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Probabilistic modeling using bivariate normal distributions for identification

of flow and displacement intervals in longwall overburden

C. O ̈ zgen Karacan n, Gerrit V.R. Goodman

NIOSH, Office of Mine Safety and Health Research, Pittsburgh, PA 15236, USA

article info

Article history:

Received 27 November 2009

Received in revised form

18 June 2010

Accepted 14 August 2010

Available online 1 September 2010

Keywords:

Longwall overburden

Gob gas ventholes

Bivariate normal distribution

Conditional probability

abstract

Gob gas ventholes (GGV) are used to control methane emissions in longwall mines by capturing

it within the overlying fractured strata before it enters the work environment. In order for GGVs to

effectively capture more methane and less mine air, the length of the slotted sections and their

proximity to top of the coal bed should be designed based on the potential gas sources and

their locations, as well as the displacements in the overburden that will create potential flow paths for

the gas.

In this paper, an approach to determine the conditional probabilities of depth-displacement, depthflow percentage, depth-formation and depth-gas content of the formations was developed using

bivariate normal distributions. The flow percentage, displacement and formation data as a function of

distance from coal bed used in this study were obtained from a series of borehole experiments

contracted by the former US Bureau of Mines as part of a research project. Each of these parameters was

tested for normality and was modeled using bivariate normal distributions to determine all tail

probabilities. In addition, the probability of coal bed gas content as a function of depth was determined

using the same techniques. The tail probabilities at various depths were used to calculate conditional

probabilities for each of the parameters. The conditional probabilities predicted for various values of the

critical parameters can be used with the measurements of flow and methane percentage at gob gas

ventholes to optimize their performance.

Published by Elsevier Ltd.

1. Introduction

For gas control during longwall mining, drilling vertical gob

gas ventholes (GGV) ahead of mining from the surface is a

common technique in the US to capture methane emissions

within the overlying fractured strata before it enters the work

environment. Although the drilling practices of GGVs may change

based on local conditions, most gob gas ventholes are drilled to

within a short distance, 30–40 ft, of the coal bed being mined and

cased with steel pipe. Commonly, the bottom section of the casing

(generally about 200 ft) is slotted and placed adjacent to the

expected gas production zone in the overburden strata. GGVs

become productive as mining advances under the ventholes and

the gas-bearing strata that surround the well fractures and

establish preferential pathways for the released gas to flow

towards the ventholes [1]. In order for GGVs to capture more

methane and less mine air, the length of the slotted sections and

their proximity to top of the coal bed should be designed based on

the potential gas sources, their locations and the fracturing and

displacements in the overburden that will create potential flow

paths for the gas.

Palchik [2] reported that, based on the measurements in the

Donetsk coal basin, the thickness of the fractured zone can vary

up to 100 times the height of the mined coal seam depending on

the size of the panel and the geology and geomechanical

properties of the layers. Hasenfus et al. [3] divided the overlying

strata into four main zones of specific properties: a gob, a highly

fractured zone, a composite beam zone and a surface layer.

Among these, gob and highly fractured zones are the prime

interests in term of gob gas productivity since they have high

permeabilities and form pathways for gas transport towards the

ventholes or towards the mining environment, provided that

there are gas emission sources within these intervals.

The characteristics of fracturing and the subsidence of overburden are revealed through predictive techniques and field

studies [4,5]. Fawcett et al. [6] described a theoretical investigation into the zones of increased hydraulic conductivity caused by

rock failure above a longwall panel. They correlated predicted

failure heights with existing experimental values. Whittles et al.

[7,8] conducted studies on the effects of different geotechnical

factors on gas sources and gas flow paths for UK longwall

operations. They studied how roof geology and its interactions

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijrmms

International Journal of

Rock Mechanics & Mining Sciences

1365-1609/$ - see front matter Published by Elsevier Ltd.

doi:10.1016/j.ijrmms.2010.08.006

n Corresponding author. Tel.:+1 412 386 4008; fax: +1 412 386 6595.

E-mail address: cok6@cdc.gov (C. O ̈ zgen Karacan).

International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
Author's personal copy

with boreholes may cause the deformation and closure of the

boreholes drilled for methane control.

It is difficult to predict production performance of gob gas

ventholes due to the involvement of multiple influential factors and

due to the unknowns related to strata fracturing and gas emission

intervals. Main factors and unknowns are drilling depths of the

boreholes, size of the casing, vacuum generated by exhausters,

atmospheric pressure conditions, location of the boreholes on the

panels, strength and thickness of formations in the overburden.

There are CFD (computational fluid dynamics) and reservoir

simulation based models [9,10] but these models are complex and

the outputs are only as good as the input data provided to them. The

latter is a major constraint before a realistic output can be obtained

from these models and predictive studies can be performed.

In order to come up with realistic but simpler solutions to GGV

productivity and design issues, Palchik [11] proposed a mathematical model for estimation of productivity of gob gas ventholes

associated with coal mining in the Torezko–Snezhnyanskya coal

district. He established a relationship between methane emission

from GGVs and the duration of well production using the

Gaussian error distribution function and the Gaussian density

function. However, he noted that in the mining area he studied,

90% of total gob gas originated from the mined coal itself and

the contribution from additional gas resources over the extracted

seam was insignificant. Thus, this analytical approach may not

represent situations where the gob gas generally originates from

the fractured strata overlying the extracted seam.

Karacan [12] developed an artificial neural network (ANN)

based model using field data to predict GGV production rates and

methane concentrations for situations where most of the gas came

from overlying strata and associated gas sources. Although this

model was successful in addressing some of the GGV design issues

and in establishing sensitivity studies, it treated the gas as a bulk

quantity. It did not differentiate between different intervals that

might be contributing to total flow to define the design and

location of different slotted casing lengths. Also, Karacan [13] used

multi-rate drawdown well test analyses techniques to evaluate the

reservoir properties of fractured gob using rate and pressure data

of GGVs. Although, this method is highly effective in reconciling

gob parameters, it is insufficient to define where is gas is coming

from due to representing the production interval as a single entity.

Despite the improvements in analytical and numerical modeling

approaches, experience suggests that it is still difficult to properly

design the GGVs and to accurately predict their productions. It is not

uncommon that the predictions may under- or overpredict the

actual values by a factor of one or two. One of the key causes of these

erroneous predictions is the difficulty in understanding the quantity

and location of the gas sources in the overlying strata.

In this study, a probabilistic approach to determine the depthdisplacement, depth-flow percentage, depth-formation and

depth-gas content is proposed using bivariate normal distributions. Conditional probabilities are calculated for fixed values of

different variables with respect to height from coal seam and

depth from surface and used to understand the flow and strata

behavior and to design casing lengths and their setting depths

accordingly to optimize GGV performance.

2. Overlying formation in Northern Appalachian coal-mining

basin

This study was performed using the data obtained from the coal

mines located in Southwestern Pennsylvania and Northern West

Virginia of the Northern Appalachian Basin. Thus, a brief description

of formations that belong to this area and forming the fractured zone

during longwall mining is necessary. A detailed description of the

Fig. 1. General stratigraphic log of the formation overlying Pittsburgh seam. This

interval constitutes the caved and fractured zones of gob.

Fig. 2. Names and locations of the tested boreholes and the associated mine

properties.

28 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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strata layers, the properties of coal-measure formations overlying

the Pittsburgh coal seam and their reservoir characteristics are given

elsewhere [5,14]. However, for completeness of this paper and the

work presented, a brief overview is given in this paper too.

From coal mining and gas control perspectives, the Monongahela Group in Northern Appalachian Basin is the most

important. It is located within the Pennsylvanian age sediments

and encompasses the stratigraphy from the base of the Pittsburgh

coal bed to the base of the Waynesburg coal bed, extending to a

thickness 270–400 ft (Fig. 1).

The general coal measures in the Monongahela Group in

Greene County shown in Fig. 1, and their thicknesses from top to

bottom of the group, can be listed as: Waynesburg main coal bed

(5.9 ft), clay (2.9 ft), sandstone (19.7 ft), limestone (4.9 ft),

sandstone and shale (60 ft), Uniontown coal bed (1–3 ft), Upper

Great limestone-Benwood (17.7 ft), Sewickley sandstone and

shale (60 ft), Lower Great limestone-Benwood (55 ft), sandy shale

(12.2 m), Sewickley coal bed (1–6 ft), sandstone (9.8 ft), the

Fishpot limestone (17.7 ft), sandstone and shale (25 ft), Redstone

coal bed (1–4 ft), limestone (9.8 ft), Pittsburgh upper sandstone

(39.4 ft), shale (0 to 9.8 ft) and Pittsburgh coal bed (4.9–12.1 ft).

3. Data profiles used in probabilistic model development

3.1. Borehole tests for data compilation and flow characterization

The flow entry tests, strata separation determinations and

strata identifications were performed during a study reported in

[15]. That study was performed in coal mines of Northern West

Virginia, which are known for their gassy nature and use gob gas

ventholes to control underground emissions by capturing the

strata gas. During the course of this project 8 GGVs were drilled

over different coal mines working in the Pittsburgh coal seam.

Fig. 2 shows the ventholes and the associated mine properties

where that they were drilled.

Table 1 gives the summary of the reported completion details of

the ventholes, their initial gas production start, the displacement

and flow profiling measurements with respect to the face location.

This table also gives the general observations reported and

measurements taken during flow profile and displacement measurements, whose experimental details are given in the following

sections. This table shows that the general observations of strata

separation locations are in general agreement with the predictions

reported in various publications based on the strength of rock

layers and their density and gamma ray log readings [2,5,16].

In this paper, the data from flow, strata separation and core

identification measurements of all ventholes were digitized and

compiled to be presented in graphical form. These graphs of

measurements are given in the following sections. These data are

also used as the raw data for bivariate normal distribution

modeling that is presented in detail later in this paper.

3.1.1. Flow entry profiles

Gas flow entry locations in gob gas ventholes were obtained

with a pulse-type logger. This system contained a cylinder in

which a mixture of Kyrpton-85 and nitrogen was stored at an

Table 1

Summary (compiled from [15]) of the completion details of the ventholes as well as initial gas production start, displacement and flow profiling measurements with

respect to the face location. The lower part of this table also gives the general observations reported during flow and displacement (subsidence) profiling.

Venthole

name

Venthole completion Start of gas

production

Subsidence

surveya

Flow profile

surveya

1-M-1 ? 0 +50/+575 +310

Fox Drilled to 18-ft to Pittsburgh seam; 310-ft slotted casing +20 +350 +10/+350

L-2 Drilled to 18-ft to Pittsburgh seam; 334-ft slotted casing 0 +195 +195

3–14 Drilled to 20-ft to Pittsburgh seam; slotted casing to 20-ft above Sewickley seam +50 +110 ?

2-M-3 Drilled to 18-ft to Pittsburgh seam; 328-ft slotted casing +10 +270 +100

2-M-4 Drilled to 90-ft to Pittsburgh seam; Bottom 29-ft open hole +70 +145 +130

L-3 Drilled to 99-ft to Pittsburgh seam; Bottom 34.5-ft open hole 0 ? ?

2-M-6 Drilled to 82-ft to Pittsburgh seam; 238-ft slotted casing; Bottom 20-ft open hole +58 +58 +80

General observations during subsidence and flow profile measurements

Venthole

name

Subsidence test Flow profile test

1-M-1 Major strata separation at 125-ft above Pittsburgh seam close to top of Sewickley sandstone. Underlying

interval had additional separations

70% of the flow entered the hole within

120-ft above Pittsburgh seam. No flow

observed within 80-ft of the coal bed

Fox Strata separations above 280-ft of the mine. Major separation at 150 above the mine In the first test, little flow was observed. In

the second test majority of flow entered

near Sewickley sandstone and Uniontown

limestone

L-2 Separations were measured 270 ft above the mine. Large displacements were located just above Sewickley

coal

No flow near the mine with an increase in

flow 70 ft above the mine near a

limestone. Major flow entry occurred from

a 10-ft sandy shale

3–14 Two strata separations of 0.8-ft at 100 and 235 ft above the Pittsburgh coal seam Short lived production to determine the

flow entry profile

2-M-3 Strata movements near Waynesburg coal at 320 ft above the mine. Majority of separations occurred below 

109-ft above Pittsburgh coal

No flow existed near the mine

2-M-4 The displacement was found nearly linear with depth above Sewickley sandstone. At least 1.5 ft cumulative

strata separation 140-ft above the mine near Sewickley sandstone

All flow entered the hole at the top of

Sewickley sandstone, 110-ft above the

mine

L-3 Strata appeared to be broken over a zone of reaching 225-ft above the Pittsburgh seam ?

2-M-6 Separations of 0.2-ft occurred near Waynesburg coal and Sewickley sandstone 85% of flow entered the hole in a 100-ft

interval from a gray shale formation to the

Sewickley sandstone

a ‘‘?’’: Unreported values or conditions; ‘‘+’’: the distance of the face past the borehole.

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 29
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0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0

Displacement (ft)

Distance above the CB (ft)

1234

Fig. 4. Combined strata separations (exclusive of surface subsidence) obtained from all ventholes.

0

50

100

150

200

250

300

350

400

0

Fraction of flow

Distance above the CB (ft)

0.5 1 1.5

Fig. 3. Combined flow profiles obtained from all ventholes.

30 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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initial pressure of 800 psi. This amount of gas was enough

approximately for 250 pulses.

This device was suspended in the borehole from the gammaray tool with a four-conductor armored cable that permitted

activation of a solenoid valve to release the krypton gas mixture

as pulses. The downhole position accuracy of this device was

found to be 70.1 ft [15].

A pulse of 85Kr tagged nitrogen was released and the formation

gas which carried the tracer gas to the gamma-ray tool and its

arrival time were recorded. The system basically recorded the

transit time of a pulse of radioactive gas to travel a known interval

while being carried in the stream of produced gas. The recorder

was driven at a preselected constant rate and also recorded the

radiation anomaly created by the passing of the krypton. The

velocities were then determined with the known time and

distance [15], from which flow rates and flow percentages coming

from each flow entry point in the borehole could be calculated.

Fig. 3 shows the results of the combined flow profile, which

has been constructed by digitizing the data from measurements of

all ventholes. In order to be able to show all flow rates in the same

scale, the data were normalized and are shown as fractions of

flow. This figure shows that only a minor percentage of flow

enters the ventholes within 50 ft of the mined coal bed. A

significant increase in the flow entry occurs at approximately

100 ft above the mined coal seam, where Sewickley sandstone is

located. The percentage of flow originating at this interval varies

between 50% and 70% of the total flow at different ventholes. As it

is shown in Fig. 4 and discussed in the next section, this interval

corresponds to where major displacements or strata separations

occur in the overburden. However, it should be noted that this

interval may not necessarily correspond to the source of the gas,

but the location of the major flow pathways. Another observation

from Fig. 3 in conjunction with the face locations at the time of

flow entry measurements is that the position of the face does not

influence the flow profile drastically. Finally, Fig. 3 shows that

almost all flow enters the venthole below at about 250 ft from the

mined coal bed.

3.1.2. Displacement (strata separation) profiles during subsidence

In order to locate the displacements and to measure the strata

separations during subsidence, radioactive tagged shaped charges

were used in the gob gas ventholes. These charges were used in

the ventholes to implant radioactive markers into the formations

to measure the subsidence and the strata separation during

mining. 60Cobalt was used as the radioactive material in the charges.

The charges had the radioactive kernels glued to the liner in two

separate configurations. Half of the charges contained radioactive

kernels located inside the cone, at 1/3 of the distance from the

0

50

100

150

200

250

300

350

400

GSHGSH

LSLS

RDCB

SS

SWCB

SWSSSWSS

WBCB

XXXXXXXX

Strata Type

Distance above the CB (ft)

0

50

100

150

200

250

300

350

400

0

Strata thickness (ft)

Distance above the CB (ft)

50 100 150 200

Fig. 5. Strata type (A) and their thicknesses (B), with respect to distance from Pittsburgh coal bed, measured in the ventholes (GSH: gray shale; LS: limestone; SS:

sandstone; RDCB: redstone coal bed; SWCB: Sewickley coal bed; SWSS: Sewickley sandstone; WBCB: Waynesburg coal bed; X: mostly shales of various kind).

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 31
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apex to the base. The others contained kernels located at 2/3 of

the apex–base distance [15].

The charges were fired through tubing. It was found that the

markers actually did not contaminate the tubing for subsequent

measurements and were implanted into the formation with a

success rate of nearly 100% to mark the formations. This

technique ensured correct measurement of the formation movements and the locations of strata separations with gamma ray

logging [15].

Fig. 4 shows the results of the combined displacement profiles

that give the amounts of strata separations, which have been

constructed from measurements of all ventholes. The values

shown in this figure are exclusive of any surface subsidence since

the measurements were made using the well head as a datum and

the displacements calculated are subtractions of strata subsidence

and well head subsidence.

Fig. 4 shows that vertical displacements due to strata

separations can be effective as high as 300–350 ft from the

Pittsburgh coal seam. The magnitude of strata separations

between this and down to almost 200–250 ft above of the

Pittsburgh seam are generally less than 1 ft, mostly around

0.5 ft. The displacement trend in this interval is almost linear

and not a function of depth. However, at 200 ft above the

Pittsburgh coal bed and below, the separations starts to depart

from low displacement values and major strata separations in the

order of 2 ft can be measured. These high separations are mostly

100 ft above the mine where Sewickley sandstone occurs. This

observation and the displacement measurements are in line with

the flow entry measurements discussed in the previous section.

Displacement trend at and below 200 ft also follows a nearly

linear trend with a 451 relationship with depth.

3.1.3. Strata type and thickness profiles

Strata identifications and their thicknesses were evaluated

from core logs and electric logs. As discussed in Section 2, this

region hosts shales, three major coal seams, and limestone and

sandstone units. The thicknesses of almost all formations change

based on location, which is a characteristic of depositional

environments that has been created in lacustrine and swamp

conditions [14].

Fig. 5a and b shows main strata types (A) and their thicknesses

(B), with respect to distance from Pittsburgh coal bed, measured in

the ventholes. In these graphs, the pictorial information given in

[15] was digitized to obtain the depth and thickness information

and plotted for quantitative information. These figures show that

Waynesburg coal bed is around 350 ft from the Pittsburgh coal

bed and its thickness is 5 ft. At the 200 ft interval, gray shale is

dominant with a varying thickness of 20–30 ft. Sewickley coal bed

and Sewickley sandstone, on the other hand, are between 75 and

100 ft from the Pittsburgh seam with thicknesses of 5–6 and

25 ft, respectively. This zone is identified as the interval where

most of the displacement occurs and where most of the gas enters

the ventholes. The formations, whose thicknesses may be up to

0

200

400

600

800

1000

1200

1400

0

Total gas content (scf/ton)

Depth (ft)

All coal beds in Pennsylvania

All coal beds above Pittsburgh seam

Pittsburgh seam

100 200 300 400

Fig. 6. Gas content-depth relationship of all coal beds in Pennsylvania, Pittsburgh seam and overlying seams.

32 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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100–150 ft, are the limestone and shale units that are located

150–225 ft from the Pittsburgh seam. Formation thickness

information was further divided by the representative Young’s

modulus data of the formations to compound them with the

formation-type information.

3.2. Gas content profiles

There are two major sources for the gas produced from GGVs.

One of them is the mined coal bed itself, and the other is the

overlying strata and its potential gas sources. In the latter case,

coal beds with enough gas potential are the most likely sources of

gas. This suggests that if the gas content of the coal bed that is

source of the gob gas is high, the gas emissions to be captured by

GGVs can be high also.

In this paper, the gas contents of all coal beds given in [17] in

Pennsylvania were compiled in a database and all the coal beds

above the Pittsburgh seam and including the Pittsburgh seam

itself were classified. Coal seam gas content and depth relationship for these classes were plotted in Fig. 6. The majority of the

data plotted in Fig. 6 was obtained from high-volatile-bituminous-A type coals, some from medium volatile coals, and the gas

content-depth relationship generally shows an increasing trend

0

50

100

150

200

0

Quantile - Normal 

TOTAL GAS (scf/ton)

0

200

400

600

800

1000

1200

0

Quantile - Normal 

DEPTH (ft)

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

Total Gas

Depth

Flow Fraction

Height from CB

Displacement

Height from CB

Thickness/Y.Mod

Height from CB

Calculated critical value from tests

S-W Test A-D Test

Lilliefors J-B Test

Data pairs for b-variate

normal distributions

50 100 150 200 200 400 600 800 1000 1200

Fig. 7. Q–Q plots for flow gas content (A) and depth data (B) obtained from all GGVs listed in Table 1. Figure C gives the critical values calculated for all data sets using four

normality tests (Shapiro–Wilk, Anderson–Darling, Lilliefors and Jarque–Bera) to see if the data are from a normal distribution (null hypothesis) for a significance level 5%,

shown by the dashed line.

Table 2

Mean (m) and standard deviation (s) of each variable and the correlation

coefficients (r) between variable pairs used in calculating joint probability

distributions.

Data pairs Correlation

Flow (fraction) Distance from coal bed (ft)

m1⁄40.699 m1⁄4196.56 r1⁄40.906

s1⁄40.244 s1⁄4100.395

Displacement (ft) Distance from coal bed (ft)

m1⁄40.746 m1⁄4205.22 r1⁄4 0.676

s1⁄40.638 s1⁄479.367

Total gas (scf/ton) Overburden (ft)

m1⁄4122.700 m1⁄4588.837 r1⁄40.558

s1⁄454.766 s1⁄4233.564

Formation thick/E (ft/GPa) Distance from coal bed (ft)

m1⁄47.484 m1⁄4154.351 r1⁄40.117

s1⁄47.921 s1⁄498.254

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 33
Author's personal copy

with depth. However, it should be also noted that gas contents

show significant variations even at the same depth.

In this work, based on the two premises for the source of gob

gas, gas content-depth data of Pittsburgh coal and the gas contentdepth data of all other coal beds overlying the Pittsburgh seam

were used in modeling the source of the gas and its gas content.

4. Probabilistic modeling approach

4.1. Bivariate normal distribution—probabilities of multivariable

events

It is generally recognized that complex geological and hydrological events always appear to be multivariate events. Events of

this nature can best be characterized by a few correlated random

variables and by their joint probabilistic properties, rather than

single-variable analyses as these can only provide limited

assessment of these events [18].

Gas emissions from gob and methane production using gob gas

ventholes can be characterized as an event with multiple episodic

phenomena, which is controlled by multiple variables, such as

distance from mining environment, gas content of strata, strata

separations, etc. A thorough understanding of the multivariate

nature of gob gas emissions and capture requires analysis of the

joint probability of two or more correlated random variables. Such

an analysis characterizes the gob formation process and better

designs the gob gas venthole completion parameters, such as the

depth and length of the slotted casing.

The normal distribution is the most commonly used distribution

to model univariate data from a population or from an experiment

[19]. In order to extend this concept to the bivariate case, let us

assume we have two variables, X and Y, each of which is individually

normally distributed. Each of these variables can be transformed into

a standard normal random variable, which can be called as Z1 and Z2.

These variables can be shown by the relationships:

Z1 1⁄4 XmX

sX

ð1Þ

Z2 1⁄4 YmY

sY

ð2Þ

where mX and mY are the mean values of X and Y, and sX and sX are

0

0.5

1

1.5

2

2.5

3

3.5

4

P(X<0.1, Y<)

P(X<0.15, Y<)

P(X<0.2, Y<)

P(X<0.25, Y<)

P(X<0.3, Y<)

P(X<0.35, Y<)

P(X<0.40, Y<)

P(X<0.45, Y<)

P(X<0.5, Y<)

P(X<0.55, Y<)

P(X<0.6, Y<)

P(X<0.65, Y<)

P(X<0.7, Y<)

P(X<0.75, Y<)

P(X<0.8, Y<)

P(X<0.85, Y<)

P(X<0.9, Y<)

P(X<0.95, Y<)

Cumulative Probability

Probability (X<,Y<40 ft) Probability (X<,Y<100 ft) Probability (X<,Y<150 ft)

Probability (X<,Y<200 ft) Probability (X<,Y<250 ft) Probability (X<,Y<300 ft)

Probability (X<,Y<350 ft)

0

0.5

1

1.5

2

2.5

3

3.5

4

P(X>0.1, Y<)

P(X>0.15, Y<)

P(X>0.2, Y<)

P(X>0.25, Y<)

P(X>0.3, Y<)

P(X>0.35, Y<)P(X>0.40, Y<)P(X>0.45, Y<)

P(X>0.5, Y<)

P(X>0.55, Y<)

P(X>0.6, Y<)

P(X>0.65, Y<)

P(X>0.7, Y<)

P(X>0.75, Y<)

P(X>0.8, Y<)

P(X>0.85, Y<)

P(X>0.9, Y<)

P(X>0.95, Y<)

Cumulative Probability

Probability (X>,Y<40 ft) Probability (X>,Y<100 ft) Probability (X>,Y<150 ft)

Probability (X>,Y<200 ft) Probability (X>,Y<250 ft) Probability (X>,Y<300 ft)

Probability (X>,Y<350 ft)

0

0.5

1

1.5

2

2.5

3

3.5

4

P(X<0.1, Y>)

P(X<0.15, Y>)

P(X<0.2, Y>)

P(X<0.25, Y>)

P(X<0.3, Y>)

P(X<0.35, Y>)P(X<0.40, Y>)P(X<0.45, Y>)

P(X<0.5, Y>)

P(X<0.55, Y>)

P(X<0.6, Y>)

P(X<0.65, Y>)

P(X<0.7, Y>)

P(X<0.75, Y>)

P(X<0.8, Y>)

P(X<0.85, Y>)

P(X<0.9, Y>)

P(X<0.95, Y>)

Cumulative Probability

Probability (X<,Y>40 ft) Probability (X<,Y>100 ft) Probability (X<,Y>150 ft)

Probability (X<,Y>200 ft) Probability (X<,Y>250 ft) Probability (X<,Y>300 ft)

Probability (X<,Y>350 ft)

0

0.5

1

1.5

2

2.5

3

3.5

4

P(X>0.1, Y>)

P(X>0.15, Y>)

P(X>0.2, Y>)

P(X>0.25, Y>)

P(X>0.3, Y>)

P(X>0.35, Y>)P(X>0.40, Y>)P(X>0.45, Y>)

P(X>0.5, Y>)

P(X>0.55, Y>)

P(X>0.6, Y>)

P(X>0.65, Y>)

P(X>0.7, Y>)

P(X>0.75, Y>)

P(X>0.8, Y>)

P(X>0.85, Y>)

P(X>0.9, Y>)

P(X>0.95, Y>)

Cumulative Probability

Probability (X>,Y>40 ft) Probability (X>,Y>100 ft) Probability (X>,Y>150 ft)

Probability (X>,Y>200 ft) Probability (X>,Y>250 ft) Probability (X>,Y>300 ft) Probability (X>,Y>350 ft)

Fig. 8. Conditional probabilities: (a) P(X1ra, X2rb), (b) P(Z14a, Z2rb), (c) P(Z1ra, Z24b) and (d) P(Z14a, Z24b), calculated for flow and distance-from-coal bed data

using bivariate normal distribution.

34 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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the standard deviations of X and Y. The standard normal bivariate

probability density function is then:

fðZ1,Z2,rÞ 1⁄4 1

2p ffiffiffiffiffiffiffiffiffiffiffiffi1r2 p exp  1

2

Z2

12rZ1Z2 þZ22

1r2

    ð3Þ

where r is the correlation between the two variables.

All of the tail or conditional probabilities that can be calculated

using the bivariate probability density function (3) can be given

as [19]

PðX1ra,X2rbÞ 1⁄4 P Z1r am1

s1

,Z2r

bm2

s2

  ð4Þ

PðZ1ra,Z24bÞ 1⁄4 CðaÞPðZ1ra,Z2rbÞ ð5Þ

PðZ14a,Z2rbÞ 1⁄4 CðbÞPðZ1ra,Z2rbÞ ð6Þ

PðZ14a,Z24bÞ 1⁄4 1CðaÞCðbÞþPðZ1ra,Z2rbÞ ð7Þ

where C is the standard normal distribution function and P is the

probability.

4.2. Normality check for the data sets

In this study, six random data sets were complied from the

field experiments and used for calculating their joint probabilities

using bivariate normal distributions to characterize the flow

and displacement intervals. These data sets are flow fraction,

displacement, strata thickness divided by Young’s modulus,

distance from the mined coal bed, gas content of the strata and

overburden depth. The random data used in bivariate normal

distribution calculations for joint probabilities were flow fraction

– distance from the coal bed, displacement – distance from the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Flow > (fraction)

Probability

Probability (X>,Y<40 ft)

Probability (X>,Y<100 ft)

Probability (X>,Y<150 ft)

Probability (X>,Y<200 ft)

Probability (X>,Y<250 ft)

Probability (X>,Y<300 ft)

Probability (X>,Y<350 ft)

Probability

Flow > (fraction)

E50

a=max

d=min

increasing c

decreasing a

increasing d

decreasing b

(slope)

c

Fig. 9. (a) Conditional probabilities calculated for different flow and distance-from-coal bed cases for the P(Z14, Z2r) region shown in Fig. 8b. (b) Locations of parameters

of logistic function given in Eq. (10) and how the change in these parameters can affect the shape of the function.

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 35
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coal bed, strata thickness divided by Young’s modulus – distance

from the coal bed, and gas content – overburden depth pairs.

In order to be able to use the above pairs in bivariate normal

distribution modeling, the univariate data sets should be checked

whether they are actually from a normal population. For this

purpose, the data (observed quantiles—X(j)) was first ordered and

the cumulative probability level of each observed quantile, which

is approximately equal to [19]

j0:5

n ð8Þ

was calculated. In Eq. (8), n is the number of observations. For

these cumulative probabilities, standard normal quantiles were

then calculated using

mþs ffiffiffi

2

p erf 1ð2p1Þ ð9Þ

where m is the mean, s is the standard deviation, erf is the error

function and P is the cumulative probability level.

Fig. 7a and b shows the quantile plots (Q–Q) for total gas

content of coal seams (Fig. 7a) and depth-from-surface data

(Fig. 7b), as an example of the normality tests applied to all data

sets used in this work. If the data are from a normal population,

the Q–Q plots should be approximately linear. Fig. 7a and b shows

that the Q–Q plots are linearly related, suggesting normal

distribution of these two data sets.

In order to further make sure that the data are from normal

distributions, four different normality tests, namely Shapiro–Wilk,

Anderson–Darling, Lilliefors and Jarque–Bera tests, were also

performed on each of the data sets [20]. These tests assume as a

‘‘null hypothesis’’ that the data are from a normal distribution and

compare calculated critical values with the selected significance

level (5% in this study). In order for the null hypothesis to be

accepted, calculated critical values should be greater than the

significance level. The calculated critical values for each of the

data sets for each test are shown graphically in Fig. 7c. This figure

shows that calculated critical values are all (except the depth data

in Lilliefors test) are greater than 0.05 and thus the data sets are

presenting the properties of a normal distribution. Therefore, the

bivariate normal distributions could be used in calculating the

joint probabilities for the data set pairs given above to

characterize the flow amount and displacements in longwall gobs.

4.3. Calculating joint probabilities using bivariate normal

distributions

Joint probabilities of four different variable pairs were

calculated for all tail probabilities of a bivariate normal distribution using Eqs. (3)–(7) and the means, standard deviations and the

correlations given in Table 2. In calculating the probabilities, flow,

0

50

100

150

200

250

300

350

400

0.0

Probability-Displacement /

Flow / (T/E) / GC

Distance from the Coal Bed (ft)

0

200

400

600

800

1000

1200

1400

Overburden (ft)

Probability (Displacement> ; Distance=<)

Probability (Flow> ; Distance=<)

Probability (T/E)> ; Distance=<)

Probability (GC> ; Overburden=<)

0.2 0.4 0.6 0.8 1.0

Fig. 10. Conditional probabilities for displacement (0.3 ft), flow (70%), T/E (10 ft/

GPa) and gas content (100 scf/ton) obtained as a function of distance from the

coalbed and overburden for P(Z14, Z2r) range.

Table 3

Parameters of the logistic equation (Eq. (10)) that were determined to calculate

conditional probabilities of different variables in the P(Z14, Z2r) range.

Distance from CB (ft) R2 d ac b

Displacement4; distance from CBr

40 0.9986 3.8389E–12 0.0185 1.7813 6.5169

100 0.9979 1.9142E–11 0.0897 1.5026 5.4938

150 0.9973 5.2739E–11 0.2333 1.2887 4.7205

200 0.9966 1.0862E–10 0.4441 1.1075 4.0799

250 0.9959 1.7154E–10 0.6439 0.9757 3.6195

300 0.9953 2.1844E–10 0.7606 0.9020 3.3551

350 0.9949 2.3997E–10 0.8003 0.8748 3.2504

Flow fraction4; distance from CBr

40 0.9964 1.0418E–11 0.0508 0.2832 4.8374

100 0.9920 5.2001E–11 0.1548 0.3841 5.5736

150 0.9952 1.6266E–10 0.3027 0.4672 6.0580

200 0.9942 4.3447E–10 0.4847 0.5409 6.2502

250 0.9932 9.9653E–10 0.6693 0.6030 6.0522

300 0.9937 6.4490E–12 0.8154 0.6483 5.5533

350 0.9956 4.7585E–12 0.9068 0.6763 5.0755

Strata thickness/Young M4; distance from CBr

40 0.9964 5.20E–11 0.0866 8.623 2.8872

100 0.9920 1.30E–10 0.2097 8.8791 2.9193

150 0.9952 2.24E–10 0.3543 9.0757 2.9439

200 0.9942 3.24E–10 0.5022 9.2433 2.9647

250 0.9932 4.09E–10 0.6244 9.3789 2.9812

300 0.9937 4.64E–10 0.7011 9.4726 2.9921

350 0.9956 4.92E–10 0.7383 9.5264 2.998

Gas content4; overburdenr

Overburden (ft) R2 d ac b

100 0.9984 2.66E–12 0.0145 57.5716 3.2447

300 0.9957 3.45E–11 0.096 71.9543 3.2503

500 0.9945 1.44E–10 0.3238 92.7746 3.8831

700 0.9962 2.45E–10 0.6424 108.3495 4.4266

900 0.9962 3.97E–10 0.8625 117.2951 4.4184

1100 0.9960 4.90E–10 0.9389 121.1653 4.3581

1300 0.9961 5.10E–10 0.9519 121.9739 4.3293

36 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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displacement, gas content and thickness data were selected as

variable X term and distance variables were selected as Y term in

Eqs. (1) and (2), respectively. Also, joint probabilities were

calculated for various X and Y values so that a distribution of

conditional probabilities could be established. The results of such

a calculation for flow and distance-from-coal bed pair are given,

as an example, in Fig. 8 for all calculated conditional probabilities.

In this calculation, a range of flow fractions between 0.1 and 0.95

and a range of distances from 40 to 350 ft were used (Fig. 8). A

similar approach was used for the other data pairs.

Although calculating conditional probabilities for all tails using

different X and Y values and presenting the results as in Fig. 8 shows

the joint probabilistic behavior of two variables and how they may

characterize an event, the predictions of conditional probabilities are

limited to the actual values used in calculations. In order to

generalize the joint probabilistic behavior of each of the data pairs,

four-parameter logistic equations were fitted to the calculated

probabilities to create functional relationships. A four-parameter

logistic equation is given in Eq. (10), and example data to which this

equation is fitted is shown in Fig. 9a, which shows the flow versus

conditional probability plot drawn using the data from Fig. 8b:

y 1⁄4 dþ ðadÞ

1þ ðx=cÞ

b 1⁄4 minþ ðmaxminÞ

1þ ðx=E50Þ

slope ð10Þ

The four-parameter logistic equation shown in Eq. (10) is one

of the basic sigmoid functions that are used for various

applications in biology and biochemistry such as growth rate

and dose–response studies [21]. The nonlinear nature of this

function makes it flexible to model data that show an ‘‘S’’ shape

character. In Eq. (10), a, b, c, d are the maximum value (a)

and minimum value (d) at asymptotes, slope at the hill of the

line (b) and value of dependent variable at 50% value of

independent variable (c) [21]. These parameters are shown

schematically in Fig. 9.

Table 3 gives the parameters of the logistic equations fitted to

different variables to calculate their conditional probabilities in

the P(Z14,Z2r) range. As this table shows, the logistic equation

was able to fit the data similar to shown in Fig. 9 with regression

coefficients of more than 0.99. Similar equation fits to variables

were performed to calculate their conditional probabilities in

other regions of probabilities given in Eqs. (4)–(7). It was seen that

all tail probabilities can be represented with sufficient accuracy

using logistic equations and that estimating the probabilities

using logistic functions can help interpolate the probabilities and

make calculation of probabilities more flexible.

5. Results and discussion—application of the technique

The bivariate normal distributions and the technique shown to

calculate joint and conditional probabilities can be used to

determine where a certain amount of flow might be inflowing

into the borehole during longwall mining and where a certain

Table 4

Conditional probabilities for strata separations more than 0.3 ft and less than 0.8 ft, and fractions of total flow more than 0.5 and less than 0.9 for all regions of the bivariate

normal distribution as a function of distance from the coal bed.

Displacement (ft)4 Distance CB (ft)r Probability Flow4 Distance CB (ft)r Probability

0.3 40 0.018 0.5 40 0.001

100 0.090 100 0.008

150 0.233 150 0.024

200 0.442 200 0.081

250 0.635 250 0.193

300 0.742 300 0.322

350 0.776 350 0.414

Displacement (ft)r Distance CB (ft)r Probability Flowr Distance CB (ft)r Probability

0.8 40 0.000 0.9 40 0.060

100 0.005 100 0.171

150 0.033 150 0.329

200 0.126 200 0.516

250 0.286 250 0.681

300 0.432 300 0.759

350 0.512 350 0.773

Displacement (ft)4 Distance CB (ft)4 Probability Flow4 Distance CB (ft)4 Probability

0.3 40 0.765 0.5 40 0.490

100 0.694 100 0.499

150 0.548 150 0.491

200 0.335 200 0.425

250 0.140 250 0.288

300 0.035 300 0.151

350 0.005 350 0.063

Displacement (ft) Distance CB (ft)4 Probability Flowr Distance CB (ft)4 Probability

0.8 40 0.547 0.9 40 0.727

100 0.545 100 0.625

150 0.513 150 0.472

200 0.411 200 0.285

250 0.251 250 0.113

300 0.110 300 0.023

350 0.034 350 0.002

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 37
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amount of strata separation might happen. The probabilistic

results given for strata thickness and gas content estimations can

also be used in relation to flow and displacement probabilities.

The ultimate benefit of applying the presented techniques is the

ability to specify gob gas venthole parameters, particularly the

location of the borehole and length of the slotted casing.

Fig. 10 shows the calculated conditional probabilities for the

occurrence of more than 0.3 ft strata separation and more than

70% of flow inflowing to the borehole. This figure shows that the

probability of more than 0.3 ft strata separation occurring below

350 ft during mining decreases from 0.8 to 0.05 below 40 ft from

the top of the coal bed. Similarly, the probability of more than 70%

of the total gas flow entering into the borehole below 350 ft

decreases from 0.45 to 0 below 40 ft. These data can be used in

conjunction with strata thickness/Young’s modulus ratio and gas

content data. For example, the probability of overlying strata

having more than 100 scf/ton gas increases to 0.7 at 900-ft

overburden depth. The probability of finding this much gas at

depths shallower than 900 ft is less than 0.7.

This data or the conditional probabilities obtained for different

values of the variables can be used to design the GGV slotted

casing lengths and to establish the depth at which they should be

located. According to the example data presented in Fig. 10, if the

casing is able to withstand the strains originating from a 0.3 ft

strata separation, then it is safe to use it below 350 ft to the coal

bed since the probability of more than 0.3-ft separations is

decreasing below that distance. At this interval, the probability of

this casing having more than 70% of the flow, however, is only 0.4

or less. For different strata-separations and flow capture goals,

conditional probabilities can be calculated using different values

for optimum design.

All tail probabilities for strata separation and flow can also be

calculated to obtain detailed information about the probabilities

of occurrence of separations and flow capture within a range of

values. In Table 4, the conditional probabilities for strata

separations more than 0.3 ft and less than 0.8 ft, and the fractions

of total flow as 0.5 and 0.9 are presented for all regions

of the bivariate normal distribution as a function of distance

from the coal bed. The data are also given in graphical form in

Fig. 11a and b.

Fig. 11a shows that the probability of a displacement or strata

separation, more than 0.3 ft, is 0.8 at a distance less than 350 ft

from the coal bed. The probability of having the same amount of

strata separation starting from the coal bed is symmetrical and

increases with distance. The intersection of these two curves

(dotted red and dotted blue) gives the probability (0.4) and

distance (180 ft in this case) where at least 0.3 ft separation

occurs in the overlying strata. The solid lines give the conditional

0

50

100

150

200

250

300

350

400

0.0

Probability-Displacement

Distance from the Coal Bed (ft)

Probability (Displacement=< ; Distance=<)

Probability (Displacement=< ; Distance>)

Probability (Displacement >; Distance =<)

Probability (Displacement > ; Distance >)

0

50

100

150

200

250

300

350

400

0.0

Probability-Flow

Distance from the Coal Bed (ft)

Probability (Flow =< ; Distance=<)

Probability (Flow =< ; Distance>)

Probability (Flow > ; Distance =<)

Probability (Flow > ; Distance >)

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Fig. 11. Graphical representation of conditional probabilities for (a) strata separation and (b) fraction of total flow given in Table 4.

38 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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probability of less than 0.8 ft separation occurring. The solid red

line in Fig. 11a shows that at most 0.8 ft separation occurs with a

probability of 0.8 at a distance of 350 ft from the coal bed and this

probability decreases as the distance to the coal bed decreases.

The interception of blue and red solid lines gives the probability

(0.25) and the location (245 ft) of at most a 0.8 ft strata

separation.

Fig. 11b shows a similar analysis for the conditional probability

of the fraction of total flow entering the borehole at various

depths. This figure shows that the probability of at least 0.5 of the

total flow entering the borehole below a given height from the

coal bed can be traced by the dotted red line. This line shows that

this probability is 0.75 at and below 350 ft, 0.3 at 200 ft and

decreases to almost 0 at and below 50 ft from the coal bed. The

intersection of two dotted lines that represent the two sides of

this distribution is where at least 0.5 of total flow enters the

borehole. This probability is 0.4 and occurs at 225 ft from the

coal bed. On the other hand, the probability of at most 0.9 of total

flow entering the borehole is 0.8 and occurs at and below 350 ft

from the coal bed. The probability of at most this much flow

entering the borehole at and below 150 ft is about 0.3. The

intersection of two solid lines shows that at most 0.9 of total flow

enters the borehole at a distance of 175 ft above the coal bed with

a probability of 0.4.

A similar example application for evaluating the probabilities

can be performed for strata thickness/Young’s modulus ratio (T/E)

and gas content of the overlying strata (GC). In this analysis 10

and 50 ft/GPa were selected as the lower and upper values for the

T/E parameter, and 100 and 250 scf/ton were selected as the lower

and upper values for strata gas content. The tabulated values of

conditional probabilities as a function of distance to coal bed and

overburden depth are given in Table 5 and graphically shown in

Fig. 12.

Fig. 12a shows that coal measure strata with a T/E ratio of at

least 10 can be located at and below 350 ft above the coal

bed with a probability of 0.35. The probability of this type of

strata decreases when close to the coal bed. The type of strata

that this T/E number can indicate may be thin and have a high

Young’s modulus. These types of formations are usually rare close

to the seam. Thus, the probability of this occurrence is low. On the

other hand, the probability of strata with a T/E of at most 50 at

and below 350 ft is almost 1.0. This probability is 0.7 less than or

equal to 200 ft from the coal bed. Increasing distances from the

coal bed increases the possibility of thicker coal-measure

formations.

Fig. 12b shows the variation of conditional probabilities as a

function of overburden depth for the existence of strata with gas

amounts of at least 100 scf/ton and of at most 250 scf/ton.

This graph shows that gas contents of at least 100 scf/ton are

more probable as the depth increases (red dotted line) to 1300 ft.

The probability of finding more than 100 scf/ton gas in the strata

at depths 1300 ft and less is 0.7. On the other hand, the

Table 5

Conditional probabilities for strata thickness/Young’s modulus ratio (T/E) of more than 10 ft/GPa and less than 50 ft/GPa and strata gas contents (GC) of more than 100 scf/

ton and less than 250 for all regions of the bivariate normal distribution as a function of distance from the coal bed and overburden depth.

T/E (ft/GPa)4 Dist. CB (ft)r Probability GC (scf/ton)4 Overburden (ft)r Probability

10 40 0.034 100 40 0.002

100 0.087 100 0.025

150 0.152 150 0.138

200 0.222 200 0.378

250 0.282 250 0.577

300 0.322 300 0.655

350 0.342 350 0.669

T/E (ft/GPa)r Distance CB (ft)r Probability GC (scf/ton)r Overburden (ft) r Probability

50 40 0.127 250 40 0.018

100 0.302 100 0.108

150 0.502 150 0.354

200 0.704 200 0.683

250 0.867 250 0.899

300 0.968 300 0.955

350 0.991 350 0.959

T/E (ft/GPa)4 Distance CB (ft)4 Probability GC (scf/ton)4 Overburden (ft)4 Probability

10 40 0.319 100 40 0.668

100 0.267 100 0.645

150 0.202 150 0.527

200 0.132 200 0.288

250 0.071 250 0.089

300 0.031 300 0.014

350 0.011 350 0.001

T/E (ft/GPa)r Distance CB (ft)4 Probability GC (scf/ton)r Overburden (ft)4 Probability

50 40 0.914 250 40 0.950

100 0.740 100 0.874

150 0.539 150 0.631

200 0.338 200 0.302

250 0.174 250 0.085

300 0.073 300 0.012

350 0.014 350 0.001

C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41 39
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probability of finding gas contents of at most (or less than)

250 scf/ton at the same depth is 1.0. The intersection of two

dotted lines shows that the depth and the probability of

encountering a gas content of at least 100 scf/ton are 650 and

0.35 ft, respectively.

6. Conclusions

This work presents probabilistic assessments of depth-displacement, depth-flow percentage, depth-formation and depthgas content relationships using bivariate normal distributions.

Using data gleaned from longwall operations in southwestern

Pennsylvania and northern West Virginia, joint probabilities were

calculated using the bivariate normal distribution. Four-parameter logistic equations were then created to define functional

relationships between all data pairs.

Joint and conditional probabilities were developed between

strata displacement and distance to coal, inflow and distance to

coal, ratio of strata thickness to Young’s modulus and distance to

coal, and gas content and overburden thickness. Graphical results

were presented for displacements of more than 0.3 and less than

0.8 ft, gas inflows of more than 50% and less than 90% of the total

flow, ratios of strata thickness to Young’s modulus (T/E) of more

than 10 ft/GPa and less than 50 ft/GPa, and gas contents of more

than 100 scf/ton and less than 250 scf/ton. For instance, the

accumulated data suggest that with a probability of approximately 40%, at least 50% of the total methane flow enters a

borehole at a location 225 ft above the coalbed. Other data show

the probability of occurrence of strata existing with a T/E ratio of

at least 10 is 0.35, but decreases with approach to the coalbed.

Finally, these results show that the probability of high gas

contents of at most 250 scf/ton increases with coalbed depth.

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0

50

100

150

200

250

300

350

400

0.0

Probability-T/E

Distance from the Coal Bed (ft)

Probability (T/E =< ; Distance=<)

Probability (T/E =< ; Distance>)

Probability (T/E > ; Distance =<)

Probability (T/E > ; Distance >)

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

0.0

Probability-Gas Content

Overburden (ft)

Probability (GC =< ; Depth =<)

Probability (GC =< ; Depth >)

Probability (GC > ; Depth =<)

Probability (GC > ; Depth >)

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Fig. 12. Graphical representation of conditional probabilities for (a) T/E ratio and (b) strata gas content given in Table 5.

40 C. O ̈zgen Karacan, G.V.R. Goodman / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 27–41
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