CRITICAL REVIEW OF NUMERICAL STRESS ANALYSIS TOOLS FOR DEEP COAL 

LONGWALL PANELS UNDER STRONG STRATA 

M. K. Larson, NIOSH, Spokane, WA 

J. K. Whyatt, NIOSH, Spokane, WA 

ABSTRACT 

Proper employment of numerical stress analysis design tools is 

based on the demonstrated ability of a model to capture key elements 

of the geologic site model and accurately simulate how these elements 

interact with a mine design. While these tools have progressed 

markedly, they are, at heart, a gross simplification of the abundant 

complexity of a natural setting and its response to mining. A generic 

deep longwall site model was developed that includes aspects of the 

geology of deep coal mines in the Wasatch Plateau and Book Cliffs 

coal fields of Utah. The site model contains a set of common features 

and observations of how these features typically respond to mining. 

This site model was the basis for evaluating use of empirical, boundary 

element and volume element stress analysis tools to analyze the 

distribution of stress around a deep longwall panel. More specifically, 

this evaluation examined shifting of stress to panel abutments and gob, 

distribution of stress in the abutment, and deformation and failure of 

bridging strata. Measurements of abutment stress changes at two 

sites in the Wasatch Plateau region were used to illustrate model 

calibration. Overall, these comparisons highlight the considerable 

differences between methods. Volume element tools can incorporate 

considerable detail and have fewer underlying assumptions, but this 

detail carries a considerable computational cost. Boundary element 

tools are much more efficient. But this efficiency also comes at a cost 

of added assumptions. These assumptions were challenged by the 

presence of a strong sandstone unit in the overburden, leading to 

boundary element results that depart significantly from volume element 

results. Empirical rules are the simplest, but are even more burdened 

by assumptions, many of which are implicit in underlying cases. 

Insight into the nature and impacts of underlying assumptions in each 

method is essential to proper use of results in mine design. 

Disclaimer: The findings and conclusions in this report are those of 

the authors and do not necessarily represent the views of the National 

Institute for Occupational Safety and Health 

INTRODUCTION 

The application of numerical stress analysis tools to ground 

control and mine design holds great promise but can also provide an 

unwarranted confidence in results. It is important that both 

practitioners and those who review and implement design based on 

numerical models have insight into the strengths and weaknesses of 

these tools. This report describes a study undertaken by the National 

Institute for Occupational Safety and Health (NIOSH) to more fully 

appreciate the application of numerical models in support of research 

on dynamic failure (bumps and bounces) in deep western coal mines. 

The study is based on a simplified generic site model with 

features typical of the Book Cliffs and Wasatch Plateau coal fields of 

Utah. This model includes a simplified stratigraphic column, estimates 

of relative properties of various strata, and most importantly, 

observations of typical large-scale response to mining of a single, 

isolated panel. In this study, the quantitative values of specific material 

properties are not as important as the ratios between values of the 

same property among strata members. These ratios can be greater in 

the Utah coal region than in the rest of the U.S [2]. Stress analyses 

are conducted with an empirical method, a boundary element program 

(LaModel) [16] and a volume element program (FLAC) [20]. Selected 

 

model results are compared, including (1) shifting of stress to panel 

abutments and gob, (2) distribution of stress in the abutment, and (3) 

deformation and failure of bridging strata. Stress measurements from 

two panels are used to demonstrate calibration of these models to 

specific sites. Subsidence and multiple-panel effects are also of 

interest but are not addressed in this paper. 

MODELING GOAL 

An important first step in developing a geomechanical model is 

careful consideration and definition of goals for the modeling exercise. 

In this case, the goal is to compare performance of empirical, boundary 

element and volume element analysis methods for a site model typical 

of deep western coal mines—that is, a coal mine under an interval of 

strong, massive strata within softer, weaker strata. A simple case is 

chosen--mining of a single panel surrounded by solid abutments (or 

very large barrier pillars). The case is further simplified by considering 

only a two-dimensional cross section located near the middle of a long 

panel. Only subcritical panels are considered in order to study stress 

transfer to abutments that might result in bump-prone conditions. 

Mining of a panel undercuts strata, causing caving and 

redistribution of vertical and horizontal stress. Some combination of 

bulking of caved material and subsidence of overburden usually result, 

allowing the caved material (gob) to support some portion of the weight 

of overlying strata. The remaining weight is carried to panel abutments 

by an arch or bridge in the overlying strata. Stress in the abutments is 

increased by this added weight, an increase often described as the 

mining-induced stress. The magnitude and distribution of mininginduced stress can be estimated based on underground observations, 

empirical relationships, or models using a variety of numerical 

methods. Whatever the method, three key quantities are estimated. 

These are (1) the proportion of load transmitted to the abutments, (2) 

the distance that mining-induced stress is transmitted into the 

abutment1

 

1 In this report, the distance from the gob that mining induced stress 

is transferred onto the coal seam is referred to as the stress transfer 

distance. 

, and (3) the magnitude and location of peak stress in the 

abutment rib. These factors are illustrated in figure 1. 

OBSERVATIONS 

Panel mining methods, both pillar and longwall, rely on caving to 

limit stress transfer to neighboring pillars and abutments. The critical 

width is the span at which arching or bridging has failed to the surface 

and maximum subsidence obtained. It can be described as the panel 

width, with reference to figure 2, when the lines extending upward from 

each rib at angle β intersect at the surface; it represents the vertical 

section geometry where caving and abutment loading conditions that 

form the boundary between subcritical and super-critical conditions. 

Crushing of yield pillars, where they are used, more gradually transfers 

stress across the entries to the solid abutment. Thus, the effective 

span in these cases includes the entries and can span multiple panels 

if they are separated only by yield pillars. Indeed, due to the greater 

mining depth, the critical width in deep western coal mines is often 

several panels. For example, the critical width at the Deer Creek mine 

occurred with mining of the third panel [2]. 
 

 

 

Figure 1. Vertical cross section across the width of a panel showing 

important concepts of stress redistribution resulting from excavation of 

a single panel. 

Figure 2. Vertical cross section across the width of mined panels 

showing geometry of supercritical (left) and subcritical (right) panels 

after Heasley [17]. The subcritical geometry is used by Mark [31] in 

ALPS. 

Maleki [30] reviewed relationships between minimum pressure 

arch width and depth of overburden, including the Holland equation 

[19] and estimates based on his own measurements of gob stress. 

These are plotted in Figure 3 against 

an empirical relationship used in ALPS (Analysis of Longwall Pillar 

Stability [31], an empirical method of assessing the stability of gate 

road pillars in longwall mining). These criteria generate a wide range 

of arch widths, particularly at depth. 

Figure 3. Maleki’s pressure arch graph [30] with ALPS assumption 

curve added. 

The weight of bridging strata and strata that cantilevers over the 

cave is shifted to pillars and abutments surrounding the panel. Since

the maximum bending stress in beams with clamped ends are 

generally greater than in cantilevers [6;39], strong, thick strata beams 

often cave in large sections that may span the width of the panel. 

Completion of caving can substantially reduce the amount of weight 

that is shifted to the abutment. 

In observations, the full extent of stress transfer is not detectable. 

The distance at which neighboring openings are impacted by shifting of 

stress is a practical quantity, but that quantity may vary according to 

site conditions. Therefore, for purposes of comparison, the distance at 

which the calculated vertical stress on the seam returns to 50% of the 

premining stress will be used as a measure. This level is arbitrary, but 

is equivalent to a 50% increase in overburden, an increase that will 

certainly be evident in driving of an entry. Observations in the 

literature refer to a “detectable” increase in stress, when determining 

as the stress transfer distance [13]. 

Observations of stress transfer distances are often remarked 

upon in discussions of ground behavior in deep western coal mines. 

Generally, distances are considered notable because of their long 

length. These observations also hint at the importance of anticipating 

these distances. These observations include: 

• DeMarco et al. [11] discuss western U.S. longwall operations 

and report that units of strong, competent strata transfer 

“considerable abutment loads over relatively large 

distances.” 

• Koehler [23] reported stresses overriding a 46-m- (150-ft-) 

wide barrier pillar to cause bump events at the Sunnyside 

No. 1 mine (also discussed by Chen et al. [9]). The mine 

was under 610 m (2000 ft) of overburden with 55-m-thick 

(180-ft-thick) sandstone about 46 m (150 ft) above the coal 

seam. 

• Barron [5] described stress transferring from the longwall 

face in a Book Cliffs mine in strong strata. Stresses induced 

by a 150-m- (500-ft-) wide face caused tailgate pillars to 

“explode without warning” 90 to 150 m (300 to 500 ft) outby 

the face. 

• Gilbride and Hardy [12] found barrier pillars “as wide as 120 

m (390 ft) or more may be necessary for pillar stability and 

abutment protection when depths reach 900 m (3000 ft) or 

more.” 

• Maleki [29] found good performance for a 150-m-wide (490-

ft-wide) barrier in the Rock Canyon and Gilson seams with 

580 m [1900 ft] maximum overburden . 

• Goodrich et al. [13] found load transfer distances greater 

than 230 m (750 ft) at the Deer Creek Mine. A pillar burst 

and other stress-induced ground conditions were apparent in 

a developed gate road as mining passed in the previous 

panel, with a full panel of intact coal seam serving as a 

barrier. Stress measurements suggested a 13% increase in 

stress was transferred over the panel. 

EMPIRICAL METHOD 

Empirical methods have been proposed to estimate the 

redistribution of stress, including a scheme that has been integrated 

into the empirical design tools ALPS and ARMPS [31;33]. This 

approach begins with a simple estimate of the load transfer distance 

using the equation of Peng and Chiang [40]: 

 (1)

where: D = distance from the gob (ft), and 

H = depth of overburden (ft). 

That is, the influence of a panel on stress conditions in 

neighboring areas disappears entirely at a distance D. The additional 

load carried within this distance in each abutment of a subcritical panel 

can be calculated from a simple geometric model of arching 

overburden (figure 2). The angle β defines a triangular “arch” that 
bounds caving. The weight of caved material is transmitted through 

the gob to the panel floor. When H tan β reaches half the panel width, 

the arch reaches the surface and abutment loading reaches its 

maximum. This geometry is called the critical span. 

Mark [32] determined β at six sites in four mines in the eastern 

U.S. coal fields. He [31] identified three of the mines as the Kitt Mine 

(Lower Kittanning Seam), the Lynch Mine (Harlan Seam), and the 

Quarto #4 Mine (Pittsburgh No. 8 Seam). He found values of β ranging 

from 10.7° to 25.2°. He concluded that a value of β = 21° would yield 

appropriately conservative estimates of side abutment load for longwall 

pillar design for these and similar cases. Significantly higher values 

have been reported. For example, a similar geometric model applied 

to subsidence at the Deer Creek Mine in the Wasatch Plateau region 

of Utah (figure 4), showed a critical span of 1.6H, or β = 38.6° [3]. The 

National Coal Board uses a similarly high value of 35° [35]. These 

results indicate that 21° may not be conservative at sites that differ 

markedly from those studied by Mark. In this report, β = 21°, the 

recommended value for ALPS method, was used to calculate 

abutment stresses. 

Figure 4. Vertical cross section across the width of a mined panel 

showing critical mining widths from the National Coal Board (A) and 

Deer Creek Mine (B); after Allgaier [3]. 

For this empirical model of a subcritical panel, the additional load 

carried in each abutment is 

 (2)

where Lss = the total subcritical panel abutment load (from figure 2), 

P = panel width, and 

γ = density of the overburden. 

Mark [32] also proposed a function to describe the distribution of 

added stress in the abutment. This function is: 

(3)

where σa= the additional vertical stress at point in the abutment caused 

by mining, and 

x = distance of that point from the abutment rib. 

For supercritical panels, Ls (figure 2) is substituted for Lss in 

equation (3), where 

 (4)

Since the premining vertical stress on the seam is, on average 

 (5)

equation (3) can be normalized with respect to premining vertical 

stress for the subcritical case as follows: 

) (6) 

Figure 5 shows the effects of varying overburden depth ( H ) and 

panel width ( P ) with constant cave angle ( β ). Panel width affects 

normalized peak stress, whereas overburden depth affects the stress 

transfer distance (only). 

The location of a 50% increase in overburden stress is given by 

(7) 

In figure 5, the location of this point along each curve is indicated 

by its intersection with the dotted, horizontal line. 

GENERIC SITE MODEL 

A site model is the starting point for any numerical modeling 

study. The site model includes key elements of the geologic setting, 

mining plan, and mechanical properties. In this case, a simple generic 

site model was developed from a review of literature describing 

geology and geomechanical response of geology to longwall mining in 

the Wasatch Plateau and Book Cliffs coal fields of Utah. These 

regions are characterized by the presence of strong, thick sandstone 

strata, a characteristic that strongly contrasts with overburden in many 

other U.S. coal fields. Agapito et al. [2] describe these strata as “thick, 

competent overburden strata that tend to bridge and interlock” and 

“very competent and strong immediate roof and floor 

sandstone/siltstone strata.” This site model is not required, or 

considered, for empirical calculations of the previous section. Those 

results implicitly assume a site model that is similar to a case or cases 

underlying the empirical method. Significant departures from 

conditions in these underlying cases in application can compromise 

results. In other words, the empirical method has its own site model. 

Generic Model Description 

The generic site model was designed to be a simple but

representative geologic column typical of deep western coal mines.

Any massive stratum with bridging potential is labeled “sandstone” 

while weaker strata are labeled “shale” or “soft shale” (including 

mudstones, siltstones and even thinly bedded sandstone). Typically, 

floor and overburden include strong, stiff and massive sandstone 

members. The immediate roof can be sandstone or shale. An 

idealized geologic column was formulated with a sandstone floor 

overlain by coal, an immediate roof of shale, a sandstone layer and 

then soft shale to the surface (figure 6). Panel width was set at 240 m 

(800 ft). A 610 m (2000 ft) thick sandstone floor was defined in 

volume-element models (the floor of a boundary element model is 

infinitely thick). 
Figure 5. Normalized abutment stress distribution curves (stress/ premining stress - 1). A, Variation of panel width; and B, variation of 

overburden thickness. The closely dotted line near the bottom of each 

graph indicates a 50% increase in vertical stress over premining 

stress. 

Figure 6. Geologic column of generic model with indicated thickness 

of members used in models. 

Properties set for each unit are listed in Table 1 (see Appendix). 

These properties were based on reports of laboratory properties on 

samples taken from the Book Cliffs/Wasatch Plateau area and 

properties used in models of specific sites in the region [1;14;21;25;27-

29;38]. The sandstone Young’s modulus was selected to be in the 

upper part of the reported range to emphasize the contrast with shale. 

The tensile strength of the immediate roof shale was chosen 

somewhat high to avoid numerical difficulties caused by localization of 

severely deformed zones resulting from very high stress 

concentrations in the volume element model. A cave and gob model 

was employed in some models to separately account for shale roof 

failure in tension. 

The sensitivity of model results to variations of thickness 

members of this column was explored by using three thicknesses of 

immediate roof shale (0, 3.0, and 15 m [0, 10, and 50 ft]) and four 

thicknesses of roof sandstone (3.0, 15, 61, and 150 m [10, 50, 200, 

and 500 ft]). For convenience in generating the volume-element grid, a 

joint interface was placed between each member. These interfaces 

were numerically “glued” to prevent slip. A ubiquitous joint model was 

used to allow slip and separation along shale bedding planes. 

MODEL DEVELOPMENT 

Implementation in a Volume Element Model 

The volume element approach discretizes individual volumes of

the rock mass, allowing for more detail in representing geology and 

mine geometry, and more complex constitutive behavior. This ability 

comes at the cost of much greater computational requirements. To 

compensate, many analyses consider only a two-dimensional section 

through the mine. The volume element approach is used by finite 

element and finite difference programs, in this case the program 

FLAC2D [20]. Three-dimensional models can also be run but are 

usually limited to a single panel because of computational limitations. 

Some of the most comprehensive three-dimensional modeling studies 

of coal longwall panels that have been published have addressed 

Australian coal mines (e.g., [4;22]). Kelly’s [22] models project the 

location and type of fracturing around a panel. Badr et al. [4] 

conducted a detailed study of yield pillar design for typical western 

longwall conditions. 

The extent of caving, where modeled, was assumed to propagate 

through the immediate roof shale to a bridging sandstone layer. This 

layer could then be tested for failure by implementation of a failure 

criterion. The gob roof as viewed in vertical section across the width of 

the mined panel was initially assumed to form an ellipse, whose 

horizontal axis was 50% larger than the thickness of the shale. 

However, preliminary model results showed shale above the curved 

gob line in the first model always failed. Therefore, caving of all 

immediate roof shale above the extracted portion of the seam is 

assumed. Stresses were initialized according to gravity loading, and 

the sides and bottom of the model were “rollered” (displacement fixed 

in the direction normal to the surface) in FLAC, creating a lithostatic insitu stress field. 

Gob stress was calculated and applied as nodal forces on the roof 

and floor of the extracted volume. The gob stress-closure equation 

followed the best-fit relation of Pappas and Mark [37] for shale to 

Salamon’s equation [42]. Nonlinear models lacking gob proved 
sensitive to the sudden application of loading, resulting in 

overestimates of plastic strain and yield. In this case, pre-excavation 

forces were applied to the gob roof and floor and incrementally 

decreased. 

Implementation in a Boundary Element Model 

The boundary element approach reduces the computational 

intensity of problems to manageable levels by reducing the problem by 

one dimension. This is accomplished by modeling the coal seam as a 

crack in an infinite elastic body (e.g., [15;43;46;47]). Depending on the 

formulation, the elastic body can be isotropic or anisotropic, or can be 

divided into a series of uniform elastic layers between frictionless 

interfaces. This last formulation is used by LaModel [16], the boundary 

element program used in this study. It has proven particularly useful in 

simulating the behavior of weak and highly stratified overburden found 

in many U.S. coalfields. A great strength of the boundary element 

method is its ability to fully consider the geometry of a mine, including 

mining of multiple seams, in relatively quick runs (compared to volume 

element methods). LaModel is one of the fastest of these codes; 

largely because it limits calculations to vertical loading (horizontal 

stresses are neither input nor calculated). Guidelines published by 

Heasley [18] for model development with LaModel were followed 

except where explicitly noted. 

One of these guidelines provides assistance in selecting layer 

thickness from the equation: 

(8) 

where E = elastic modulus of the overburden, 

υ = Poisson’s ratio of the overburden, 

Es = elastic modulus of the seam, 

h = seam thickness, 

d = extent of the coal yielding at the gob edge, and 

H = seam depth. 

One interesting feature of this equation is that the product of 

elastic modulus and thickness is a constant. That is: 

(9) 

Heasley’s calibration guidelines [18] produce values for E and t, 

but an equivalent result can be achieved by varying E and t according 

to equation 9. 

Often it is beneficial to use multiple modeling methods to study a 

problem. For example, one may wish to compare LaModel results with 

those from a model with a massive elastic overburden. This might be 

another boundary element program like MULSIM/NL (multiple seams, 

nonlinear) [46;47] or a simple FLAC model. In such a case, it is 

important to realize that properties that seem similar are not always 

equivalent. Overburden elastic modulus is a case in point, especially 

for LaModel, because of its dependence on lamination thickness. In 

fact, MSHA [34], based on Heasley’s formulation [15], showed how any 

combination of homogeneous and laminated moduli can be made 

equivalent for a particular panel width by choosing the lamination 

thickness (t), according to: 

(10) 

where t = lamination thickness, 

Ehomogeneous = Young’s modulus of a homogeneous overburden, 

Elaminated = Young’s modulus of a laminated overburden, 

L = half width of the longwall panel, and 

υ = rock mass Poisson’s ratio. 

Note that this relationship depends on panel width as well as 

material properties. Put another way, if panel behavior is welldescribed by an overburden mass modulus, the panel width must be 

considered when translating to a laminated layer modulus for LaModel. 

Equation 10 can be reconfigured to find the invariant overburden 

product (equation 9) for LaModel that is equivalent to a given mass 

modulus as follows: 

(11) 

Conversely, if a rock mass is well-described by LaModel, an 

equivalent overburden mass model should vary elastic modulus to 

keep the product of modulus and panel width constant. This 

relationship underscores how much the laminated overburden model 

departs from the homogeneous elastic body often employed in volume 

element and other boundary element methods. 

LaModel uses only a fraction of the parameters defined for the 

volume element model since fundamental assumptions limit the range 

of material behavior. For instance, roof interfaces have no strength, 

and the layers are elastic and cannot fail. Abutment and pillar yielding 

are defined by the empirical Mark-Bieniawski equation [33] applied to 

pillars. This both simplifies and limits this approach. Simplification 

speeds model development and execution. However, the user must 

understand how this limits the range of behavior that can be modeled. 

Overburden properties are defined in LaModel for a set of strata 

layers of uniform thickness separated by frictionless interfaces. All of 

the strata beams have the same properties. Thus, overall properties 

equivalent to the generic model layers have to be estimated. This was 

attempted in two ways – by finding an equivalent bending stiffness and 

by finding a weighted average for a given layer thickness. In the 

equivalent bending stiffness method, each member of the roof column 

was treated as a cantilever and sufficient force was applied at the end 

of the cantilever to achieve one unit of displacement. The required 

forces were combined and used in the cantilever displacement 

equation [39] to determine an equivalent Young’s modulus. Second, a 

weighted average was found, weighted according to member height. A 

range of layer properties and interval thicknesses were tested 

(summarized in table 2). Gob stress-closure was modeled using 

equations by Zipf [47] and Heasley [16], with parameters fit to match 

the Salamon equation used in the FLAC model. 

Table 2. LaModel overburden parameters. 

Condition Description

Equivalent overburden to FLAC model 

LaModel rock properties determined with both equivalent stiffness and 

weighted thickness methods for each layer 


thickness. 


LaModel layer interval 7.6, 15, 31, 61, 150, 305, and 610 m (25, 50, 


100, 200, 500, 1000, 2000 ft) 

MODEL RESULTS 

Three key model results are (1) shifting of stress to panel 

abutments and gob, (2) distribution of stress in the abutment, and (3) 

deformation and failure of bridging strata. These results are compiled 

and compared with in-mine observations and the empirical model 

underlying ALPS and ARMPS. 

(1) Shifting of Stress 

The proportion of stress that shifts to abutments is a basic model 

result and has a large impact on other results. Figure 7 shows the 

range of proportion of the overpanel weight that is shifted to the 

abutment by modeling tool. The range of results for FLAC and 

LaModel includes all cases of shale and sandstone thickness 

described earlier for a panel width of 240 m (800 ft) and overburden 

thickness of 610 m (2000 ft). The ALPS column shows the range at 

this same overburden thickness for a range of β from 21° to 38.6°, 

the latter value from Allgaier [3], where 21° and 38.6° are located at the 

bottom and top, respectively of the indicated range. The volumeelement results show very high transmission of stress to the abutment, 
implying minimal loading of the gob. LaModel results show the largest 

range of results. Thinner layers increase roof flexibility and, thus, shift 

load from abutments to gob. The empirical method is consistent with 

thicker layers in the boundary element model while both fall slightly 

short of abutment loading calculated with the volume element method. 

(2) Abutment Stress Distribution 

The distribution of stress shifted to the abutment is another

important result. Stress shifted to the immediate rib can drive yielding 

and, possibly, outbursting of coal. Stress carried deeper in the 

abutment may impact neighboring excavations and is a particular 

concern for excavation of new entries, etc. Typical results are 

illustrated in figure 8. In the figure, overburden elastic properties were 

determined by the weighted thickness method. Results calculated with 

overburden properties determined by the equivalent stiffness method 

were similar and, therefore, not included in the figure. One of the most 

powerful FLAC results is the contrast between an immediate thick 

sandstone roof and a shale interval in the immediate roof. The shale 

limits peak stress in the coal and increases depth of yielding, moving 

stress away from the abutment rib. The assumed stress distribution 

used in the ALPS method predicts a much lower peak stress, but this 

peak is located at the ribline (recall figure 5). 

The total area under the curve in an abutment stress plot is 

equivalent to the transferred load. Figure 7 shows that the stress 

transferred to the abutment should be greatest for the volume element 

model, and least for the most flexible boundary element cases. 

Figure 7. Chart showing the range of proportion of overpanel weight 

shifted to abutments for three modeling methods for a panel width of 

240 m (800 ft) and an overburden thickness of 610 m (2000 ft). The 

locations of numbers in the LaModel column represent average result 

for that layer interval, where the interval is in ft. 

This is 

not evident in figure 8, largely because the right-hand “tail” of the 

curves has been truncated. In the truncated portions of the curves, the 

volume element and thick layer boundary element models have higher 

stress levels. 

Figure 8. Vertical stress profile on abutment for the generic model 

case of 61 m (200 ft) of roof sandstone. Gob is modeled. A, No 

immediate roof shale; B, Immediate roof shale thickness is 15 m (50 

ft). 

The distance that these models can transfer significant stress 

levels was characterized by the location at which the mining induced 

stress surcharge decayed to 50% of the overburden stress. The 

ranges of the 150% total stress level is plotted for a number of cases 

by method in figure 9, and the range of the 200% total stress level is 

similarly plotted in figure 10 for comparison. 

Figure 9. Distance from gob where vertical stress on coal is 150% of 

premining vertical stress. Results include immediate shale thickness 

of 0, 3.0, and 15 m (0, 10, and 50 ft) and all LaModel layer intervals 

modeled. In these figures, the 

LaModel results with the smallest interface spacing plot at the bottom 

of the indicated range. LaModel results with interface spacing closer to 

overburden depth plot at the top of the indicated range. 
E 

Figure 10. Distance from gob where vertical stress on coal is 200% of 

premining vertical stress. Results include immediate shale thickness 

of 0, 3.0, and 15 m (0, 10, and 50 ft) and all LaModel layer intervals 

modeled. 

(3) Bridging Strata 

The behavior of strata bridging or arching over the mined panel 

determines how much load is shifted to the abutments and influences 

how this load is carried within the strata. Overlying strata will also 

deform as they are undercut, a relationship often plotted as a ground 

reaction curve. In pillared panels, the slope or stiffness of this curve, in 

comparison to the post-peak curve of the pillar, determines whether 

one mode or mechanism of pillar bump occurs (figure 11). A ground 

reaction curve can be developed in FLAC or LaModel, but not from the 

empirical relationships described earlier. 

Figure 11. Post peak bump model comparing the descending slopes 

of a pillar with a flatter ground. reaction curve. The area of the shaded 

region represents the energy released kinetically (after Crouch and 

Fairhurst [10]). 

Ground reaction curves have historically been determined by one 

of two methods. Starfield and Wawersik [44] used a perturbation 

method in which a strip or element is displaced or converged by a 

small specified amount. Equilibrium conditions are recalculated and 

the stress on the strip is determined. Local mine stiffness at that 

element is given by the decrease in stress per increase in compressive 

strain. In short, this method provides a single value for stiffness that 

corresponds to an infinitesimal closure of an entry or panel. 

Salamon [41] replaced a pillar by a jack exerting the same forces, 

and then incrementally lowered pressure in the jack to create a ground 

reaction curve that may vary in stiffness as pressure is reduced. He 

then applied a rigorous mathematical approach to determine stability. 

In this study, we used the approach of a jack which incrementally 

reduces the applied pressure to the roof and floor of the gob. A 

variation of this method was used in LaModel. In these calculations, 

the Young’s modulus of the gob elements was reduced in various 

stages to reduce stress transfer between the roof and floor. While this 

did not provide a uniform reduction of force, it did provide a nearly 

linear stiffness reduction that was close to the desired result. 

Ground reaction curves were determined at mid-panel with FLAC 

and LaModel for the case of no immediate roof shale and 15 m (50 ft) 

of roof sandstone. In the case of FLAC, vertical stresses on the floor 

and roof of the cave were determined before excavation and stored in 

arrays. These stresses were applied in place of the excavated gob 

and then incrementally decreased. The model was allowed to come to 

equilibrium in each increment. Applied stress was then plotted against 

closure strain (figure 12). 

In LaModel simulations, Young’s modulus of the coal was 

incrementally reduced to simulate an incremental reduction of applied 

stress to the excavation roof and floor. When the layer interval was set 

low (for example, 7.6, 15 or 31 m [25, 50, or 100 ft]), it was possible for 

the closure to exceed seam height, at which point the curve was 

terminated. 

Figure 12 shows examples of these ground reaction curves. The 

initial ground reaction curve stiffness calculated with FLAC is similar to 

that from LaModel for layer thicknesses of 150 m, 305 m, and 610 m 

(500 ft, 1000 ft, and 2000 ft). However, the FLAC model stiffness 

changes suddenly as stress is almost entirely removed, indicating 

strata collapse (the “elbow” in this curve in Figure 12). Such collapses 

can be an important failure mechanism where overburden includes 

strong strata (e.g., [7;45]). The corresponding state of failure 

calculated by FLAC for this case (no immediate roof shale and 15 m 

[50 ft] of roof sandstone) is shown in figure 13. The sandstone failed in 

tension, and the soft shale failed significantly, both in shear and 

tension along ubiquitous joints. In this case, the sandstone in the roof 

likely prevented caving completely to the surface. It is interesting to 

note that the angle of failure was 16° to 18°, similar to the average 

ALPS angle, β = 21°. In addition, the failure surfaces tended to 

break horizontally along the ubiquitous joints. 

LaModel cannot follow this type of behavior since it is limited to 

elastic overburden behavior. Thus, it may be necessary to check for 

possible collapse when using LaModel. For example, Newman [36] 

verified his LaModel result by confirming that the maximum selfsupporting span of a strata beam was greater than panel width. 

MODEL EVALUATION 

The most important conclusion that can be drawn is that the 

empirical, volume element and boundary element methods reviewed 

differ in the ranges of behavior that can be accessed. This is 

especially evident where failure of bridging overburden might occur. It 

also applies to any impact of horizontal stress which is not modeled by 

the LaModel program. The analyst must insure that all important 

behavior modes are considered, although this consideration may make 

use of a variety of calculations and models. 

A second conclusion is that a comparison to real ground behavior 

is needed to focus the modeling effort. In this study, a wide range of 

results were produced from small variations within a simple site model 

and choice of analysis method. Ground behavior information can 

include observations, like reports of long stress transfer distance, or 

direct measurements. Quantitative data might include pressure 

measurements made in the gob or abutment. Gob measurements are 

relatively rare. Those that have been made indicate that gob can carry 

significant loads [8;26], particularly where vertical joints are frequent 
[26]. Of course, vertical joints were not explicitly considered by any of 

the models considered here. Thus, their contribution can only be 

accounted for by “calibrating” models. 

Figure 12. Ground reaction curves at mid-panel for cases of no 

immediate roof shale and roof sandstone thickness = 15 m (50 ft). 

Figure 13. State of zones in FLAC model showing failure of 

sandstone and soft shale for case of Mohr-Coulomb sandstone (15 m 

[50 ft]) and coal (1.5 m [10-ft]), and complete excavation of coal. The 

legend indicates whether a zone is elastic, has yielded or failed, has 

yielded or failed in the past but is currently elastic, and kind of yield or 

failure. This view does not show the entire model. 

Borehole pressure cell measurements in pillars and abutments 

are easier to make and are somewhat more plentiful. A calibration is 

demonstrated for two such cases from the literature in figures 14 and 

15 [5;24]. These cases contain gate road entries not included in the 

models. However, aside from local perturbations around entries, each 

can be optimized to provide the best possible approximation of the 

measured stress distribution. In this case, the quality of fit increases 

with complexity from empirical, to boundary element, to volume 

element, which is hardly surprising. The choice of model and model 

parameters would follow based on closeness of fit and the relative 

effort required by alternative methods. 

Figure 14. Mining-induced stress around the 6th Right gate roads at a 

Utah Mine [24]. A, ALPS assumed stress distribution function and 

measurements; B, LaModel results, including calibrated models with 

measurements; and C, FLAC results with measurements. 

A third conclusion is that, for this generic site model, there is no 

“equivalent” LaModel boundary element model that follows easily from 

a volume element model with a strong stratum. This conclusion arises 

from strong differences in the way overburden is modeled by these

methods. One reflection of this difference, is that the product of 

lamination thickness and modulus, not the modulus itself, is a key input 

to LaModel. Note that for each calibration case in figures 14 and 15, 

two LaModel runs with variations of overburden modulus and 

lamination thickness that satisfy equation (9) have identical plots. As 

such, the default layer thickness of 50 ft can be used for all models, 

provided appropriate adjustments are made to overburden modulus. 

This may, however, result in a modulus that departs markedly from a 

realistic range of properties as determined by other methods. 

DISCUSSION AND CONCLUSIONS 

Empirical, boundary element, and volume element modeling 

programs are often used in assessing the performance of mine plans. 

The selection of a modeling tool is important and essential in the 

modeling process. Making the right choice depends on understanding 

the relative merits of these tools and their abilities to simulate ground 

response to mining in the context of a particular site – in this case, a 

generic site model typical of deep western coal mines. The most 

remarkable element of this model is inclusion of an interval of strong, 

bridging strata. 
Figure 15. Mining-induced stress around the 9th East gate roads at a 

Utah Mine [5]. A, ALPS assumed stress distribution function and 

measurements; B, LaModel results, including calibrated models with 

measurements; and C, FLAC results with measurements. 

The most important consideration is whether a design project 

considers all possible modes of ground response to mining. For 

empirical methods, this concern is best addressed by comparing 

conditions at underlying cases with those for the application. A 

mismatch may compromise empirical method results. It is also 

important to recognize and, if necessary, compensate for the fact that 

the boundary element program, LaModel, does not consider the effects 

of horizontal stresses and failure of the overburden. 

This work also showed that the assumptions and features of 

these volume and boundary element programs differ too markedly for a 

model constructed in one to be “converted” into the other through 

application of the same input parameters. Thus, care must be taken 

when taking input parameters from past analyses using different 

modeling programs. New models should be calibrated to field 

observations and, ideally, measurements of critical behavior. 

Finally, for the site model considered here, the degree of 

sophistication correlated well with ability to match borehole pressure 

cell measurements at a typical site. This is not necessarily a general 

result, but is consistent with the greater level of detail available in 

volume element models. The fidelity required of a model is necessarily 

constrained by uncertainty in input and calibration measurements. 

However, it is essential that all significant ground behaviors, especially 

failure modes, are considered in the course of a design project. 

Ignoring a possible failure mechanism simply because it is ruled out by 

a program’s underlying assumptions is not acceptable practice. At the 

same time, the level of effort required to achieve desired detail with a 

volume-element model may require much larger preparation and 

computation times, possibly making this approach impractical or, in 

some cases, practically impossible. FLAC models described here took 

up to 14 days to solve with a modern workstation. In such cases, 

application of a variety of simpler models and calculations addressing 

various aspects of panel behavior may be advantageous. 

ACKNOWLEDGMENTS 

The authors would like to thank Dr. Keith Heasley for discussions 

that strengthened this paper. 

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APPENDIX 

Table 1. Estimated properties of materials as used in the generic model. 

Property Soft shale Sandstone Shale Coal 

Young’s modulus, GPa . . . . . . . . . . . . . . . . . . . . . 10.3 34.6 13.8 3.45 

Young’s modulus, psi . . . . . . . . . . . . . . . . . . . . . . 1.50 million 5.00 million 2.00 million 0.50 million 

Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.35 0.25 0.35 0.30 

Density, kg/m3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310 2310 2310 1280 

Density, lb/ft3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 144 144 80 

Cohesion, MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 33.8 20.5 7.09 

Cohesion, psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2970 4910 2970 1030 

Friction angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30° 25° 30° 30° 

Dilation angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5° 5° 5° 5° 

Tensile strength, MPa . . . . . . . . . . . . . . . . . . . . . . 2.07 5.03 6.89 2.07 

Tensile strength, psi . . . . . . . . . . . . . . . . . . . . . . . . 300 730 1000 300 

Ubiquitous joint angle . . . . . . . . . . . . . . . . . . . . . . 0° -- 0° -- 

Ubiquitous joint cohesion, MPa . . . . . . . . . . . . . . . 1.4 -- 1.4 -- 

Ubiquitous joint cohesion, psi . . . . . . . . . . . . . . . . 200 -- 200 -- 

Ubiquitous joint friction angle . . . . . . . . . . . . . . . . . 25° -- 25° -- 

Ubiquitous joint dilation angle . . . . . . . . . . . . . . . . 5° -- 5° -- 

Ubiquitous joint tensile strength, MPa . . . . . . . . . . 0.83 -- 0.83 -- 

Ubiquitous joint tensile strength, psi . . . . . . . . . . . 120 -- 120 --