APPENDIX A._OVERCORE STRAIN DATA FROM HOLLOW INCLUSION CELLS

Figure A-1 shows the overcore strain data collected for mean the bit had passed the gauges. The ordinate (ver-
each HICell measurement. The abscissa (horizontal axis) tical axis) is the measured value of strain in units
is the distance between the drill bit and the centerline of of microstrain. Oscillations of measured strain from
the rosettes measured in inches. Negative values indicate gauge 9 in these plots is probably a fault in the recording
the bit was approaching the gauges, while positive values channel.

05

1.0 F e Gauge 1 VGauge 6
mGauge2 Gauge 7
1.55 0 Gauge3 @Gauge 8
4 Gauge 4 vGauge 9

-2.0 0 Gauge 5
Litt oo} oj jf 1 i
eS a 68

MICROSTRAIN, 103

[- Top, looki
40 into borehole

aed)
-15- 2,3 ,
15 ;

2.0 64,5
-osLLLILititiiiiit
9-8 -7-6 -5-4-3-2-1012345

DISTANCE TO GAUGES, in

Figure A-1.—Overcore readings from HiCells. A, S1H1C1; B,
$1H2C1; C, SH3C1; D, S1H3C2; E, S2H1C1.

it
678


Six HICells and six BDG’s were installed at site 1; of
these, four HICells and three BDG’s were successfully
overcored. Appendix A contains plots of all successful
HICell overcore strain measurements. HICell gauge re-
sponse during overcoring appears normal with the excep-
tion of gauge 9, where oscillations in the data were
probably caused by a fault in the recording channel.
Fortunately, gauge 9 was a redundant angle gauge.
Appendix B contains plots of the BDG overcoring data.

Four HICells were installed at site 2, but only one was
successfully overcored. Sectioning the retrieved HICell
overcores with the gauges still in place showed that the
failures were caused by air bubbles in the gauge cement
and improper bonding of the gauge to the borehole wall.
Because three nonparallel boreholes are needed to obtain
sufficient data from a BDG to estimate the stress state and
only two boreholes were drilled, it was not possible to
determine stresses with the BDG at site 2.

Biaxial testing of HICell overcores for Young’s modulus
and Poisson’s ratio also proved troublesome; only a single
test was completed. Partial data were obtained from four
other overcores, including the results of two uniaxial tests.
As a consequence, the estimate of Young’s modulus
needed for data reduction was highly uncertain. Anisot-
ropy added to the uncertainty; this topic is discussed in the
"Anisotropy" section. Figure 9 shows a biaxial test being
performed in the mine.

RESULTS

The in situ stresses were calculated from measured
HICell strains and BDG displacements using a data re-
duction program modified by Larson (4). Both the manu-
facturer’s original program and the modification by Larson

Figure 9.—Connecting overcored HiCell to strain indicator to
conduct biaxial chamber test.

assume linearly clastic and isotropic rock behavior. Both
use a multiple variable linear regression model to find a
best fit to the HICell strain measurements. This approach
views the nine HICell strain measurements available for
determination of the six independent components of stress
as an overdetermined system and successively eliminates
three redundant gauge measurements to obtain the best fit
or the solution with the least variance. Only redundant
measurements may be eliminated. For example, only one
of the two axially positioned gauge measurements can be
discarded in the process of seeking a best fit.” A similar
approach was applied to the BDG measurements of bore-
hole wall displacement.

The best-fit cartesian components of stress obtained
from all HICell measurements at sites 1 and 2 and the
BDG measurements at site 1 are given in table 1. The
results show a consistent ordering of the normal stresses
at site 1. With the exception of the measurement from
S1H3C2,§ the vertical stress is always the largest normal
stress; the east-west stress is intermediate in value; and the
north-south stress is the smallest normal stress. The east-
west direction is perpendicular to strike; the north-south
direction is parallel to strike. The shear stresses tend to
be an order of magnitude smaller than the normal stresses
and vary in algebraic sign, although the north-vertical or
south-vertical shear stresses from each HICell are always
negative. A shear stress sign reversal implics a reversal in
direction of action.

7A HICell has two axial gauges (1, 7), three circumferential gauges
@, 6, 8), and four angle gauges (3, 4, 5, 9) oriented at 45° to the cell
axes.

In the notation, the letter and the first digit indicate site, the second
letter and the second digit indicate borehole, and the third letter and the
third digit indicate cell. Thus SIH3C2 means site 1, borehole 3, cell 2.

Table 1.—Best fit stress components for each test
at sites 1 and 2, pounds per square inch

(Compressive stress Is positive)

Shear
Normal North- East-  North-
Test North-East) Ver «Southwest south
south west tical. «ands and_—aand
east- ver- ver
west tical tical
SITE 1
SIHIC1.. 2,218 2,245 «4,868 =—-327,—844 7
S1H2C1 2674 2,771 3,551 920-471-171
S1H3C1 1,962 2,419 3,838 132 1,187-379
S1H3C2.. 2,575 7,506 4,409 «858 -1,373 -1,193
BOG .... 2830 4,750 5,170 1,590 130-300
SITE 2
S2HiC1 .. 4,288 2,921 5153379475



deg
ft

in

UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT
degree pet percent
foot psi pound per square inch

inch



APPENDIX B.-OVERCORE PLOTS FROM BOREHOLE DEFORMATION GAUGES

Figure B-1 shows the overcoring response of BDG at —_ gauge; however, the ordinate represents the change in the
site 1 for boreholes 1, 2, and 3. Again, the abscissa isthe diameter of the borehole as measured in inches.
distance between the drill bit and the centerline of the

-25b KEY

L O Vertical
~30 0 -60°

A+60°
-35

bo af

f
y
oad ao

DIAMETER CHANGE, 10+ in

&

-35 ae es Ca a eS
6 54-32-1012 3 4 5 6
DISTANCE TO GAUGE, in

Figure B-1.—Response of BDG to mining, site 1. A, Hole 1; B,
hole 2; C, hole 3.


An alternative view guided by rock mechanics principles
(7)is to average the axial strain measurements before pro-
ceeding to an analytical estimate of the three normal
stresses and the shear stress Om, Oy Oz and o,, in a
plane perpendicular to the borehole axis (the z-axis).
The z-direction shear stresses o,, and o,, can then be
estimated from the three shear strains calculated from the
three three-gauge rosettes in a HICell. The latter calcu-
lation may be expressed analytically as a least squares fit
of the two shear stresses to the three measurements.

The alternative approach is based on average axial
strains and may lead to significant differences in results
when the axial strains are substantially different from one
another. A theoretical analysis of strain relief using over-
coring shows that the axial strains are independent of posi-
tion about the circumference of the HICell. Ideally, the
axial strain measurements should be equal in isotropic
tock, Indeed, a criterion for measurement validity could
be based on equality of the axial strains, In actual prac-
tice, it would be quite unusual if the axial strains were
equal; averaging the axial strains seems reasonable. An

unanswered question is how different the axial strains can
be before the data are deemed invalid.

The data presented in table 5 illustrate the effects of
different treatments of axial strain in the reduction of the
HICell data from site 2. The four K-factors? used in the
data reduction were set to 1 for the comparisons. Inspec-
tion of the results given in table 5 show that reversal of
normal stress ordering relative to data from site 1 was
present in the original least squares, best fit approach.
The results using the high values of axial strain are quite
close to results from the original analysis. The reason
is that the low value is eliminated during the process of
obtaining a best fit. However, the reversal does not occur
when the axial strains are averaged before the least
squares approach is applied. Averaging not only removes
the reversal, but substantially reduces the north-south
stress as well, The same results occur when the low value
of axial strain in the site 2 data are used. The results from
axial strain averaging from site 2 are much closer to those
from site 1 (table 3).

Table 5.—Different axial strain treatments and results from site 2, pounds per square Inch

(Comprossive stress is positive)

‘Shear
Treatment Normal North-south East-west North-south
North-south —_East-west Vertical and and vertical and vertical
east-west
Original 4,442 2,658 5,789 105 787 +735
Low . 2,862 2,978 5,684 -199 710 534
Average 2,649 2,964 5,639 417 698 878
High .. 4273 2,596 5,706 221 853 567

SUMMARY AND CONCLUSIONS

The measured and calculated shaft pillar stresses at
site 1 and measured stresses at site 2 are summarized in
table 6. Although the data are sparse, several conclusions
scem justified.

1, The measured normal stresses at site 1 are less than
those estimated from finite-element simulations of actual
mining. The finite-element results are therefore pessimis-
tic and somewhat conservative.

2. Differences in normal stresses between sites 1 and 2
are the same magnitude as differences within site 1, so
that horizontal gradients of normal stress between sites 1
and 2 cannot be determined reliably. The shear stresses

are an order of magnitude smaller than the normal
stresses and show a considerable percentage variation that
masks the smaller absolute differences,

A more detailed investigation of the physical appro-
priateness of the least squares, best fit approach seems
warranted. Also needed is a more elaborate analysis of
axial strain effects on the data reduction procedure for
anisotropic rock,

°K-factors are defined as correction factors that compensate for the
gap between the rock and the strain gauges (4).


specimens taken from a borehole or from rock masses the
size of an ore body or larger. Thus, variability occurs over
a multiplicity of scales and reflects interactions among
applied load, geologic structure, and rock type. Applied
loads are primarily gravitational and tectonic.

Pair-wise differences are commonly observed among
stress measurements (1) from the same borehole, (2) from
different boreholes at the same drill site, (3) from different
drill sites on the same level, and (4) from drill sites on
different levels in the same mine. Such differences may
result from differences in the types of gauges used or data
reduction procedures, as well as actual differences in in
situ stresses.

Over distances of about 1,000 ft, approximately the
linear dimension of a finite-element model of a typical
hard-rock mine stope, it may be practical to take into
account stress field changes. Thus, a difference in stress
between two sites may be important in constructing a
numerical model if the distance between these sites is
about the same as the linear dimensions of the area of the
mine being modeled. For example, a finite-element model
of a stope in a mine where levels are 150 ft apart would
be about 1,000 ft high. A reasonable assumption would be
that the vertical stress gradient caused by gravity loading
alone would be about 1.2 psi per foot of depth, The

difference in the vertical stress component (gravity)
between the top and bottom of the model would then be
about 1,200 psi. If the stope were centered in a
moderately deep mine, say, at the 3600-ft level, then the
difference in vertical stress between the top and the
bottom would be more than 25 pct of the vertical stress at
the center of the stope. If the same stope were centered
at the 7200-ft level, the difference would only be about 13
pet and perhaps of no great concern.

Horizontal gradients in stress should also be considered
in numerical analyses of mine opening stability. However,
there appears to be little discussion in the technical lit-
erature of how horizontal stress gradients might be incor-
porated into such an analysis. When considering vertical
stress gradients, there is a simple physical basis for deter-
mination, that is, gravity loading. The situation is much
less clear for horizontal gradients where the specific
physical cause is not known. Measurements must then be
relied upon to reveal the presence of horizontal stress
gradients. However, if the differences in stresses between
sites about 1,000 ft apart horizontally are similar to the
differences between measurements taken from the same
borehole or at the same site, then it is questionable
whether there is any justification for including horizontal
stress gradients in a model analysis.

WORKING HYPOTHESIS

A simple hypothesis for approaching the problem of
determining stress gradients is to consider the model
region as the neighborhood of a mathematical point. Let
the state of stress o at point x,y,z, and time t, be
characterized by the six independent components 0,4) dy,
F,23 Oxy) Fy OF, Of the stress tensor referred to a fixed
cartesian coordinate system. The stress field at nearby
point xy,z and later time t may be estimated from known
data by a Taylor series expansion (equation 1) about the
given point. Thus,

AKYZt) = FC YorZorto) + (80/Ax)dx + (0/ay)dy
+ (80/a2)dz + (a0/at)dt
+ higher order terms, (1)
where o represents any one of the six stress components.

The derivatives in equation 1 are evaluated at the given
point.

If the in situ stress state varies slowly over geologic
time, then

(80/at) = 0 1)

over the time spans of engineering interest, and only spa-
tial variations in the in situ stress field need to be
determined. Negligible variations over time are implied
in most in situ stress measurements, although time-
dependent, postmining stress changes are considered in
some cases, for example, in salt mines. Lateral variation
is also considered negligible in most determinations of
the premining stress state.
The change in stress Ac between xy,z and x,J,y2, is

Ao = (80/8x)dx + (30/ay)dy + (80/82)dz. (3)

The smaller the stress gradients 30/8x, c/ay, and
40/82, the greater the distance (as measured by dx, dy,


INTRODUCTION

The U.S. Bureau of Mines and the Homestake Mining
Co. are engaged in a cooperative effort involving studies of
rock mechanics and the stability of pillar mining near the
Ross shaft at the Homestake Mine, Lead, SD. The project
is part of Bureau research into advanced concepts for
modern mine design in support of the Bureau’s goal to en-
hance recovery of the Nation’s mineral resources. The re-
search goal is to advance design technology in the complex
three-dimensional realm of underground hard-rock mining.

Large-scale, catastrophic motion about a shaft is clearly
unacceptable. However, shaft wall displacements that can
be accommodated operationally without serious interrup-
tion of the hoisting schedule may be tolerable. Thus, the
key to a successful engineering plan for shaft pillar min-
ing is to ensure that movement in a rock mass is within
expectations.

An effective approach to studying rock mass movement
combines laboratory testing for rock properties, numerical
modeling and simulations of various pillar extraction se-
quences, measurements of stress and displacement in a
mine, and monitoring shaft reactions to mining. Meas-
urements and monitoring provide data for model valida-
tion, calibration, review, revisions, and possible extension
of the model from two to three dimensions. A simple
model is preferred at the outset, but when mine measure-
ments and monitoring data indicate that the complexities
of mine geology and mine geometry require a more com-
prehensive model, then revisions and extensions are
necessary.

The Ross shaft pillar is in high-grade ore on the west
side of the shaft and extends 100 ft north and south of the
shaft. Vertical extent is indefinite. Extensive mining in
the past led to the formation of the shaft pillar and un-
doubtedly modified the original stress ficld about the
Homestake Mine. Stress measurements in the pillar pro-
vided the data needed to compare the results of finite-
element simulations of historical mining to the present
(posthistorical, prepillar mining) stress state. The present
stress state is important because future mining in the pillar
and model simulations depend on the stresses at the start
of pillar mining.

This Report of Investigations is the first of several
related to the Ross shaft pillar study. It discusses the
results of stress measurements taken at two sites on the
3650 level of the Homestake Mine, where a large percent-
age of the shaft pillar ore is found. Site 1 (fig. 1) is
located near the center of the Ross pillar about 250 ft west
of the shaft and 20 ft south of the shaft centerline. Site 2
(fig. 1) is located approximately 750 ft southeast of site 1
in a relatively remote region of the mine.

The immediate objectives were to determine (1) the
present state of stress in the Ross shaft pillar and (2) the
horizontal stress gradients in the same area. The overall
objective was to advance the state of the art of mine
design, The Homestake Mine site provided an opportunity
for full-scale testing of concepts that have industry-wide
applicability.

ACKNOWLEDGMENTS

The in situ stress measurement project involved the
efforts of many people. Special thanks go to manage-
ment and technical personnel at the Homestake Mine,
particularly Al Winters and Carl Schmuck, who provided
access to the mine, and to Wayne Corso, who provided

engineering support. Thanks also go to Dave Benjamin
and Paul Hyndman of the Bureau’s Western Field Op-
erations Center, Spokane, WA, who assisted with mapping
the geology of the mine sites.


IN SITU STRESS MEASUREMENTS NEAR THE ROSS SHAFT PILLAR,
HOMESTAKE MINE, SOUTH DAKOTA

By J. C. Johnson, W. G. Pariseau,? D. F. Scott,? and F. M. Jenkins*

ABSTRACT

In situ stresses are important input data for the design of safe, stable stope layouts and extraction
sequences. However, it is commonly assumed that normal and shear stress gradients in a stress field
are negligible and, consequently, that stresses are uniform throughout the region of analysis. To
evaluate these assumptions and to provide input data for an analysis of a shaft pillar mining plan, in situ
stresses were measured at the Homestake Mine, Lead, SD, using overcoring on hollow inclusion cells
and borehole deformation gauges.

These measurements were used with conventional least squares, best fit data reduction schemes. The
results showed that normal stresses within the shaft pillar were less than, but comparable to, those
estimated from finite-element modeling. Shear stresses were an order of magnitude less than normal
stresses and varied considerably from site to site. Possible causes of this variability include effects of
scale, elastic moduli, anisotropy, and axial strain. Because of the variability, it is not possible to
determine the magnitude of horizontal stress gradients.

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

?Mining engineer, Spokane Research Center; McKinnon professor of mining engineering, University of Utah, Salt Lake City, UT.
3Geologist, Spokane Research Center.

‘Staff engineer, Division of Health, Safety, and Mining Technology, U.S. Bureau of Mines, Washington, DC.


Table 6.—Stresses near Ross shaft pil

at sites 1 and 2, pounds per square Inch

(Compressive stress is positive)

Shear
Site and calculation North-south East-west North-south
North-south Vertical and and vertical and vertical
east-west
SITET
2,387 3,735 4,167 189 “597 -453
Finite-element model... 2,577 4,882 6,135 -258 76 256
SITE2
‘Axial strain average .... 2,649 2,964 5,639 417 698 878
REFERENCES

1. Pariseau, W.G. Research Study on Pillar Design for Vertical Cra-
ter Retreat (VCR) Mining (contract JO215043, Univ. UT). BuMines
OFR 44-86, Oct. 1985, 233 pp.; NTIS PB 86-210960.

2. Jenkins, F. M., and R. W. McKibbin. Practical Considerations of
In Situ Stress Determination. Paper in Application of Rock Character-
ization Techniques in Mine Design (ed. by M. Karmis). Soc. Min. Eng.
AIME, Littleton, CO, 1986, pp. 33-39.

3. Commonwealth Scientific and Industrial Research Organization
(Warwick, Queensland, Aust.). Field Manual C.S.LR.O. HI Stress Gage.
1983, 98 pp; distributed by Geokon, W. Lebanon, NH.

4. Larson, M. K. STRESsOUT—A Data Reduction Program for In-
ferring Stress State of Rock Having Isotropic Material Properties: A
User's Manual. BuMines IC 9302, 1992, 163 pp.

$. Duan, F. Numerical Assessment of the Importance of Anisotropy
to Steeply Dipping Vertical Crater Retreat Stopes. M.S. Thesis, Univ.
UT, 1986, 178 pp.

6. Amadei, B. Rock Anisotropy and the Theory of Stress Measure-
ments. Springer-Verlag, New York, NY, 1983, 478 pp.

7. Duncan Fama, M. E., and M. J. Pender. Analysis of the Hollow
Inclusion Technique for Measuring In Situ Rock Stress. Int. J. Rock
Mech. Min. Sci. & Geomech. Abstr., v. 17, 1980, pp. 137-146.



True north

Mine north
° 20
Ld
Scale, ft
LEGEND

a Fracture along bedding
b Fracture, 1-2 in wide open
c Bedding
d Fracture
B1 BDG overcore
C1

HICell overcore

Poorman Formation

Poorman Formation

Poorman Formation

Homestake Formation

Figure 3.—Geologle structures and overcoring locations at site 2.

BACKGROUND

In situ, or premining, stresses are the loads applied to
excavations in rock. These stresses play a major role in
numerical analyses of the safety and stability of mine
openings. It is therefore important to determine pre-
mining stresses as accurately as possible within the
practical constraints of cost and data variability.

In situ stress measurements require specially trained
personnel and equipment and are relatively expensive to
obtain. The cost often limits the number of measurements
to those that can be taken from boreholes collared from
a single drill station. Mapping a highly variable in situ
stress field over a large region is not practical despite
advances in equipment and techniques that allow three-
dimensional stress states to be determined from a single

gauge. However, if the premining stress field is homoge-
neous, then a single, reliable stress measurement is suf-
ficient to define the in situ stress field. The difficulty
is that stress ficld homogeneity, or lack of it, cannot be
known until a sufficient number of measurements are
made,

In practice, the number of measurements actually col-
lected is usually far less than what is needed for achieving
a high confidence level in statistical analyses. The reason
is large measurement variability. In the present context,
variability refers to differences in the outcome of in situ
stress measurements and is closely linked to heterogeneity
of the rock mass as well as experimental error. Rock
mass heterogeneity ranges from laboratory-sized test


REPORT OF INVESTIGATIONS/

In Situ Stress Measurements Near the Ross
Shaft Pillar, Homestake Mine, South Dakota

By J. C. Johnson, W. G. Pariseau, D. F. Scott, and F. M. Jenkins

NITED STATES DEPARTMENT OF THE INTERIOR

BUREAU OF MINES


homogencous rock mass, the normal stresses in equation
11 are principal stresses. Thus, with compression positive,

Oy = Oy (12a)
oy = oy (12b)
and On = Oy (12¢)

At the Homestake Mine, an analysis of previous stress
measurement data (1)° leads to the calculations

oy = 1252, (13a)
oy = 2,078 + 0.53 z, (13b)
and oy, = 121 + 0557, (13e)

where z = depth below ground surface. These calcula-
tions are consistent with simplified equation 11 and suggest
vertical loading by gravity only, but with horizontal loads
in excess of those caused by gravity alone. At the 3650
level near the Ross shaft pillar, the stresses calculated
from equation 13 are 0, = 4,563 psi, 0% = 4,013 psi, and
oy = 2,129 psi.

The in situ stress hypothesis represented by equation 9
assumes linear premining stress gradients and implies that
the stress gradients are independent of position. Thus, a
linear regression analysis is an alternative approach to
defining the coefficients in equation 9. A companion anal-
ysis of variance of measurements taken within a given site
and between sites could also be done, provided that the
number of measurements justified a statistical approach.
Unfortunately, this is seldom the case.

MINE MEASUREMENTS

The stress relief overcoring method was used for all
stress measurements, Hollow inclusion cells and borehole
deformation gauges were used to provide a check on
results. Difficulties in gauge cement and biaxial chamber
tests were encountered. The working environment was
severe; time prevented taking as many measurements as
desired.

PROCEDURES

Nearly horizontal 1.5-in-diam boreholes were drilled
about 30 ft deep at cach site. Drift dimensions near the
measurement sites are about 20 ft wide by 10 ft high.
Thus, to reduce the influence of stress concentration at a
gauge site, the first gauge was installed 20 ft into a hole.

Commonwealth Scientific and Industrial Research
Organization (CSIRO) hollow inclusion cells (HICells)*
(fig. 4) and the Bureau’s borehole deformation gauge
(BDG) (fig. 5) were used to collect all stress meas-
urements. The HICell contains three three-gauge strain
rosettes and allows for the determination of the three-
dimensional stress state from a single overcore. The BDG
requires three overcores, one each in three nonparallel
boreholes. Gauges of both types were installed in bore-
holes at site 1, but only two boreholes were drilled at
site 2.

Italic numbers in parentheses refer to items in the list of references
preceding the appendixes at the end of this report.

Reference to specific products does not imply endorsement by the
US. Bureau of Mines.

The location of all gauges and holes at site 1 and site 2
are shown in figures 2 and 3, respectively. Installation and
data reduction were according to procedures described in
references 2 and 3. After gauge installation, stress relicf
was obtained by overcoring (figs. 6-7). Figure 8 shows the
drill station in a drift. Strain measurements were continu-
ously recorded during the overcoring process. A heating
unit was used to maintain drill water at a constant temper-
ature during overcoring and thus prevent spurious thermal
strains from appearing in the measurements.

hee oe

ae
ee

Te

Figure 4.HICell assembled for installation. When forced to
the bottom of prepared hole, rod and plunger extrude epoxy to
bond device to rock.


dz) must be between points before a significant change in
stress occurs. From equation 1,

Aa = 9(%Y,2) - O%YosZ) (4)

If stress measurements are made at four sites that are not
all in the same plane, then equations 3 and 4 can be solved
for the stress gradients, In matrix notation, the system of
equations is

{40} = [d]{do}, (2)

where {Ao} = (A,o A,o d,o)' and is a 3- by 1-column
matrix of stress changes between the original site and each
of the three additional sites 1, 2, and 3. The parentheses
imply a 1- by 3-row matrix, superscript ' means to
transpose, and {30} = (90/8x 40/ay 80/42) and is a
3- by 1-column matrix of the unknown spatial gradients of
a stress component. The coefficient matrix [d] contains
the coordinate distances between measurement sites.

1% Yio
Y2-Yo
Y3-Yo

1-2
YZ |, ©
%-T

[d] = | 2%

3%

where x,,y,,2, = coordinates of measurement site 1 and so
forth. Equation 5 must be solved for the spatial gradients
of each of the six stress components.

Once all the stress gradients are known, a stress
component in the model at arbitrary point xy,z is given by
the linear form

o = ay + ax t+ ay + ayz, (0)

where the coefficients are known constants. Equation 7
remains the same with a change in the origin of the
coordinates. Thus, the in situ stresses are given in matrix
notation as

{o} = [cl{p}, ®)

where {0} = (04 Oy 0, Oy Cm Op)‘ and is a 6
by 1-column matrix; [c] = the coefficient matrix, {p}
= (1 xy 2)! and is a 4- by 1-column matrix; and x,y,z are
coordinates with respect to a new origin (XYZ) rela-
tive to the original measurement site. The possibility of
using a new origin point is a modeling convenience.

The coefficient matrix [c] contains stress gradient
information determined previously and has the dimensions
of a 6 by 4 matrix. The coefficients in typical row r of {c]
are

1 = Fo + (80/8x)X + (80/8Y) Yo

+ (80/82),2, (9a)
Cy = (80/8), (9b)
&3 = (80/8Y)o, (9c)

and yy = (30/82) (9d)
where the stress gradients are determined from equation
5 and the subscript , implies evaluation with respect to the
original measurement site. The evaluations are simple
arithmetic calculations using measurement data.

Equation 8 allows for a simple assignment of initial
(premining) stresses to all elements in a finite-element
model. For example, when linear displacement (constant
strain) elements are used, the coordinates x,y,z are simply
the coordinates of an element centroid, ¢, is the measured
state of stress at point x,Yo;% and the gradients are
derived from equation 5. If horizontal stress gradients are
absent, then

xx = [Pq + (804/92) oZq] + (80,,/8z)z. (10)

The other five stress components are calculated in a
similar manner.

In a homogenous rock mass loaded by gravity only, the
premining shear stresses are zero, and further simplifica-
tion of equation 8 is possible. Only the vertical and two
horizontal stresses and their vertical gradients remain.
Thus, the premining stress state must be given by

oy = a + by, (11a)
OW = a + by, (11b)
and Oy = a3 + bz, (Ac)

where a,, a, and a, correspond to the constant terms in
brackets in equation 10; b,, b,, and b, correspond to the
gradient terms; subscript , = vertical; and subscripts ,
and ,, = horizontal and are mutually perpendicular. Under
the simplified conditions of gravity loading only and a


Poorman Formation

Hole 4
N 00° E}
-3 (uy)
eo, LEGEND

a Quartz lens, 2 ft thick
b, i,j | Bedding planes

c Clean fracture

d Bottom limb of fold

e Top limb of fold

f Small fold
Joints

BDG overcore
HICell overcore

True north

Mine north

0 20
Ld

Scale, ft i
Poorman Formation

Figure 2.Geologic structures and overcoring locations at site 1.


CONTENTS

Abstract ...
Introduction .
Acknowledgments .......
Geology ....
Background .
Working hypothesis .. .
Mine measurements
Procedures
Results ..
Discussion of results . . .
Variability ..
Elastic moduli

Summary and conclusions .
References senee
Appendix A.—Overcore strain data from hollow inclusion cells...
Appendix B.—Overcore plots from borehole deformation gauges

ILLUSTRATIONS

Plan view of 3650 level, Homestake Mine, Lead, SD ...
Geologic structures and overcoring locations at site 1
Geologic structures and overcoring locations at site 2
HICell assembled for installation .

Calibration of BDG prior to installation .
Specially designed thin-walled diamond set bits used to drill 6in holes .......
Disking phenomenon, indicating high vertical stress, displayed i in 6-in-diameter core taken 5 to 10 ft from

NAWEY NE

-

9. Connecting overcored HICell to strain indicator to “conduct biaxial chamber test .
A-1, Overcore readings from HICells ..... . .
B-1. Response of BDG to mining, site 1 ..

TABLES

Best fit stress components for each test at sites 1 and 2.
Best fit principal stresses for each test at sites 1 and 2..
Average cartesian stress components from sites 1 and 2.
Principal stresses from averaged and pooled results
Different axial strain treatments and results from site 2 .
Stresses near Ross shaft pillar at sites 1 and 2

AYSyYNr

Page

RRARBEBERSameauwunne

10
11

12
14


Measurement $1H3C2 is an exception. Although the
north-south normal stress is still the smallest, the vertical
and cast-west stresses are in reverse order relative to the
other measurements from site 1. Large vertical shear
stresses are also indicated, but these are about one-half
the smallest normal stress,

At site 2, the vertical stress is the largest normal stress,
but the north-south and east-west stresses are in reverse
order. Again, large vertical shear stresses are noted.

The results in terms of principal stresses are shown in
table 2, Major principal stresses (compression) range
from a low of 3,623 psi at SIH2C1 to a high of 8,297 psi
at S1H3C2, Intermediate principal stresses range from
1,930 to 5,065 psi; minor principal stresses range from
1,718 to 2,614 psi. Principal stresses at site 2 are near the
mid-ranges of the stresses from site 1.

The major principal stress lies within 30° of vertical, so
that the intermediate and minor principal stresses lie in
planes within 30° of horizontal. There are two exceptions
to this pattern, One is at S1H3C2, noted previously; the
other was obtained from the BDG. High shear stresses in
both cases rotate the major compression to within 24° of
horizontal.

The data in table 2 suggest that the intermediate
principal stress is oriented perpendicular to strike and that
the minor principal stress is parallel to strike. The same
suggestion is implicit in the cartesian stress components
shown in table 1. However, variability in orientation is
large, and no strong conclusions are warranted,

There are two ways to average site measurements to
characterize the stress state at a given site. The first is to
average the cartesian stress components arithmetically; the
second is to pool all HICell strain measurements from the
site and calculate a least squares fit to the pooled data.
The results of both methods may be compared with the
stresses calculated from finite-element modeling of past
mining sequences. The difference in the average site 1
and site 2 stress states then defines the stress gradients
according to the hypothesis presented earlier.

Table 3 shows the arithmetically averaged HICell data
from table 1 and the same cartesian components of stress
obtained from a best fit to the pooled strain data from
site 1. Both sets show the same ordering of normal
stresses noted previously. The vertical stress is the larg-
est; the smallest normal stress is parallel to strike. The
tendency for the shear stresses to be an order of magni-
tude less than the normal stresses is seen again in table 3.
The averaged data from site 2 are the same as the data in
table 1 because there is only a single measurement avail-
able from site 2. The finite-element modeling results
follow the same trend, but show larger normal stresses
than either averaged or pooled results.

Table 2.—Best fit principal stresses
for each test at sites 1 and 2

(Compressive stress is positive)

Magnitude, Orientation, deg
Test! psi Dip direction Dip,
from azimuth +down
SITE 1

SIHICT:
. 4917 287 82
2,522 316 7
1,892 226 4
3,623 48 73
2,759 246 “16
2,614 154 5
4,571 73 60
1,930 158 3
1,718 66 30
8,297 257 23
4,212 130 55
1,981 178 -25
5,764 59 24
5,065 69 66
1,921 151 4

SITE 2

S2H1C1:
Opes 5,664 337 61
oy 3,897 190 25
oes 2,152 94 “14

1g, = major principal stress; 2, = intermediate prin-
cipal stress; c, = minor principal stress.

The principal stresses and directions calculated from
the results shown in table 3 are given in table 4. There
are noticeable differences in magnitude and orientation of
the principal stresses when the stresses are calculated from
arithmetic averages of the cartesian components of stress
and a least squares best fit to the pooled strain measure-
ments. The numerical reason for the differences is that
higher vertical shear stresses are present in the arith-
metically averaged data. The finite-element principal
stresses are greater than the pooled and averaged results,
as expected from the cartesian information provided in
table 3. Orientation of the principal stresses obtained
from using the finite-element analysis is closer to the re-
sult obtained by pooling orientations, but is more closely
aligned with strike than are orientations derived from
either pooled or arithmetically averaged results.


stresses are misleading because the orientations of the
principal axes are not fixed.

There are only two HICell measurements available to
study within-hole variability at site 1 and none at site 2.
The measurements from site 1 indicate that the order of
the largest and the intermediate normal stresses was
reversed and that these stresses varied by about 50 pet
from their mean stress components. The high degree of
variability over a distance of several feet in the same hole
brings the validity of the measurements into question.
However, the results may also be an artifact of the data
reduction scheme.

There are four HICell measurements available to study
within-site variability at site 1. When the east-west stress
measurements from S1H3C2, which are unusually high, are
ignored, deviations from the arithmetic mean are about
16 pet. Stress variability at site 1 is therefore character-
ized by no more than a 1,300-psi spread in normal stress.
Differences between high and low values of normal stress
fall within this range. The spread is the same magnitude
as the increase in vertical stress caused by gravity loading
of a finite-clement mesh about 1,100 ft high.

The contact between the Homestake and Poorman For-
mations may have influenced the results at site 1. Stress
measurements taken on both sides of a geologic contact
may shed light on how changes in rock properties affect a
stress field at a material discontinuity. A fundamental
issue in rock mechanics concerns such changes.

The difference in arithmetically averaged vertical
stresses between sites 1 and 2 is about 1,000 psi or about
+11 pet of the mean of the vertical stresses at the two
sites. However, the between-site variability of the other
normal stress is problematic because of the reversal in
ordering. The situation raises questions similar to those
regarding within-hole variability. As it is, the observed
difference between north-south stresses is almost 2,000 psi,
while the difference between the east-west stresses is
1,400 psi. The ranges are +44 and +23 pet of combined
means from the two sites, respectively. Thus, horizontal
stress variability, or differences between sites on the same
level spaced about 750 ft apart, is somewhat greater than
vertical stress variability within site 1 and between sites
land 2.

ELASTIC MODULI

For isotropic materials, the pertinent elastic moduli for
reducing strain measurements to stresses are Young’s
modulus (E) and Poisson’s ratio (v). Values of 6.5 x 10°
psi and 0.20 were used to obtain all results reported here.
The amount of data from biaxial and other tests to
establish rock properties, although limited, suggests that
both values should be higher. At fixed v, the calculated

stresses are directly proportional to E. Increasing E by
20 pet increases all stress components by 20 pet. The
effect of increasing v is more subtle. Trial calculations
and comparisons at fixed E using the same set of strain
measurements and a value of 0.37 for v show an increase
in normal stresses up to 18 pet, but a decrease in shear
stress. The best choice of elastic moduli is complicated
further by the question of anisotropy.

ANISOTROPY

The computer program that reduces HICell strain
measurements to stresses is based on the assumption of
isotropic rock. However, the metasedimentary formations
at the Homestake Mine have directional characteristics
associated with foliation. Laboratory test data (5) show
three mutually orthogonal material directions that are
perpendicular to foliation, parallel to foliation downdip,
and parallel to foliation along strike.

The rock is orthogonally anisotropic; nine independent
moduli characterize the elasticity of each formation. The
principal values of Young’s modulus in the Poorman
Formation range from 7.2 x 10° psi perpendicular to the
foliation to 13.7 x 10° psi parallel to foliation. Young’s
modulus at an angle to the foliation may be greater or less
than the principal values. The same is true of the shear
modulus. Poisson’s ratios range from 0.15 to 0.23.

Foliation is least pronounced in the Homestake For-
mation; principal values of Young’s modulus range be-
tween 9.0 x 10° and 12.8 x 10° psi, and Poisson’s ratios
range between 0.14 and 0.19.

If the rock were isotropic, then a higher Young's
modulus (greater than 6.5 x 10° psi) would be justified in
the data reduction computations. However, it is possible
that there is a relative maximum or minimum modulus at
angles to the foliation and that some values fall outside the
range given here. The assumption of isotropy and its
effect on the in situ stress state must therefore await a
more detailed analysis and a data reduction scheme that
accounts for anisotropy. Amadei (6) has proposed such a
scheme, but a field measurement procedure has not been
established as a standard practice.

DATA REDUCTION USING LEAST
SQUARES APPROACH

The least squares, best fit approach to calculating in
situ stresses from the measured HICell strains is not
necessarily the best one. Substantial differences in stresses
may occur between the first trial using data from all nine
gauges in a HICell and the third, "best fit" trial using data
from only six gauges. The best fit stress state is thus a
result based on a purely statistical criterion.


Library of Congress Cataloging in Publication Data:

In situ stress measurements near the Ross shaft pillar, Homestake Mine, South
Dakota / by J.C. Johnson ... [et al}.

p. cm, — (Report of investigations; 9446)
Includes bibliographical references (p. 15).
1. Rock pressure—Measurement. 2. Rock mechanics—South Dakota—Lead

Region. 3. Homestake Mine (S.D.). I. Johnson, J. C. (Jeffrey C.). Il, Series:
Report of investigations (United States. Bureau of Mines); 9446.

TN23,.U43 [TN288] 622 s—dce20[622'.25]  92-23496 CIP



Report of Investigations 9446

In Situ Stress Measurements Near the Ross
Shaft Pillar, Homestake Mine, South Dakota

By J. C. Johnson, W. G. Pariseau, D. F. Scott, and F. M. Jenkins

UNITED STATES DEPARTMENT OF THE INTERIOR

BUREAU OF MINES


is needed at the controls.

Figure 7.—Disking phenomenon, indicating high vertical
Horizontal configuration for

stress, displayed in 6-in-dlameter core taken 5 to 10 ft from

opening.

experienced dri

Figure 6.—Specially designed thin-walled diamond set bits
used to drill 6-in holes. Both HICell and BDG were alternately

Installed and overcored in same holes.

Figure 5.—Callbration of BDG prior to installation. Instruction
manual must be followed carefully to ensure accurate results.



Table 3.—Average cartesian stress components from sites 1 and 2, pounds per square inch

(Compressive stress is positive)

Shear
Averaged North-south Eastwest North-south
North-south Vertical and and vertical and vertical
east-west
SITE 1
Arithmetic. 2,357 3,735 4,167 189 “597 453
Pooled : 2,142 2,188 3,807 266 222 -39
Finite-element model . . 2,577 4,882 6,135 258 76 256
SITE 2
Arithmetic. 4,238 2,321 5,153 37 694 715
Pooled 4,238 2,321 5,153 37 694 715
Table 4.Principal stresses from averaged and pooled results
(Compressive stress Is positive)
Magnitude, Orientation, deg Magnitude, Orientation, de,
Averaged! psi Dip direction Dip, Averaged? psi Dip direction Dip,
from azimuth +down from azimuth +down
SITET SITE 2
‘Arithmetic:
4,685 70 54 5,664 337 61
3,326 276 33 3,897 190 25
2,248 178 13 2,152 94 “14
3,837 89 82 5,664 337 61
2,420 134 5 3,897 190 25
"3 1,880 44 5 2,152 94 14
Finite-element
model:
a. 6,156 56
2: 4,909 174
% 2,529 84

1g, = major principal stress; 0, = intermediate principal stress; 0, = minor principal stress.

DISCUSSION OF RESULTS

Factors that influence the interpretation of results in-
clude the following:

1. Factors influencing variability at different scales,

2. The numerical values of Young’s modulus and
Poisson’s ratio used to calculate stresses from the HICell
strain measurements,

3. The assumption of isotropic rock in the Homestake
Mine, and

4, The least squares, best fit data reduction procedure.

Although data are sparse and do not allow a rigorous,
detailed statistical analysis, some discussion is warranted.

The last item above is a particularly interesting topic that
does not appear to be addressed in the technical literature.

VARIABILITY

Tf other factors remain constant, differences in the
stress state from a given site must be caused by heteroge-
neity of the rock mass along the same borehole and
among boreholes collared at the same site. Geologic var-
iability on the scale of feet induces variability among stress
measurements on the same scale. Variability is properly
measured within a cartesian frame; differences in principal