UNITED STATES BUREAU OF MINES

UNITED STATES DEPARTMENT OF THE INTERIOR

RI 9618 REPORT OF INVESTIGATIONS/1996

Rock Mechanics Study of Shaft Stability

and Pillar Mining, Homestake Mine,

Lead, SD

(In Three Parts) 

3. Geomechanical Monitoring and Modeling Using UTAH3

By W. G. Pariseau, J. C. Johnson, M. M. McDonald, and M. E. Poad
U.S. Department of the Interior

Mission Statement

As the Nation’s principal conservation agency, the Department of the

Interior has responsibility for most of our nationally-owned public

lands and natural resources. This includes fostering sound use of our

land and water resources; protecting our fish, wildlife, and biological

diversity; preserving the environmental and cultural values of our

national parks and historical places; and providing for the enjoyment

of life through outdoor recreation. The Department assesses our

energy and mineral resources and works to ensure that their

development is in the best interests of all our people by encouraging

stewardship and citizen participation in their care. The Department

also has a major responsibility for American Indian reservation

communities and for people who live in island territories under U.S.

administration.
Report of Investigations 9618

Rock Mechanics Study of Shaft Stability and Pillar

Mining, Homestake Mine, Lead, SD

(In Three Parts) #. Geomechanical Monitoring and Modeling

Using UTAH3

By W. G. Pariseau, J. C. Johnson, M. M. McDonald, and M. E. Poad 

UNITED STATES DEPARTMENT OF THE INTERIOR

Bruce Babbitt, Secretary

BUREAU OF MINES

Rhea Lydia Graham, Director
International Standard Serial Number

ISSN 1066-5552
CONTENTS

Page

Abstract ...................................................................................... 1

Introduction .................................................................................... 2

Geomechanical model (UTAH3) ..................................................................... 3

Input data ................................................................................... 3

Mesh development ............................................................................. 4

Model verification ............................................................................. 5

Finite-element model calibration ..................................................................... 5

Rock properties ............................................................................... 5

Preexcavation stresses .......................................................................... 5

Displacement data ............................................................................. 6

Regression analyses ............................................................................ 7

Application of calibrated model ...................................................................... 8

Shaft stability ................................................................................. 9

Wing pillars .................................................................................. 9

Central pillar ................................................................................. 9

Shaft framing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

ILLUSTRATIONS

1. Homestake Mine development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Section and plan views of two-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Three-dimensional, finite-element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Plan view of coarse mesh about shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5. Plan view of insert mesh about shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6. Comparison of coarse and insert meshes in calculations of shaft wall safety factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7. Stresses on elements composing tunnel walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8. Shaft extensometer layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9. Example of extensometer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

10. Correlation coefficient as function of extensometer anchor displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

11. Ross shaft pillar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

12. Shaft wall safety factor as function of depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

13. Shaft wall displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

14. Shaft wall safety factor versus depth at various stages of mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

15. Calculated extent of yield zone after central pillar mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

16. Shaft wall displacement versus depth at various stages of mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

17. Shaft strains versus depth at various stages of mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

TABLES

1. Project chronology ........................................................................... 2

2. Preexcavation and posthistorical mining stresses ...................................................... 6

3. Finite-element mining sequences ................................................................. 7

4. Regression analysis of calculated extensometer displacements on measured displacement ......................... 7
UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT

Metric Units

cm centimeter m meter

h hour Mbyte megabyte

kbyte kilobyte mm millimeter

kg kilogram t metric ton

kPa kilopascal

U.S. Customary Units

ft foot st short ton

in inch tr oz troy ounce

Disclaimer of Liability

The U.S. Bureau of Mines expressly declares that there are no warranties expressed or implied that apply to the software

described herein. By acceptance and use of said software, which is conveyed to the user without consideration by the Bureau of Mines,

the user hereof expressly waives any and all claims for damage and/or suits for or by reason of personal injury, or property damage,

including special, consequential, or other similar damages arising out of or in any way connected with the use of the software described

herein.
McKinnon Professor of Mining Engineering, University of Utah, Salt Lake City, UT. 1

Mining engineer, Spokane Research Center, U.S. Bureau of Mines, Spokane, WA. 2

Research civil engineer, Spokane Research Center, retired. 3

Supervisory mining engineer, Spokane Research Center. 4

Rock Mechanics Study of Shaft Stability and Pillar

 Mining, Homestake Mine, Lead, SD

(In Three Parts) 

3. Geomechanical Monitoring and Modeling Using UTAH3

By W. G. Pariseau, J. C. Johnson, M. M. McDonald, and M. E. Poad 1 2 3 4

ABSTRACT

A U.S. Bureau of Mines case study of pillar recovery in high-grade ore near the Ross shaft at the Homestake

Mine, Lead, SD, has demonstrated the usefulness of the finite-element method for evaluating shaft pillar mining

plans and shaft stability. In this study, borehole extensometers and other instruments were installed to provide data

for model verification and to monitor shaft displacement. Results of a recalibrated two-dimensional model (UTAH2)

confirmed the premining stability evaluation. However, after mining began, concern developed because of cracks

and other signs of ground motion that appeared at considerable distances from the area of active pillar mining. An

intense three-dimensional modeling effort using the computer modeling program UTAH3 was initiated. The results

again showed the observed effects were within expectations and that the shaft would remain safe. Three-dimensional

analyses of alternate pillar mining scenarios indicated that more of the shaft pillar ore reserve could be recovered than

previously thought on the basis of two-dimensional analyses. This report focuses on startup and implementation

of the three-dimensional model, calibration of the three-dimensional model, and application of the model to shaft

pillar mining.
2

Table 1.—Project chronology

Phase Topic Beginning date

1 ...... 2-dimensional premining analysis ...... April 1987.

2 ...... Installation first instruments ........... October 1987.

3 ...... Pillar mining begins ................. March 1988.

4 ...... 2-dimensional model validation ........ March 1989.

5 ...... 3-dimensional model 1 .............. August 1990.

6 ...... 3-dimensional model 2 .............. August 1991.

7 ...... Additional instrumentation ............ July 1994.

8 ...... 3-dimensional model 3 .............. March 1995.

INTRODUCTION

Because of the importance of shaft pillar design to the mining In the second RI of the series (Part 2. Mine Measurements and

industry, research was undertaken at the Homestake Mine, Lead, Confirmation of Premining Results), instrument calibration and

SD, to investigate the extraction of valuable reserves within the updating of the original two-dimensional, finite-element model

Ross shaft pillar. The study was a cooperative effort and involved were described (Pariseau and others, 1995b). This RI focused on

the U.S. Bureau of Mines (USBM), the Homestake Mining Co., a vertical section that allowed for sequential, lift-by-lift simulation

and the University of Utah, Salt Lake City, UT. Table 1 shows of cut-and-fill extraction of the ore reserves in the shaft pillar and

the chronology of the main phases of the project. addressed several numerical modeling concerns that arose during

The Homestake Mine is located in the northern Black Hills of ft) wide pillar within the original 60-m (200-ft) wide shaft pillar.

South Dakota. Figure 1 shows the general layout of the mine, A mechanized cut-and-fill method would be used to mine stopes

which is the oldest and deepest in North America. Development north and south of the central pillar. The first 5-m (15-ft) high lift

extends to the 8000 level (feet below the surface), with the Yates was taken below the 3650 level after ramp access was developed.

and Ross shafts providing access. About 8,398 kg (270,000 tr oz) Almost immediately, worrisome indications of ground control difof gold and 2,177 kg (50,000 tr oz) of silver are recovered from ficulties appeared in the form of cracking near the 3200 level, the

1.5 million t (1.7 million st) of ore milled per year. Most of the site of original damage in the 1950's. Concrete slabs in the 3650-

ore reserve in the Ross shaft pillar lies between the 3200 and 3800 level pump room began to crack so that realignment of pipes and

levels on the west side of the shaft. Stoping methods are mainly pumps became necessary. Some sticking of shaft conveyances

mechanized cut-and-fill and vertical crater retreat. occurred.

Pillar mining began below the 3650 level in late 1988. Shortly Although only a small tonnage of ore in the shaft pillar had

afterward, movement was observed on the 3200 level, where the been mined, the evidence seemed to indicate a serious and

shaft had been damaged in the early 1950's. In fact, it was the growing threat to the Ross shaft. Concerns were expressed by

experience in the early 1950's that led to definition of the existing company, union, and regulatory officials. Pillar mining was

shaft pillar. Additional pillars within the shaft pillar were then sharply curtailed while the advice of external consultants was

defined in response to the threat of renewed ground movement. sought. Serious doubts were expressed about the results of the

The first RI of this series (Part 1. Premining Geomechanical numerical model, which had indicated that the shaft would remain

Modeling Using UTAH2) described the general objectives of the stable after mining the ore reserve 60 m (200 ft) beyond the shaft.

study, site geology, practical shaft stability criteria, and the However, no plausible explanation was put forth to explain how

approach taken to the problem (Pariseau and others, 1995a). such a small amount of actual mining [about 9 m (30 ft)] in two

Discussed in detail were two-dimensional, finite-element lifts) below the 3650 level could produce such severe effects near

simulations of (1) historical mining leading to the present shaft the 3200 level, which was more than 90 m (300 ft) distant. The

pillar and (2) future mining of the ore reserve in the shaft pillar. need to take some action to ensure the safety of operations led to

The results indicated that the Ross shaft remained in elastic the definition of two additional "wing" pillars within the original

ground, and thus, no large, catastrophic ground failures were shaft pillar, despite the fact that these pillars would substantially

likely. Hence, the pillar mining plan was considered safe. reduce the pillar ore reserve.

the earlier premining study.

The approach to the particular problem of model calibration

was (1) to install instruments prior to stoping for the purpose of

measuring rock mass response to the first lifts taken in the shaft

pillar and (2) to install instruments near the shaft to monitor shaft

stability. Stope instruments provided early data for model

validation, calibration, and updating. It was expected that these

shaft instruments would then provide objective measurements of

ground movement around the shaft in response to pillar mining

and would also warn of any potential threat to shaft stability

independently of numerical model results.

The initial pillar mining plan was to leave a central 18-m (60-
3

Continued research and investigation revealed that long-term strain) analyses are limited (Pariseau and others, 1990).

creep of the shaft walls had reached the limit of shaft guide Figure 2 illustrates these limitations. Computer simulations of 5

adjustment. Thus, sticking of shaft conveyances, although stope and parallel drift excavation in vertical sections produce

coincidental with the start of pillar mining, was not a direct horizontal, tunnel-like openings and do not take into account

consequence of taking the first lifts below the 3650 level. stress concentration about a vertical shaft. No estimate of out-ofChipping the concrete behind the shaft buntons allowed the plane deformation is possible. Stopes, shafts, and raises excavated

buntons to rebound and shaft guide adjustments to be made. in horizontal sections (plan view) produce vertical, shaft-like

Two hypotheses were advanced to explain the observed openings that greatly exaggerate horizontal dis- placements and

damaging effects of stress. One was that graphitic schist, which is give no information about vertical motion. Also associated with

weak in shear, or possibly an unknown fault could be causing plane strain analyses is a tacit assumption that the material

instability in the shaft. Some exploratory two-dimensional directions of anisotropic rock, which are associated with foliation

analyses were done in which graphitic schist was represented. A strike and dip, coincide with the plane of analysis. Therefore,

number of effects were obtained by varying the extent, elastic vertical shear stresses are neglected. Out-of-plane stress and dismoduli, and strengths of the schist. However, additional site placement measurements are also not useable for calibration of

investigations (inspection of drill holes and excavation walls and two-dimensional models.

readings of extensometers installed to monitor stability) did not Because of these limitations and the importance of shafts to

reveal the presence of a fault or enough graphitic schist to justify mine safety, the feasibility of conducting a three-dimensional,

incorporating either of these geologic features into the finite- finite-element analysis was investigated. Experience gained in

element model. Therefore, it was concluded that the geology was three-dimensional analyses of undercut-and-fill stoping at the

not the source of the observed damage. Lucky Friday Mine (McMahon and Pariseau, 1989) and of

The second hypothesis was that the geometry of the two- vertical crater retreat mining at the Homestake Mine (Pariseau and

dimensional model was simply not an adequate approximation of Duan, 1989; Pariseau and others, 1989) were quite helpful in

the actual shaft pillar geometry. Two-dimensional (plane addressing the new issues associated with shaft pillar mining.

GEOMECHANICAL MODEL (UTAH3)

Essential requirements of a geomechanical model for number of measurement techniques used in various parts of the

simulating shaft pillar mining at the Homestake Mine include (1) mine by several investigators (Johnson and others, 1993), (3)

elastic behavior to accommodate initial application of rock and fill properties estimated from laboratory tests and scale

load, (2) yielding beyond the elastic limit (elastic-plastic factors determined for the shaft pillar region (Pariseau, 1986), and

deformation), (3) accommodation of arbitrary initial (pre- (4) the historical and planned mining sequence.

excavation) stress states, (4) knowledge of directional rock Rock property scale factors are decimal fractions used to

properties associated with foliation and schistosity (anisotropic multiply laboratory values of rock properties to obtain rock mass

elastic moduli and strengths), (5) knowledge of the different rock values. A scale factor of 1 implies that the rock mass property of

types and geologic structures, and (6) capability for sequential interest is equal to the value determined from laboratory tests. In

cuts and fills to follow proposed and historical mining sequences the absence of prior information, a scale factor of 0.5 may be

in three dimensions. These requirements were met by a version reasonable (Heuze, 1980). Separate scale factors for elastic

of the finite-element code UTAH3, which has been in use since moduli and strengths were used in this study. In the two1975 and is similar in concept to UTAH2 (Pariseau, 1978). dimensional analyses described in part 1 of this series (Pariseau

UTAH2 is now in the public domain (Pariseau and others, 1991), and others, 1995a), these were 0.36 and 0.60, respectively, and

while UTAH3 is not. Of critical importance to implementation were obtained by back-analysis, that is, a comparison of measured

of the geomechanics model was an efficient mesh design, one that displacements with model displacements, in conjunction with a

was reliable and economical. simple energy-scaling rule of thumb. This "rule" postulates that

INPUT DATA strengths to be adjusted as the square root of the modulus scale

Input data for both codes are similar and include (1) stope criterion that gives equal modulus and strength scale factors. In

geometry and geologic descriptions obtained from mine maps any case, the modulus scale factor can be determined from

and sections, (2) preexcavation stresses obtained from a regression analyses of calculated and measured displacement

Use of the Ross shaft began December 1934 (Bjorge and others, 1935). 5

strain energy density at failure is independent of scale and requires

factor. An equally acceptable rule-of-thumb is a strain-to-failure

changes in the elastic range.
4

Back-analysis in three dimensions may result in smaller scale increases computational time. Overnight turnaround time was

factors than would back-analysis in two dimensions because desired as a practical matter. Longer times tended to increase the

calculated displacements are likely to be less in the former. For probability of lost runs and reductions in productivity.

example, changes in wall displacement in a circular tunnel are Primary stopes at the Homestake Mine were about 15 m (50

twice those in a sphere of the same diameter excavated in the ft) wide; the level interval is 46 m (150 ft). These dimensions

same material under the same preexcavation hydrostatic stress lead to a minimum mesh width of 230 m (750 ft) with elements

field (Love, 1944). Consequently, the rock mass modulus about 6 m (20 ft) on edge along the stope walls. There are about

needed to bring displacement of a calculated, three-dimensional 900 elements and 1,700 nodes in a typical vertical section in the

spherical model into agreement with measured displacement shaft pillar region. The entire mesh contains over 34,000 nodes

would be one-half that needed to enforce agreement with a two- and elements. Figure 3 shows the dimensions of the Ross shaft

dimensional cylindrical model. Generally, back-analysis using a pillar mesh. The overall mesh was constructed by manually

three-dimensional model is expected to produce smaller scale partitioning a generic section through the shaft pillar region,

factors and lower rock mass moduli and strengths than a two- enlarging this section, and then repeating the enlarged section

dimensional model, other factors being equal. A limited along strike until the outer dimension criterion was met. Figure

parameter study suggested the use of 0.25 and 0.50 for modulus 3 also shows the main geologic features of the generic section and

and strength scale factors, respectively, in the three-dimensional the mesh refinement used in the proposed pillar mining area.

analysis of shaft pillar mining at the Homestake Mine. The three- There are 34 element slabs in the mesh.

dimensional strength scale factor is five-sixths the two- The central element slab contains the Ross shaft. Figure 4 is

dimensional factor; both are numerically consistent with the a plan view of the mesh in the vicinity of the shaft. The area is

scaling criterion for simple strain energy density. about 24 m (80 ft) wide and 30 m (100 ft) long. A more refined

Geologic information in the form of rock type distribution was "insert" mesh was constructed for this important region. The

part of the input data. Major formations at the Homestake Mine purpose was to obtain more accurate results in the vicinity of the

are the Precambrian Poorman Formation in the footwall and the shaft than allowed for by the larger and "coarser" stope-scale

Ellison Formation in the hanging wall. Ore bodies are localized mesh. Inspection of the coarse mesh near the shaft shows

in the Homestake Formation (Slaughter, 1968). This information elements the size of the shaft itself, which cannot, therefore,

was incorporated in the model by simply tracing formation indicate localized yielding in the shaft wall. Shaft wall yielding

contacts as shown on mine sections, which were on 15-m (50-ft) in the coarse mesh would engulf the shaft before the computer

centers. Contact location was subsequently converted to model could detect movement. Figure 5 shows the element

numerical input data using a large digitizing table after the finite- partitioning of the shaft insert mesh in plan view. The shaded

element mesh was developed. elements in figure 5 define the shaft wall after excavation. There

MESH DEVELOPMENT than 5,000 elements in the insert mesh, which extends from top

Mesh development for numerical reliability was governed by boundaries of the shaft insert mesh match companion nodes in the

several guidelines. The first was that there should be at least five larger coarse mesh.

mesh points along the least dimension of a stope or stope Procedures simulating a stope lift in pillar mining required

combination; the second was that outer mesh dimensions should running the coarse mesh first and then repeating the same process

be at least five times the greatest stope dimension; and the third using the insert mesh. Node displacements from coarse mesh

was that the element aspect ratio should be no greater than five. output were then specified as input at the companion nodes of the

These guidelines were based on much com- putational experience insert mesh. A means of checking the validity of the procedure

using an iterative equation solver, which greatly reduces the is to use a duplicate of the coarse mesh as the insert mesh. Results

amount of memory available for fast computations (fast memory). (element stresses, strains, node displacements) from a given run

In this regard, UTAH3 requires about 1 kbyte of fast memory per using the original coarse mesh should agree with the final results

node; a 10,000-node mesh requires about 10 Mbytes of fast using the two-run insert mesh procedure.

memory. A symmetric elimination equation solver using banded Figure 6 illustrates the differences between the shaft wall safety

storage would require about 50 times as much memory. Fast factors obtained with the original coarse mesh and those obtained

random access memory (RAM) is not essential, but virtual using the refined insert mesh. The safety factor data in figure 6

memory g r e a t l y refer to rock mass property values, not laboratory 

are five nodes along the least dimension of the shaft and more

to bottom of the larger coarse mesh. Nodes on the outer
5

values. The lowest safety factor for the coarse mesh for the shaft is not closely followed during this procedure. A more

wall (about 1.6) was generated after shaft excavation and refined mesh would be required if accurate stress concentration

historical mining but prior to pillar mining. This figure suggested measurements in the stope walls were needed.

caution. The corresponding insert mesh safety factor was very Computer run times per cut in the 34,000-node coarse mesh

nearly 1.0 and was a cause for concern. Differences between ranged from several days using the virtual memory capability of

coarse and insert mesh shaft wall displacements showed a smaller a Prime computer to about an hour on an IBM 3090 to a few

percentage difference than observed in the case of safety factors. minutes using a Cray X-MP machine. Workstation (HP 9000)

The insert mesh technique has several advantages over a single run times were two to three times longer, about 2 to

mesh: less drastic mesh gradation and better numerical 3 h, than IBM 3090 run times.

performance, smaller size and faster run time, less input and

output data. The main use of this technique is for analyses MODEL VERIFICATION

involving openings of greatly different sizes in the same mesh, for

example, shafts 4.5 m (15 ft) wide and stopes 45 m (150 ft) wide. Comparisons between the two-dimensional code UTAH2 and

The disadvantage is the need to do two runs. Insert meshes may the three-dimensional code UTAH3 were done to verify the threebe used in both two- and three-dimensional analyses. dimensional model and to establish confidence in the consistency

Once the mesh was partitioned into elements, mine sections of results. The main procedure used a transitional tunnel problem

showing the extent of past mining were overlain at regular in which a tunnel-like opening was excavated in an initially

intervals along strike of the coarse mesh. Elements occupying old stressed rock mass (1) in a plane strain analysis using UTAH2, (2)

stopes were identified by number and placed in cut element files, in a single-element slab using UTAH3, and (3) end-to-end in the

which, when sequenced in time, served as historical mining input coarse mesh using UTAH3. No displacement of the outer mesh

data. Elements occupying stopes identified in the ore reserve of faces perpendicular to the tunnel axis was allowed in the threethe Ross shaft pillar were identified in the same way; when se- dimensional meshes. Figure 7 compares the stresses in the

quenced in time, these elements served as pillar mining input elements composing the tunnel walls. The results from the threedata. Stope geometry dimensional analyses are in close agreement with results from the

two-dimensional analysis, which verifies procedural and code

consistency.

FINITE-ELEMENT MODEL CALIBRATION

In principle, the calibration procedure for the three- that is, orthotropic rock properties. A generalized Hooke's 

dimensional Ross shaft pillar model was the same as for the earlier law relates stress and strain in the elastic range of deformation.

two-dimensional model. Regression analyses of calculated and Quadratic, pressure-dependent yield criteria limit the range of

measured displacements obtained within the elastic range of purely elastic deformation (Pariseau, 1972). Associated flow

deformation provided a scale factor for elastic moduli. Extent of rules are used in the postelastic range. Each of the three major

yielding beyond the elastic limit served to identify a strength scale formations at the Homestake Mine (Poorman, Homestake, and

factor. The same scale factor for elastic properties was applied to Ellison) thus require nine independently specified elastic moduli

all elastic moduli; similarly, the same scale factor for strength was and strengths; 54 are required in all. Laboratory test values are

applied to all strengths (Pariseau and others, 1985). Rock mass given in part 1 of this series (Pariseau and others, 1995a). Some

properties for use in the calibrated model were then obtained by igneous dikes occur in the shaft pillar region; these were assigned

multiplying laboratory test values of moduli and strengths by the properties appropriate to the Homestake Formation. Old stopes

appropriate scale factor. Model calibration also depended on the are filled with sand and waste rock. Properties for these materials

preexcavation stress state, which then required verification for were assumed to be isotropic and were estimated on the basis of

complete model calibration. experience. Both had negligible influence on shaft stability.

ROCK PROPERTIES PREEXCAVATION STRESSES

The finite-element code UTAH3 allows for independent The preexcavation stress state before the first underground

anisotropic elastic and strength properties. The anisotropy mining was done at the Homestake Mine was characterized

allowed provides for three mutually orthogonal material axes, by normal stress factors Fv (vertical stress), FH (stress
6

Table 2.—Preexcavation and posthistorical

mining stresses, kilopascals

North-south, Fh East-west,

FH

Vertical

Preexcavation .... 14,700 27,700 31,500

Posthistorical:

 Measured ...... 16,300 25,600 28,700

 Model ......... 17,800 33,700 42,300

perpendicular to strike), and Fh (stress parallel to strike) in Remote extensometers extended away from the shaft pillar into

kilopascals. In this analysis, F = 28.275 Z, F = 14,317 + 11.99 undisturbed ground where little motion was expected. v H

Z, and F = 834 + 12.44 Z. Z is depth below the surface in Displacement data shown in figure 9A confirm this expectation. h

meters. Stope extensometers were installed in the vicinity of first pillar

This characterization was obtained from several stress mining to obtain calibration data early in the life of the project.

measurements taken in different sections of the mine by different Stope extensometers generally produced large displacements,

investigators using borehole deformation gauges and doorstopper more than 10 mm (0.4 in), and were first used to calibrate the

cells. Stress measurements were also made using hollow two-dimensional model (Pariseau and others, 1995b). Figure 9B

inclusion and borehole deformation gauges in the course of the shows a recording of an extensometer installed in a cross-cut rib

current shaft pillar study. These data pertain to the stress state that near a stope on the 3650 level where pillar mining began. In this

existed after much historical mining but prior to any shaft pillar regard, stope extensometer installation preceded pillar mining by

mining. A similar set of stresses was calculated through a finite- several months. The response of the hanging wall stope

element simulation of historical mining that included sinking of extensometer was similar to that shown in figure 9B and included

the Ross shaft in an unperturbed stress field but with no pillar large step-like increases over a period of several months before

mining. reaching the recording limits of the instruments (Pariseau and

The finite-element simulation of historical mining used the others, 1995b).

figures above as initial stresses. Initial normal stresses Both types of extensometer data illustrated in figure 9 show

(preexcavation) and posthistorical normal stresses (prepillar relatively large displacements that were common to all points on

mining) are shown in table 2. Generally, shear stresses were an a given extensometer. At the same time, relative dis- placements

order of magnitude less than normal stresses. Additional details between points of a given extensometer were much smaller. For

are given in Johnson and others (1993). The data in table 2 show example, the stope data in figure 9B show about 36 mm (1.4 in)

that the finite-element stresses were higher than the measured of common displacement but only about 3 mm (0.1 in) of relative

stresses on the 3650 level of the shaft pillar prior to pillar mining. displacement between points along the instrument hole. This

Higher prepillar mining stresses were expected to induce higher phenomenon is most simply explained by motion of the reference

postpillar model stresses and thus to introduce some conservatism point at the hole collar that, in turn, is associated with yielding of

in the analysis. Additional adjustments to scale factors and initial the cross-cut wall in the neighborhood of the hole collar. Indeed,

stresses for model calibration were not justified because of cracking and sloughing of development walls near stopes was not

uncertainties in the data, which included large variations in stress unusual or unexpected. The common displacement in shaft

measurements (20% to 40% above and below mean values) and extenso- meter data and motion of the shaft extensometer collars

the assumption of isotropy in stress measurements. are less easily explained because there was little visual evidence

DISPLACEMENT DATA correspond to the measured displacements. Only displacements

Displacement data for model calibration were obtained from calculated "readings." The finite-element simulation sequences of

multipoint borehole extensometers (MPBX's) installed near shaft historical and shaft pillar mining are given in table 3.

and stope walls (figure 8). Shaft extensometer displacements

were generally small, a few millimeters. Figure 9A is an example

of shaft extensometer response over a 3-year period following

installation in November 1987. Positive displacement indicates

e x t e n s i o n relative to the collar. 

of yielding. Creep was indicated, but whether associated with the

rock mass, with extensometer cement, with other aspects of

installation, or some combination is unknown. The example data

in figure 9 are the best available; many plots defied interpretation

and much of the data were lost because of power stoppages,

equipment failures, electrical noise, and so forth.

Extensometer output allows for direct conversion into

measurements of relative displacement between anchor points

located at regular intervals along the instrument hole and the hole

collar point. A positive relative displacement indicates elongation

parallel to the hole. Finite-element simulation of the mining

sequence allows for calculation of relative displacements that

that occurred after installation of an extensometer were used in the
7

Table 3.—Historical and shaft pillar mining sequences

 Cut

 number

Mining simulation

Historical sequence (past mining north and south of shaft pillar):

 1 ....... Sinking of the Ross shaft.

 2 ....... Stoping below 3650 level.

 3 ....... Stoping between 3650 and 3500 levels.

 4 ....... Stoping between 3500 and 3350 levels.

 5 ....... Stoping between 3300 and 3250 levels.

 6 ....... Stoping above 3250 level.

Pillar mining (calibration and updating sequence):

 7 ....... Excavation 3650 hanging wall drift, November 1987.

 8 ....... June 1988.

 9 ....... July 1988.

 10 . . . . . . December 1988.

 11 . . . . . . February 1989.

 12 . . . . . . April 1989.

 13 . . . . . . May 1, 1989.

 14 . . . . . . May 10, 1989.

Pillar mining (further updates, including north and south of central

 pillar):

 15 . . . . . . July 1989.

 16 . . . . . . October 1990.

 17 . . . . . . November 1992.

 18 . . . . . . December 1992.

 19 . . . . . . January 1994.

 20 . . . . . . August 1994.

Pillar mining (future central pillar extraction):

 21 . . . . . . Estimated summer 1995, 3650 level sill to +50.

 22 . . . . . . Estimated winter 1995, 3650 level, +50 to +100.

 23 . . . . . . Estimated summer 1996, to 3500 level.

 24 . . . . . . Estimated winter 1996, 3500 level sill to +50.

 25 . . . . . . Estimated summer 1997, 3500 level, +50 to +100.

 26 . . . . . . Estimated winter 1997, to 3350 level.

REGRESSION ANALYSES

Elastic behavior allowed for regression analyses using a linear

model. The regression variables were calculated displacements

(y-axis) on measured displacements (x-axis). The slope of the

regression line is the ratio of Young's modulus associated with

measured displacements to the modulus used in the finite-element

calculations, provided that the rock mass behavior is entirely

elastic. This ratio is, in fact, an elastic modulus scale factor.

Inelastic behavior in the mine and in the model introduce some

degree of nonlinearity. The square of the correlation coefficient

(r ) is a measure of the variance that may be explained by a 2

simple, linear model; a correlation coefficient (r) greater than 0.71

explains more than 50% of the data variance. Unfortunately, very

few case histories in rock mechanics compare calculated with

measured displacements, so there is no large body of evidence nor

consensus of expert opinion as to what may be considered a

reliable fit of model results to mine measurements. A correlation

coefficient of 0.7 or higher may be considered a satisfactory fit,

especially in view of the effects of inelastic phenomena about

many mine openings. Regression analyses thus provide guidance

toward determining an appropriate scale factor for model

calibration, but such analyses are not absolute.

Table 4 shows the results of several regression analyses of

calculated extensometer displacements on measured

displacements. These displacements are relative between

extensometer points in both model and mine data and are

accumulated differences between the time of installation

Table 4.—Regression analysis of calculated extensometer displacements on measured displacements

Extensometers used in Number Regres- Regres- Regression Maximum Maximum Exten- 

 Line regression analysis of data sion line sion line correlation measured calculated someter

Number points slope intercept coefficient displace- displace- lengths, Type Number

r ment, mm ment, mm m 

1 ...... Shaft . . . . . . . . . . . E3 27 0.84 0.01 0.96 3 3 30

2 ...... Shaft ........... E1, E9, E10 81 0.25 0.00 0.65 2 0.5 30, 60

3 ...... Stope .......... E14, E17, E18 17 2.43 0.07 0.74 6 20 18

4 ...... Combination of

 2 and 3 ........ E1, E9, E10, 98 2.68 &0.02 0.81 6 20 18, 30, 60

E14, E17, E18

5 ...... Combination of

 1, 2, and 3 ..... E1-E3, E9, 152 1.90 &0.01 0.71 6 20 18, 30, 60

E10, E14, E17,

E18

6 ...... Combination of

 1 through 5 ..... E1-E3, E7- 206 1.78 0.00 0.68 6 20 18, 30, 60

E10, E14, E17,

E18
8

and pillar stoping time. Displacement changes were calculated in guarantee that the only points in the elastic domain are present in

accordance with the mining sequences in table 3, beginning with the data. Overestimated (model) displacements may result from

the historical sequence and proceeding through the pillar mining rock mass model strengths that are too high, as well as elastic

calibration sequence. The number of points in table 4 was moduli that are too low. In addition, experience with

determined by the number of pillar mining steps, the number of extensometer data shows that the goodness of fit indicated by the

extensometers, and the number of anchor points per correlation coefficient may be influenced by the magnitudes of

extensometer. the measured displacements (Pariseau and Duan, 1989). Figure

The shaft extensometer regression data in line 1 of table 4 10 shows such an effect in data obtained from an earlier study of

shows a very high correlation coefficient (r = 0.96) and a vertical crater retreat mining at the Homestake Mine (Pariseau,

regression line slope of 0.84, which suggests that a somewhat 1986). Inspection of figure 10 shows that displacements greater

lower modulus scale factor should be used (84% of the input 0.25 than about 2 mm (0.08 in) are needed to achieve a correlation

modulus scale factor). However, the shaft extensometer coefficient greater than 0.70. Smaller displacements lead to low

regression in line 2, which has a lower correlation coefficient (r = correlation coefficients. The same phenomenon appears in the

0.65) but which is based on many more data points, suggests a shaft pillar data; large stope displacements lead to high correlation

much lower scale factor. The shaft displacement data in both coefficients, while the smaller shaft displacements are not well

lines 1 and 2 show a maximum displacement of about 3 mm (0.1 correlated. Thus, as a practical matter, much of the rather small

in) over lengths of 30 to 60 m (100 to 200 ft), that is, less than amount of displacement from the shaft extensometers may simply

one part in ten thousand, which is a uniaxial strain equivalent of be noise. Despite such complications, some justification for an

0.01%. increase in the elastic properties scale factor could be made.

The stope extensometer regression in line 3 shows much larger An accurate, independent determination of rock mass strength

displacements. Calculated displacements reach 20 mm (0.8 in) would help to clarify the situation. However, the nonlinearity

over a length of 18 m (60 ft), which amounts to roughly one part introduced by yielding in the model and in the mine pose

in one thousand, or about 0.1% uniaxial strain equivalent. The substantial analytical difficulties that remain to be resolved. The

correlation coefficient (r = 0.74) is in the satisfactory range, but strength scale of 0.5 adopted for the Ross shaft pillar project was

the calculated displacements are much larger than those measured. based on (1) the extent of yielding inferred between the stope

This overestimation is reflected in the regression line slope of hanging wall and the development drift on the 3650 level during

2.43, which indicates that the input scale factor of 0.25 should be calibration of the two-dimensional model (Pariseau and others,

increased by 143%, to 0.61. 1995b) and (2) a simple analytical relationship between two- and

Combinations of shaft and stope extensometer regressions are three-dimensional back-analyses of long cylindrical and spherical

shown in lines 4, 5, and 6 using an increasing number of data excavations. Although better fits of the model results to mine

point pairs (98 to 206). The regression line slopes decrease from measurements were desired, correlation coefficients of about 0.7

2.68 to 1.78, indicating a modulus scale factor ranging from 0.67 were considered satisfactory for the initial calibration of the threeto 0.45. The correlation coefficient decreases from 0.81 to 0.68. dimensional shaft pillar model. Thus, scale factors of 0.25 and

These results suggest a modulus scale factor that is equal to the 0.50 for elastic moduli and strengths were adopted for the Ross

strength scale factor (0.50). The results also suggest that the shaft pillar model.

alternative strain-to-failure criterion may be applicable. However,

the case is not clear because there is no

APPLICATION OF THE CALIBRATED MODEL

The most important application of the calibrated model was an pillar mining were calculated in advance to detect any substantial

assessment of shaft stability with respect to the effects of pillar departure from model forecasts as pillar mining proceeded.

mining. The model was also applied to the functioning of the Safety factor distributions and changes in safety factor induced

two horizontal wing pillars in the main shaft pillar, the role of the by various pillar mining sequences were the most important tools

central pillar, and potential deformation of shaft framing. Figure for interpreting model results in terms of shaft stability. Direct

11A shows the central pillar in a plan view of the 3650 level; model output in the form of numerical values of displacements,

figure 11B shows the wing pillars in a vertical east-west section strains, and stresses induced by mining are not directly associated

through the original shaft pillar. Ex- tensometer readings with objective criteria of stability. The 

expected at various stages of future
9

safety factor actually used for stability assessment was the ratio of in the first three-dimensional, finite-element simulations of

strength to stress, where strength is stress at the limit of elastic historical and planned pillar mining that called for a central pillar

deformation. Any tendency to increase stress beyond the elastic and two horizontal wing pillars to be left within the main shaft

limit would result in inelastic yielding, that is, elastic-plastic pillar.

deformation. The distribution of the shaft wall safety factor as a function of

Specifically, the yield condition (Y) that marks the limit to depth at various stages of mining is shown in figure 12. In this

purely elastic deformation is Y = J + I, where n = 2 in the regard, the shaft wall safety factor is the lowest factor of safety in (n/2)

present analysis, J = an anisotropic expression that reduces to a any element at the shaft wall. This element (27) tended to be the

dimensionless second invariant of deviatoric stress in the isotropic same regardless of depth and is identified in the insert mesh

case, and I = an anisotropic form that reduces to a dimensionless illustrated in figure 5. Figure 12 shows that most of the change

first invariant of stress in the isotropic case (Pariseau, 1972). in the shaft wall safety factor had occurred in association with

When n = 1 and isotropy prevails, one obtains the Drucker-Prager historical mining. Simulation of the original pillar mining plan,

yield condition. The isotropic form of the yield condition has a in which the central pillar and the two wing pillars were left,

parabolic shape (n = 2) in principal stress space and depends on showed little change subsequent to historical mining. Nowhere

all three principal stresses. No assumption is necessary along the shaft route did the safety factor become less than about

concerning the direction or magnitude of the intermediate 1.0; the shaft wall remained in elastic ground.

principal stress, which is often the case in many two-dimensional As determined by the shaft wall motions, model results are

models based on the Mohr-Coulomb criterion. This is the same illustrated in figure 13. The bulges in figure 13 draw attention to

yield condition that was used in the two-dimensional model the influence of pillar mining on shaft wall position.

(Pariseau and others, 1995a; 1995b). When the elastic limit is

reached, Y = 1. The greatest value of (J) possible is thus (J) WING PILLARS 1/2 1/2

max

= (1 & I) , which depends on element stresses. The actual value -n

is (J) , which also depends on the element stress state. The The two wing pillars accounted for about 25% of the original 1/2

act

element safety factor is then the ratio of (J) to (J) . Uniaxial shaft pillar ore reserve. The wing pillars were introduced into the 1/2 1/2

max act

compressive, tensile, and shear failures are special cases that may pillar extraction plan shortly after the first lifts were taken below

arise in response to the evolving element stresses; they are the 3650 level. At that time, cracking was again initiated near the

handled automatically during the model run. site of original shaft damage on the 3200 level and in the pump

Structural criteria for shaft bunton safety were not developed, room on the 3650 level. No three-dimensional model results

but a simple calculation shows that a 6-mm (0.25-in) axial were available for design guidance at the time.

compression of a 6-m (20-ft) long mild steel beam causes the Subsequent development of the three-dimensional model and

stress to reach the elastic limit. This calculation suggests that shaft simulation of alternative shaft pillar mining sequences in which

wall closure greater than 6 mm (0.25 in) may result in structural the wing pillars were recovered showed few differences in safety

damage to shaft framing. Of course, actual shaft wall motion is factor when compared with sequences in which the wing pillars

a complex, three-dimensional phenomenon and varies in time as were left. As before, most of the changes in safety factors in the

mining proceeds. The accuracy required to use finite elements to shaft wall occurred with historical mining; pillar mining induced

predict structural damage was considered beyond the accuracy of only a small change, and the wing pillars had little effect.

the geomechanics model. Model results did indicate several The fact that model results continued to show that observed

centimeters of shaft wall displacement. However, the relative ground motion and extensometer readings were within

displacements between opposing shaft walls were much smaller. expectations gave credibility to the suggestion that the wing

The opening appeared to "float" with the rock mass in response pillars were not needed to ensure shaft stability. This result was

to historical and pillar mining. timely enough to be incorporated into the decision to extract the

SHAFT STABILITY a future date.

The Ross shaft was considered safe and stable with respect to CENTRAL PILLAR

large, possibly catastrophic, ground motions if shaft wall

deformations induced by mining in the shaft pillar remained in The decision to create a central pillar was based on a

the elastic range. Yielding, if present, could not encompass the premining analysis of a two-dimensional plan view that involved

shaft. Some localized yielding in the skin of an opening could be simulation of historical mining north and south of the original 60-

tolerated because such yielding could be controlled easily by m (200-ft) wide shaft pillar followed by simulations of various

bolting, screening, or other conventional ground-control s h a f t pillar mining sequences.

measures. These safety and stability criteria were met 

upper, second wing pillar and to recover the lower wing pillar at
10

Extraction of the full width of the original shaft pillar im- the safety factor was reduced to 1. Figure 15 shows that the

mediately west of the Ross shaft produced large displacements at calculated yield zone was confined to elements at the shaft wall

the shaft walls. The central pillar reduced shaft wall and was limited to a few elements near the shaft corners.

displacements associated with pillar mining to about 9.5 mm from Although absolute displacement of the shaft wall is not a

24.1 mm (0.38 in from 0.95 in) or about 60% in simulations reliable index to stability, displacement calculations assist in

using an 18-m (60-ft) central pillar. The central pillar was visualizing shaft wall motions and provide a basis for estimating

introduced into the original pillar extraction plan from the outset what extensometer readings will be. The major component of

and also accounted for about 25% of the total ore reserve. shaft wall displacement is toward the pillar mining region, that is,

A plan was developed to extract the central pillar following in an east-west direction. Figure 16 shows calculated east-west

recovery of the ore reserves in the shaft pillar. Mining would displacement of the southeast corner of the shaft wall. The

proceed from the lower levels upward using mechanized cut-and- negative displacement in figure 16 is westward toward pillar

fill methods as before. The geomechanics model would be mining. Each mining phase results in additional displacement,

recalibrated at the same time existing stope instruments were that is, historical mining resulted in about 1 cm (0.5 in) of

repaired and new instruments installed. Extensometers would be displacement, pillar mining roughly doubled shaft wall

used to measure pillar displacements induced by mining on either displacement, and simulation of central pillar mining indicated an

side of the central pillar between the 3650 and 3350 levels. additional 1 cm (0.5 in) of shaft wall displacement.

Unfortunately, operational demands prevented implementation Mining-induced shaft displacements will be superposed on

of the plan before the stopes on either side of the central pillar had long-term creep and will eventually require adjustment of shaft

been completed. Consequently, mining of the central pillar had guides to ensure that conveyances will operate smoothly. The

to be simulated using the scale factors for the original rock exact timing is uncertain, but it can be estimated by monitoring

properties, that is, 0.25 and 0.50 for elastic moduli and strengths, and inspection.

respectively. The simulations were carried out before the first lift

was taken in the central pillar. Table 3 shows run number and SHAFT FRAMING

cut, beginning with the original Ross shaft cut (CT1), proceeding

through historical mining (CT6) and pillar mining on the north Relative displacements between shaft walls provide a measure

and south sides of the central pillar (CT20), and through mining of shaft deformation that may be transmitted to shaft buntons and

the central pillar (CT26). other frame components. Shaft strains were defined as relative

Figure 14 shows the distribution of the shaft wall safety factor displacements per unit of shaft dimension (width and breadth in

at the end of the Ross shaft cut (CT1), at the end of historical a horizontal plane). Figure 17 shows these strains as a function

mining (CT6), just before the central pillar was mined (CT20), of depth after the shaft cut, at the end of historical mining, before

and after the central pillar was extracted (CT26). Most of the central pillar mining, and at the end of central pillar mining (CT1,

decrease in safety factor occurred in conjunction with historical CT6, CT20, and CT26, respectively). Figure 17 also shows that

mining. Stress relief associated with pillar mining actually the greatest compressive strain, 300 microstrain, is associated with

increased shaft wall safety factors between the 3200 and 3650 historical mining, whereas an extension strain of 500 microstrain

levels. Below the 3650 level, a small additional decrease is shown at the end of simulated central pillar mining. These

occurred. This decrease was enough to cause some shaft wall ele- estimated deformations are below an elastic limit of steel, for

ments to respond inelastically as example, 1,000 microstrain.

DISCUSSION

Repeated analyses using different combinations of elastic measurements and observations. However, the computational

moduli and strength scale factors might lead to a better overall fit effort would be enormous, and there is no guarantee that the

of model displacements and yield zones to extensometer readings results would be much different or more reliable. In this regard,

and yield zones actually seen in development openings and stope the more general back-analysis procedures (including the inverse

walls. More elaborate back-analyses that include the characterization problem, as well as the direct approach used here)

preexcavation stress state could also be done for the purpose of are reviewed by Gioda (1985), who points out that "it is

achieving better agreement with mine impossible to carry out a meaningful 
11

calibration if the values of the measured data are too close to the The three-dimensional geometry and geology of hard-rock

tolerance of the measuring device." Gioda also notes that "a mines continues to defy rapid preparation of meshes and easy

careful choice of the type of quantity to be measured and the visualization of results. While increased storage capacity and

locations where the measurements are performed" is essential. In speed of computer hardware makes 24-h turnaround time feasible

this regard, the resolution of an extensometer may be a fraction of for medium-scale finite-element meshes (.30,000 nodes),

a millimeter in the laboratory, but in mining practice, sensitivity advances in low-cost, user-friendly meshing and visualization

appears to be much less. tools remain a pressing need.

CONCLUSION

In conclusion, this study has demonstrated, through an and design where the cost is justified by the importance of the

integrated program of laboratory testing, stress analysis, and mine mining objective, in this case, the safe and profitable recovery of

measurements, that three-dimensional, finite-element analysis is ore in shaft pillars.

an engineering tool for mine stability analyses

ACKNOWLEDGMENTS

The Ross shaft pillar project involved the efforts of many assistance during all phases of the project. A grant of computer

individuals. Special thanks go to Allen Winters, vice-president time from the UTAH Supercomputing Institute, which is funded

and general manager, and his excellent team at the Homestake by the State of Utah and IBM Corp., is also gratefully

Mine, for providing access to the mine and acknowledged.

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