the 75th and
 

25th percentiles of the data, and the dots are outliers. 

The same types of summary data for the non-coal data set are 

shown in figures 9 and 10. The depth ranges are similar to coal 

data, except in South Africa where the non-coal data includes a 

number of measurements from extremely deep gold mines. In 

general, the modulus values for the non-coal measurements are 

about twice as great as those from the coal measurements. 

Eastern Western UK & South NSW Qld India United States Germany Africa Australia

Modulus (psi 10 ) 6 * 

0 2 4 6 8 10 

Figure 8. Range of elastic modulus of the coal stress 

measurements, by region. 

Depth, (ft ) 

0 2,000 4,000 6,000 8,000 

North Northern South Africa India 

America Europe 

Figure 9. Range of depths of the non-coal stress 

measurements, by region. 

Modulus (psi 10 ) 6 * 

0 5 10 15 20 

India 

Figure 10. Range of elastic modulus of the non-coal stress

measurements, by region.
 

 

 

 

 

 

 

 

 

 

 

 

 

Data Weighting: In order that the coal data not be 

overwhelmingly influenced by the Australian data, all the data was 

weighted by the following fomula: 

Weight of an individual measurement= 1/ (nR)

0 5 (8) 

where nR is the number of measurements from a particular region. 

The result is that an individual data point from a region with 

few measurements will count more heavily than one from a region 

with a lot of measurements, but overall the heavily populated 

regions will still have more influence. For example, the data set 

contains 40 measurements from the eastern U.S. and slightly more 

than 4 times as many measurements from NSW. In the weighted 

analysis, an individual U.S. measurement is given a weight about 

two times as great as a measurement from NSW, but in aggregrate 

all the NSW measurement have twice as much influence over the 

final equation as do all the eastern U.S. measurements. 

As a check, all of the analyses were run using both weighted 

and unweighted data, and it was found that the results did not 

differ significantly. The weighted results will be reported here in 

this paper. 

Depth Gradients and Modulus Effect for the complete data 

sets: Multivariate regression was conducted using the statistical 

package STATA. The first analyses looked at the relationship 

between depth and stress, with no other variables. The form of the 

model is thus: 

SHmax = B0 + B1 (Depth) (9) 

In this analysis, B1 is the gradient of the maximum horizontal 

stress with depth (in psi/ft), and the intercept B0 can be interpreted 

as the “excess stress” that is not associated with the depth gradient. 

The results are shown in table 1: 

Table 1: Regression results using Equation (9) as the model. 

Excess stress 

B0 (psi 

(MPa)) 

Depth 

gradient B1 

(psi/ft 

(MPa/m)) 

r-squared 

Coal data 1057 (7.3) 1.06 (0.024) 0.33 

Non-coal data 1386 (9.6) 1.24 (0.028) 0.56 

Combined data set 1159 (8.0) 1.18 (0.026) 0.50 

In other words, for the combined data set, we can explain half 

of the variation simply by the depth (and a constant). All the other 

variability—regional location, modulus, proximity to the entry and 

other measurement errors (Gadde et al., 2006), is only responsible 

for the other 50%. Although depth is not as strong a predictor in 

for the coal data as it is for the non-coal data, the depth gradients 

are all also approximately equal to the vertical stress gradient of 

1.1 psi/ft (0.025 MPa/m). The similarity between the coal and 

non-coal equations is striking—these are two completely 

independent data sets, drawn from similar parts of the world. 

When the analyses were run with just the modulus, instead of 

depth, the r-squared values were all reduced. In the case of the 

coal data, the reduction is only from 0.33 to 0.28, but for the noncoal data (and the complete data set) the r-squared is reduced from 

about 0.5 to approximately 0.20. These results indicate that 

modulus effect is most pronounced in layered, coal measure 

geologies, but that even there depth explains about as much of the 

variation in the data as the modulus does. 

In the next set of analyses, the effects of modulus and depth 

are explored simultaneously. The regression equation that is used 

is: 

SHmax = B0 + B1 (Depth) + B2 (Modulus) (10) 

In this model, the excess stress consists of two components, the 

intercept (B0) and plus the modulus term (B2*E). (Note that with 

all these models, the excess stress is independent of the depth. 

Statistical analyses confirmed that there was minimal interaction 

between the modulus term and the depth within this data set.) 

The results in Table 2 show that adding the “modulus factor” 

improves the r-squared values considerably, from 0.33 to 0.52 for 

the coal data, and to above 0.60 for the non-coal data set. Note 

that the values for the depth gradient drop slightly but are still 

close to 1.1 psi/ft (0.025 MPa/m). The biggest change from table 

1, particularly for the non-coal data, is a reduction in the constant. 

In other words, it appears that the two elements of the excess stress, 

the intercept and the modulus term, are closely related. 

Table 2. Regression results using equation (10) as the model. 

Intercept 

B0 (psi 

(MPa)) 

Depth gradient B1 

(psi/ft(MPa/m)) 

Modulus 

factor 

B2(10-3) 

r-squared 

Coal data 100 (0.7) 0.87 (0.020) 0.44 0.52 

Non-coal data 284 (2.0) 1.16 (0.026) 0.19 0.62 

Combined data 368 (2.6) 1.12 (0.025) 0.21 0.58 

It is also worth pointing out that the relative importance of the 

depth effect and excess stress change with depth. For example, for 

a coal mine at 300 ft (90 m) of cover and a modulus of 3 x 106

 psi 

(4.3 GPa), the total predicted horizontal stress is 1693 psi (11.9 

MPa). The excess stress accounts for 1,432 psi (9.9 MPa), and the 

depth effect contributes just 261 psi (1.8 MPa) to this total. If the 

depth increases to 1,500 ft (450 m), the predicted stress is now 

2737 psi (18.9 MPa), and all the increase is due to the depth effect. 

Region-by-region analyses: The next step was to determine 

predictive equations in the form of equation (10) for each of the 

individual regions. It is reasonable to expect that different regions 

might have different relationships between stress, modulus, and 

depth, owing to different characteristics of the crust, tectonic 

forces, etc. The results of the individual analyses are shown in 

table 3. 

Several observations can be made on the data in table 3: 

•	 The r-squared for the individual equations are between 

about 0.50 and 0.70 for nine of the eleven data sets, which 

is not significantly better than what was achieved by 

analyzing the entire data set. 
 

 

 

•	 The most consistent coefficients are the depth gradients 

(B1), and these are also generally the most statistically 

significant coefficients. The range is from about 0.6 to 1.8 

psi/ft (0.013 - 0.040 MPa/m) for the larger data sets. 

•	 The modulus factors are also fairly consistent, ranging 

from about 250 to 600 x 10-3. However, in four of the 

smaller data sets, these coefficients are not significant. 

Moreover, the modulus effect is more pronounced in the 

coal measure rocks. 

•	 The excess stress, determined using the mean modulus 

value for each region, was remarkably consistent, 

averaging about 1,000 psi (7 MPa) for the eight largest 

data sets. The range was from a low of 400 psi (2.8 MPa) 

in Queensland coal mine measurements, to a high of 1,600 

psi (11 MPa) in the eastern U.S. coal measurements. 

Overall, while the equations for some regions (Western U.S., 

NSW) seem reasonably reliable, in many cases the small size of 

the individual data sets is probably responsible for poor 

correlations. 

Unified analyses, controlling for different regional depth 

gradients: A statistical technique is available that allows us to 

combine the power of using the largest possible data set, while 

simultaneously allowing for some regional variation. This is 

accomplished by allowing the coefficients for the depth gradient to 

vary region by region3

. 

3

The technique involves creating interaction terms involving dummy variables (Wooldridge, 2006, pp. 244-252). In this case, a dummy variable is defined 

based on region, and then that dummy variable is interacted with the variable “depth.” 

______________________________________________________________________________________________________ 

 

Table 3. Regression results for regional subsets of the data, using equation (10) as the model. 

n Intercept B0 

(psi (MPa)) 

Depth gradient B1 

(psi/ft (MPa/m)) 

Modulus factor 

B2(10-3) r-squared 

East U.S. coal 42 369* (2.6) 1.34 (0.030) 0.302 0.55 

West U.S. coal 20 -915 (-6.4) 0.66 (0.015) 0.62 0.71 

UK/Ger coal 52 -249* (-1.7) 0.55 (0.012) 0.51 0.50 

South Africa coal 22 866 (6.1) -0.03* (0) -0.01* 0 

India coal 5 376* (2.6) 1.29 (0.029) -0.04* 0.79 

NSW coal 170 -633 (-4.4) 1.78 (0.040) 0.56 0.71 

Qld coal 64 -210 (-1.5) 1.40 (0.031) 0.34 0.51 

U.S./Can non-coal 115 -273 (-1.9) 1.45 (0.033) 0.27 0.70 

N. Europe non-coal 47 905 (6.3) 1.24 (0.028) 0.02* 0.55 

South Africa non-coal 14 327* (2.3) 2.03* (0.046) -0.04* 0 

India non-coal 16 -826 (-5.8) 2.20 (0.049) 0.26 0.56 

*Signifies statistic is not significantly different from 0 at the 95% confidence level. 

Table 4. Stress prediction parameters for equation 10 determined for the individual coal regions using the unified analysis 

regression technique. 

n Intercept B0 

(psi (MPa)) 

Depth gradient B1 

(psi/ft (MPa/m)) 

Modulus factor 

B2(10-3) 

East U.S. coal 42 -298 (-2.1) 1.64 (0.037) 0.41 

West U.S. coal 20 -298 (-2.1) 0.78 (0.018) 0.41 

UK/Ger coal 52 -298 (-2.1) 0.71 (0.016) 0.41 

South Africa coal 22 -298 (-2.1) 1.11 (0.025) 0.41 

India coal 5 -298 (-2.1) 0.44 (0.010) 0.41 

NSW coal 170 -298 (-2.1) 1.84 (0.041) 0.41 

Qld coal 64 -298 (-2.1) 1.36 (0.031) 0.41 

The final equations are in the form of equation (10), but while 

all of them use the same coefficients for the components of the 

excess stress , each region has its own depth gradient. The results 

are shown in table 4. 
 

 

 

The regression analysis used to obtain table 4 achieves an rsquared of 0.69. Accounting for nearly 70% of all the variation in 

such a large and diverse data set is an impressive accomplishment. 

The following further observations can be made on these results: 

•	 The greatest depth gradient, approaching 2.0, was found 

for the NSW measurements, which corresponds to the 

prediction based on the WSM; 

•	 The results for the eastern U.S., Queensland, and the 

Western U.S. are also in good agreement with WSM 

predictions, and; 

•	 The depth gradient for Europe is lower than was 

expected based on the WSM. Previous researchers have 

hypothesized that extensive past mining in the UK and 

German coalfields may account for the discrepancy 

(Muller, 1991). 

The equivalent results for the non-coal data are shown in 

table 5. It is significant that all but one of the regional depth 

gradients determined for the non-coal data are very similar to the 

gradients found for the coal data. The exception is India where the 

coal data set is very small. 

Table 5. Stress prediction parameters for equation 10 

determined for the individual non-coal regions using the 

unified analysis regression technique.

 n 

Depth gradient 

B1 (psi/ft 

(MPa/m)) 

Modulus 

factor 

B2(10-3) 

U.S./Can non-coal 115 1.51 (0.034) 0.21 

N. Europe non-coal 47 0.98 (0.022) 0.21 

South Africa. non-coal 14 0.95 (0.021) 0.21 

India non-coal 16 1.54 (0.035) 0.21 

*The constant (B0) was not statistically significant at the 95% 

level. 

Figure 11 is a type of “residual plot” that helps evaluate the 

validity of the regression equation. It compares the maximum 

horizontal coal mine stresses predicted by table 4 with the 

measured ones for all regions except Australia. The Australian 

data appears on figure 12. Both figures show that the residuals 

(discrepancies) are quite evenly distributed for all regions and 

stress levels, which indicates that the regression results are valid. 

0 1000 2000 3000 4000 5000 

Predicted Stress 

7000 

US-East 

6000 US-West 

UK/Germany 

5000 SAfrica 

India 

4000 prediction 

3000 

2000 

1000 

0 

Measured Stress 

Figure 11. Residual plot for table 4, non-Australian data. 

Figure 13 plots the coal stress measurements against the depth 

of cover. Two predicted stress gradients derived from Table 4 are 

shown in the figure, in addition to the vertical stress gradient. The 

gradient for the “normal” grouping, which includes the eastern US, 

Australia, South Africa, and India, is based on an average gradient 

of 1.6 psi/ft (0.037 MPa/m) and an average modulus of 2.5*106

 psi 

(18 GPa). The “low gradient” grouping includes the western U.S. 

and the UK and German coalfields. The average stress gradient 

for this group is 0.93 psi/ft (0.021 MPa/m) and the average 

modulus is 3.2*106 psi (22 GPa). 

Analysis Using Modulus and Tectonic Strain Factor (TSF): A 

number of analyses were conducted using model represented in 

equation (7), and employing the TSF concept as defined by 

Dolinar (2003) and Nemcik et al. (2005). In these analyses, when 

only the modulus was used as the dependent variable, the rsquared values were not much different than those obtained when 

modulus was regressed against SHmax (less than 0.30 for the coal 

data, and less than 0.2 for the non-coal data). In an alternative 

analysis, individual TSF values were determined for each region. 

This analysis found the highest TSF in NSW, with a value of about 

0.85, while the TSF determined for the Queensland, U.S., and 

European coalfields was about 0.45. The r-squared for this 

analysis was only 0.50, however, considerably lower than the 0.69 

obtained with table 4. 

Analysis of the minimum principal stress, Shmin: No data was 

available from Australia for analysis of the minimum principal 

stress. The results of the analyses for the other regions are shown 

in Table 6. The model that was employed is shown in equation 11: 

 Shmin = B0 + B1 (Depth) + B2 (Modulus) (11) 

The same “unified” regression technique used to obtain tables 

4 and 5 was used in the analysis. The r-squared for the regression 

using the coal data was 0.54, and it was 0.67 for the non-coal data. 

The constant was not statistically significant in either equation. 
 

 

 

 

 

 

Table 6. Stress prediction parameters for equation 10 

determined for the individual non-coal regions using the 

unified analysis regression technique. 

n 

Depth gradient 

B1 (psi/ft 

(MPa/m)) 

Modulus 

factor 

B2(10-3) 

East U.S. coal 42 1.34 (0.030) 0.15 

West U.S. coal 20 0.56 (0.013) 0.15 

UK/Ger coal 52 0.42 (0.009) 0.15 

South Africa coal 22 0.20 (0.005) 0.15 

India coal 5 0.42 (0.009) 0.15 

U.S./Can non-coal 115 0.89 (0.020) 0.12 

N. Europe Non-Coal 47 0.48 (0.011) 0.12 

South Africa non-coal 14 0.44 (0.010) 0.12 

India non-coal 16 1.03 (0.023) 0.12 

7000 

6000 

 Measured Stress (psi) 5000 

4000 

3000 

2000 

1000 

0 

NSW 

QLD 

prediction 

0 1000 2000 3000 4000 5000 6000 

Predicted Stress (psi)

Figure 12. Residual plot for table 4, Australian data only. 

MEASURED STRESS (PSI) 

0 1000 2000 3000 4000 5000 6000 7000 

0 

500 

1000 

1500 

2000 

2500 

3000 

DEPTH OF COVER (FT)

US-East 

US-West 

UK/Ger 

SAf 

Vertical Stress 

NSW 

QLD 

India 

Norm al Gradient 

Low Gradient 

3500 

4000 

Figure 13. The coal stress measurements data base, showing predicted ranges of stress for regions with “normal” and “low” 

depth gradients (see text for more details). 

DISCUSSION 

At the beginning of the last section, several predictions were 

made about the horizontal stress measurements based on the WSM 

“critically stressed crust” model. In nearly every instance, the 

prediction was confirmed by the analysis: 

•	 The depth was as important as the modulus in predicting 

the horizontal stress in the coal mine data set, and it was 

a much better predictor in the non-coal data set. When 

both factors were combined, the accuracy of the 

predictions improved significantly. 

•	 The calculated depth gradient of 1.6 times the vertical 

stress for the eastern U.S., and 1.4 times the vertical 

stress in the Bowen Basin, was within the range of what 

was predicted for coalfields located in stable, a-seismic, 

mid-plate areas. The depth gradient that was calculated 

for the northern European coalfields of the UK and 

Germany was a little lower than expected, but even there 

it was still 0.9 times the vertical stress. 
 

 

 

 

 

 

 

 

 

 

 

 

 

•	 The greatest depth gradient was found to be in the 

seismically active compressive regime of the Sydney 

Basin, and one of the lowest depth gradients was in the 

active extension regime of the western U.S. coalfields. 

It is reassuring that the depth gradients that were determined 

empirically seem to match up well with those that have been 

identified in the deep crustal measurements made by the WSM. 

The findings of this study indicate that we neither have to look for 

nor explain a discontinuity in stress at the transition between the 

“near surface” and the “deep” crust. 

The study also confirmed that the “excess stress” is one 

significant component of the near-surface stress that does not fit 

directly into the WSM model. The excess stress typically adds 

500-1,500 psi (3.5-10.5 MPa) to the horizontal stress that would be 

predicted by the depth gradient alone. At depths of less than 

1,000 ft (300 m), the excess stress is usually responsible for at 

least 50% of the measured maximum horizontal stress. 

Some important characteristics of the excess stress that were 

observed in the analysis are: 

•	 Its magnitude, averaging about 1,000 psi (7 MPa), is 

quite constant over a broad range of independent data 

sets; 

•	 Its magnitude does not noticeably increase with depth. 

•	 The value of the modulus factor was about half as large 

in the non-coal data set as it was in the coal data set. 

However, since the average modulus in the coal data set 

is about twice the average non-coal modulus, the overall 

excess stress is about the same in both geologic regimes. 

•	 The modulus factors determined from the analysis of the 

minimum principal stresses are consistently about half as 

large as those calculated for the maximum principal 

stress. In other words, the modulus effect is just as 

anisotropic as the vertical stress gradients. 

Taken together, these observations indicate that the source of 

the excess stress is likely the same current tectonic regime that is 

responsible for the other aspects of today’s stressfields, not 

“residual stress” or some other cause that would likely vary from 

locality to locality. One possible explanation is that while the 

“critically stressed crust” model normally assumes that the 

cohesion on fault planes is very small relative to the stresses at 

depth, near the surface that assumption may not be valid (Zoback 

et al., 2003). The cohesional strength of critically stressed faults 

may be enough to allow the crust to carry 1,000 psi (7 MPa) of 

stress more than would be the case if the cohesion actually was 

zero. 

One further observation is that the modulus factor was more 

important to predicting the stress in the coal mine data set than it 

was in the non-coal data. In fact, for the coal data, the modulus 

was almost as powerful a predictor of the stress as the depth. It 

makes sense that in layered, sedimentary coal measure strata there 

would be a strong tendency for the stiffer rock units to carry more 

load. On the other hand, the horizontal strain within the rock is 

certainly not, in general, constant regardless of depth. 

While the stress prediction equations that were developed 

during this study are quite robust statistically, it should be 

recognized that there are significant unexplained variability in the 

data. Greater variability would certainly be expected near the 

surface in mountainous terrain, where topographic effects are 

likely to be substantial. In situ stresses are also likely to be less 

predictable in seismically active coalfields like NSW and the 

western U.S.. Site specific stress measurements are still the only 

technique that can provide assurance of the local stress conditions. 

A final comment is that there now seems to be no justification 

for employing the “Poisson’s effect” or other static earth theories 

in any aspect of the analysis of in situ stress. The horizontal 

stresses measured underground are caused by the large plate 

tectonic forces that are currently being carried by the earth’s crust. 

Poisson’s effect can play a role when loads are actively applied to 

the ground, as in a longwall tailgate (Frith and Colwell, 2006), but 

there is no theoretical or empirical basis for using it to explain any 

aspect of the in situ stresses that develop over geologic time. 

CONCLUSIONS 

Using the largest data base of in situ stress measurements from 

coal mines ever assembled, this study found that the data fits the 

“critically stressed crust” model based on plate tectonics theory 

surprisingly well. Past studies had already shown that plate 

tectonics explained the orientation of the stresses observed 

underground, now it seems it explains their magnitudes as well. 

In particular, the maximum horizontal stress was found to increase 

with depth at rates of approximately 0.8-2.0 times the vertical 

stress. The study also identified a second component of the 

measured stress, called the “excess stress,” which adds 

approximately 500-1,500 psi (3.5-10.5 MPa) to the total. The 

excess stress is apparently independent of the depth. 

The study makes a major contribution by providing equations 

for estimating the maximum horizontal stress in several major 

coalfields around the world. More importantly, the study provides 

a framework for understanding the source of the horizontal 

stresses encountered underground every day. By showing that 

these “near surface” stresses are tied to those found in the deep 

crust, it links the worlds of rock engineering and geophysics. 

Hopefully, future research can build on this foundation to develop 

better tools for understanding and predicting in situ stress, and for 

using those predictions to design safer underground structures. 

Disclaimer 

The findings and conclusions in this report have not been 

formally disseminated by the National Institute for Occupational 

Safety and Health and should not be construed to represent any 

agency determination or policy. 

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