ARTICLE IN PRESS

International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147

Contents lists available at ScienceDirect

International Journal ofRock Mechanics & Mining Sciences

1365-16

doi:10.1

� Corr

E-m

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

Stress field determination from local stress measurementsby numerical modelling

G. Li a, Y. Mizuta b,�, T. Ishida c, H. Li d, S. Nakama e, T. Sato e

a Mewbourne School of Petroleum & Geological Engineering, University of Oklahoma, OK 73019-1004, USAb Department of Eco Design, Faculty of Engineering, Sojo University, Kumamoto 860-0082, Japanc Department of Civil and Earth Resources Engineering, Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8540, Japand Carbon & Energy Management, Alberta Research Council, Canada T6N 1E4e Japan Atomic Energy Agency, Mizunami, Gifu 509-6132, Japan

a r t i c l e i n f o

Article history:

Received 30 January 2007

Received in revised form

19 June 2008

Accepted 17 July 2008Available online 19 September 2008

Keywords:

Stress determination

Stress measurement

Numerical modelling

Boundary element method

Hydraulic fracturing

09/$ - see front matter & 2008 Elsevier Ltd. A

016/j.ijrmms.2008.07.009

esponding author. Tel.: +8196 326 3782; fax:

ail address: ymizuta@eco.sojo-u.ac.jp (Y. Mizu

a b s t r a c t

A back analysis using a three-dimensional (3D) boundary element method (BEM) is used to calculate

the far-field stress state from local stresses measured in situ. The far-field stresses are decomposed into

tectonic and gravitational components and account for the influence of localized faulting and

topography. Therefore, the far-field stresses are taken to consist of a constant term, a term that varies

linearly with depth, and a hyperbolic term, with one of the principal stresses being vertical. A BEM for

inhomogeneous bodies is introduced to calculate elastic gravitational stresses, which is necessary for

determination of the far-field stresses.

An application to the stress field determination for the Mizunami underground research laboratory

(MIU) is carried out. Based upon the local stresses generally measured by conventional hydraulic

fracturing (HF), the unknown stress state at MIU is estimated and compared with the measurements

carried out recently by the improved HF method with flow rate measurements at the position of

straddle packer. After calculating the far-field stress state by BEM back analysis, 3D-finite difference

methods (FDM) forward analysis was carried to calculate the in situ stresses at certain locations. The 3D

FDM results roughly coincide with the measured results.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In many civil and mining engineering projects, especially thoseinvolving underground excavation, the initial state of stress, i.e.,the state of stress prior to any excavation or construction, can beessential to understand. This state may be complex owing to(a) heterogeneity in rock type and rock structure and (b) thegeologic history. Recent experience with underground researchlaboratories (URL) has highlighted the necessity of understandingthe ambient stress state for designing and constructing reposi-tories for high-level radioactive wastes, especially with regard toexcavation stability and hydraulic suitability. Such projectsrequire stress evaluations for rock masses with widths of severalkilometres and depths greater than 1 km [1]. In some cases, therock at the depth of repository candidates is influenced substan-tively by topography. Analytical solutions for stresses exist for arange of idealized two-dimensional (2D) topographies [2], butaccurate analytical solutions are not available for an elasticmedium with an irregular three-dimensional (3D) surface. Such

ll rights reserved.

+8196 3111769.

ta).

terrains may be analysed with numerical models, such as finitedifference methods (FDM) or finite element methods (FEMs).A problematic aspect of both these methods, however, is that themodel boundary conditions are applied to boundaries of arbi-trarily defined geometry, and it is not clear what these should be.Hence, the central question is how the boundary conditionsof a numerical model of a rock mass should be assigned. Incomparison, boundary element methods (BEMs) have the advan-tages that no artificial truncated boundaries are necessary andopen regions deep underground can be treated rather accurately.

We present here a non-linear numerical inverse method forevaluating the in situ state of stress in a rock mass. It accounts fortectonic stresses, topography and rock mass inhomogeneity.Unlike back analysis by FEM or FDM with strain components ordisplacement components on model boundaries adopted asunknowns, we carried out a back analysis by using the BEM andtreated the far-field stress state as the unknown. The BEM backanalysis results can then serve as the input boundary conditionsfor FEM or FDM analysis, two popular and powerful methods well-suited for non-linear or large-strain problems. With the numericalmodel boundary conditions determined, we can easily study avariety of key issues, such as excavation-induced stress or thestress perturbations caused by faults. Finally, we applied this

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G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147 139

method to evaluate the stress field of the Tono district, Japan, acandidate for an underground repository site for high-levelradioactive wastes. The numerical model is 2.4 km�3.2 km inarea. Stress measurements in nine different locations obtained byvarious methods were adopted for the back analysis.

2. Overview of methods

2.1. Back analysis

The in situ stress field can never be completely measured, andas a result methods for evaluating it involve simplifying assump-tions. The most common practice is to assume lateral confinement(i.e., no horizontal displacement anywhere due to gravitationalloading). The following equation often is used to describe theinitial state of stress in a uniform soil or rock mass below ahorizontal free surface:

s0zz ¼ rgh; s0

xx ¼ s0yy ¼

n1� nrgh (1)

where r is the mass density of the material, g is the gravitationalacceleration, n is the Poisson’s ratio, and h is the elevation belowthe surface which assumes a negative value [3]. The terms s0

xx ands0

yy are the two horizontal principal stresses, and s0zz is the vertical

stress at depth h. Actual field measurements, however, show thatthe horizontal stresses commonly do not follow this relationship,and in many places are several times larger than the vertical stress[3,4]. Furthermore, the horizontal stress state is not uniform,especially in a seismically active region such as Japan [5]. Thecurrent stress state reflects a series of geologic events, thestiffness of a rock mass, as well as gravitational and tectonicstresses [6]. Finally, rock mass heterogeneities and discontinuitiescause local stress variations. In light of these factors, assumption(1) is likely to be inappropriate in many cases, and factors liketectonic stresses must be considered.

The following equation provides a more general relationship,with the constraints that one of the principal stress components isvertical and varies linearly with depth:

s0 ¼ Sþ arghþb

hþ akrgh (2)

where

sm;1xx

sm;1yy

sm;1xy

..

.

..

.

..

.

..

.

..

.

..

.

sm;nxy

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

¼

b1xx;xx b1

xx;yy b1xx;xy g1

xx;xx g1xx;yy g1

xx;zz l1xx;xx l1

xx;yy l1xx;xy

b1yy;xx b1

yy;yy b1yy;xy g1

yy;xx g1yy;yy g1

yy;zz l1yy;xx l1

yy;yy l1yy;xy

b1xy;xx b1

xy;yy b1xy;xy g1

xy;xx g1xy;yy g1

xy;zz l1xy;xx l1

xy;yy l1xy;xy

..

. ... ..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

. ... ..

.

..

. ... ..

. ... ..

. ... ..

. ... ..

.

bnxy;xx bn

xy;yy bnxy;xy gn

xy;xx gnxy;yy gn

xy;zz lnxy;xx ln

xy;yy lnxy;xy

2666666666666666666666666664

3777777777777777777777777775

Sxx

Syy

Sxy

axx

ayy

azz

kxx

kyy

kxy

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

(5)

s0 ¼

s0xx s0

xy 0

s0yx s0

yy 0

0 0 s0zz

2664

3775

S ¼

Sxx Sxy 0

Syx Syy 0

0 0 0

264

375

a ¼axx 0 0

0 ayy 0

0 0 azz

264

375

k ¼

kxx kxy 0

kyx kyy 0

0 0 0

264

375

Here s0 stands for the in situ measured stress state, S is thetectonic (regional) stress state, a is the 3-by-3 matrix ofcoefficients due to gravitational loading with ax ¼ ay ¼ n/(1�n)and az ¼ 1, k is the 3-by-3 matrix of coefficients for a lineardistribution of horizontal stresses [7], and a and b are constants,determined from measured data by a simplex method, that allowfor a non-linear distribution of certain stresses with depth [4]. Inthis equation, a has dimensions of length, and b is dimensionless.We emphasize that {Sx, Sy, Sxy, ax, ay, az, kx, ky, kxy} are consideredas coefficients of far-field stress components. The right-hand sideof Eq. (2) contains a constant term, a term that varies linearly withh, and a non-linear term with a hyperbolic coefficient b/(h+a). Asmentioned later, the values of a ¼ 904.6 m and b ¼ 812.1 m weredetermined to minimize the sum of the differences of the stressvalues given by Eq. (2) and the stress values measured.

The calculated stress component scij at an arbitrary point can

be expressed as

scij ¼ bij;pqSpq þ gij;pqapq þ lij;pqkpq (3)

where the superscript c refers to a stress component calculated atthat point and repeated indices imply summation. bij,pq, gij,pq andlij,pq are calculated stress components due to each independentunit stress state established by the coefficients of far-field stresscomponents Spq, apq and kpq, respectively. If the measured stresscomponent at an arbitrary point is expressed as sm

ij , the followingequation is obtained by putting sm

ij ¼ scij in Eq. (3):

smij ¼ bij;pqSpq þ gij;pqapq þ lij;pqkpq (4)

Applying Eq. (4) to all stress components at all measurementpoints, we can obtain the equation relating all field measurementstress components with the unknown coefficients of far-fieldstresses

In Eq. (5), the superscript n in the last row of the left columnstands for the number of underground measurement points, andthe superscript m means that the stresses in the left column are allfield measurement stress components. Since, in practice, thenumber of equations is usually more than six, i.e., the number of

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G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147140

the unknown coefficients Sx, Sy, Sxy, kx, ky and kxy, a unique solutioncannot be guaranteed. A least-squares procedure is adopted hereto find the best solution of the equation.

2.2. Calculating elastic gravitational stresses by introduction of a

boundary element method for inhomogeneous bodies

To calculate bij,pq, gij,pq and lij,pq in Eq. (5) at an arbitrary pointdue to each independent unit stress state, numerical modelling isnecessary for terrains with an irregular 3D surface, as discussedabove. Thus, the central question is how to assign boundaryconditions to the model of a rock mass. In general, the FEM andFDM are more popular than the BEM for dealing with intricategeological conditions (e.g., inelasticity, heterogeneity, and aniso-tropy). However, certain artificial boundary conditions have to beassumed for FEM and FDM such as prescribed displacementboundary conditions or prescribed stress boundary conditions,and it is hard to evaluate the potential influence of the boundaryon the calculated model.

The BEM, on the other hand, is well suited for unboundedproblems such as excavations or constructions in rock mass sinceno artificial boundaries are necessary; furthermore, it saves timein numerical modelling since the geometrical dimension isreduced by one [8]. BEMs include the fictitious stress method(FSM), displacement discontinuity indirect method (DDM), andthe direct boundary integral method [9]. The indirect methodswere extended by Kuriyama and Mizuta [10,11] to establish 3Dprocedures using triangular leaf elements. Martel and Muller [12]showed how the displacement discontinuity BEM can be used tocalculate elastic gravitational stresses beneath topographic sur-faces by modelling the topographic surface as a large traction-freecrack. They implemented the approach in 2D plane-strainanalyses in homogeneous bodies but noted that the methodcould be extended to three dimensions.

To meet needs for large underground excavations or construc-tions, we present a 3D BEM for calculating elastic gravitationalstresses for a homogeneous rock mass. Fig. 1 shows a conceptualmodel for calculating the stress state in a homogeneous linearelastic region with a topographic surface. Considering principlesof linear elastic theory and continuum mechanics, we assume ahorizontal arbitrary plane above the highest elevation of thesurface. Anything under this plane is assumed as one homo-geneous linearly elastic body. The stress state at an arbitrary pointunder the topographic surface can be calculated by subtractingthe perturbation to the ambient stress state due to the overburdenbetween the horizontal arbitrary plane and the surface, from theambient stress state under the assumed horizontal referenceplane. The total stresses are obtained as the sum of the ambientstresses state and the perturbation to it, for example,

ðsix�x� Þt ¼ ðs

ix�x� Þin þ ðs

ix�x� Þ0. (6)

Fig. 1. Conceptual model for calculating the stress state with a topographic surface (The

with boundary elements; (b) the ambient stress state under the assumed horizontal re

where ðsiz�z� Þt; ðti

y�z� Þt; ðtiz�x� Þt are equal to zero for the free

topographic surface. An initial stress state under the assumedhorizontal arbitrary plane prior to the removal of the overburdencan be calculated analytically using Eq. (2). The perturbationstresses due to the overburden from the arbitrary plane to thetopographic surface can be easily solved with the adoption offictitious stresses. Here, body forces are described in a globalcoordinate system where the z-axis is vertical. The topographicsurface stresses are described by an element-based local coordi-nate system. The origin of the local coordinate system is at thecentre of the element, with z*-axis normal to the element. In thisapproach, the effect of horizontal ambient stresses is establishedby Sx, Sy, Sxy, ax, ay, kx, ky and kxy. For any boundary element i alongthe topographic surface, the following equation can be set up:

ðsiz�z� Þin ¼

PNj¼1

Aijz�x�P

jx� þ

PNj¼1

Aijz�y�P

jy� þ

PNj¼1

Aijz�z�P

jz�

ðtiy�z� Þin ¼

PNj¼1

Aijy�z�x�P

jx� þ

PNj¼1

Aijy�z�y�P

jy� þ

PNj¼1

Aijy�z�z�P

jz�

ðtiz�x� Þin ¼

PNj¼1

Aijx�z�x�P

jx� þ

PNj¼1

Aijx�z�y�P

jy� þ

PNl

j¼1

Aijx�z�z�P

jz�

9>>>>>>>>>>=>>>>>>>>>>;

(7)

where Pjx� ; Pj

y� ; Pjz� are fictitious stresses introduced to facilitate

the numerical solution to the problem and are fictitiousquantities, Aij is the influence coefficient relating the effect of aunit fictitious stress at element j to the traction at element i, N isthe total number of the boundary elements, and ðsi

z�z� Þin; ðtiy�z� Þin;

ðtiz�x� Þin are the stress components induced by the fictitious

stresses at all N elements. On the other hand, since

ðsiz�z� Þin ¼ �si

z�z�

ðtiy�z� Þin ¼ �ti

y�z�

ðtiz�x� Þin ¼ �ti

z�x�

9>>=>>;

(8)

where siz�z� ; ti

y�z� ; tiz�x� are the three components of the initial

ambient stresses at element i, and can be determined by the depthfrom the assumed arbitrary plane and inclination of the element.Thus, we obtain a system of 3N algebraic equations with 3N

fictitious unknown stresses. Once the three fictitious stressesPi

x� ; Piy� ; Pi

z� are solved for, other stress components such asðsi

x�x� Þin; ðsiy�y� Þin and ðti

x�y� Þin induced by the above fictitiousstresses can be calculated from the same expression by using Aij

x ,etc. By the same method, all of the six stress components of theinitial stresses at a specified point k under the assumed arbitraryhorizontal plane ðsk

x�x� Þ0, etc., can be calculated through anequation with the same form as Eq. (7) but with an appropriateset of new influence coefficients.

Rock masses are typically heterogeneous and contain disconti-nuities such as faults. To adequately estimate the state of stressproperly, the effects of inhomogeneity must be considered incertain cases [13]. Fig. 2 shows an inhomogeneous rock mass

depth D is actually infinite): (a) the stress state under the topographic surface tiled

ference plane and (c) the perturbation due to the overburden.

ARTICLE IN PRESS

Fig. 2. Conceptual model for calculating the stress state in an inhomogeneous body containing two subregions R1 and R2 (The topographic surface and the interfaces with

local coordinate systems are tiled with boundary elements).

Fig. 3. Axi-symmetric model containing two subregions.

G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147 141

containing two subregions R1 and R2. The two subregions are eachassumed to be homogeneous, isotropic and linearly elastic. Thecommon portions of the two subregions define the interfacebetween the subregions. The local x*, y*, z* coordinate systemsassociated with the interfaces appear as shown in Fig. 2. The localcoordinates x�R1, z�R1 and x�R2, z�R2 are oppositely directed along theinterfaces, i.e., x�R1 ¼ �x�R2 and z�R1 ¼ �z�R2. The areas must extendsufficiently far away from the area of interest such that thebehaviour in that area is not greatly affected by perimeter of theR1�R2 interface. The upper subregion can be solved by the methodabove for the homogeneous half-plane with a free topographicsurface. The lower subregion with the interface can be treated asthe same problem expressed by Eq. (7), except that the freetopographic surface is replaced by the interface between the twosubregions. Tractions and displacements must be continuousalong the interface between the two subregions, we can have thefollowing equations for the three stress componentssj

z�z� ; tjy�z� ; t

jz�x� at element j along the interface

s½R1 �z�z� ðjÞ ¼ s½R2 �

z�z� ðjÞ

t½R1 �y�z� ðjÞ ¼ �t

½R2 �y�z� ðjÞ

t½R1 �z�x� ðjÞ ¼ t½R2 �

z�x� ðjÞ

9>>=>>;

(9)

and

u½R1 �z� ðjÞ ¼ �u½R2 �

z� ðjÞ

u½R1 �y� ðjÞ ¼ u½R2 �

y� ðjÞ

u½R1 �x� ðjÞ ¼ �u½R2 �

x� ðjÞ

9>>=>>;

(10)

for the displacement. The minus signs in Eqs. (9) and (10) are aconsequence of the opposite directions of the local coordinates ofsubregions R1 and R2 along the interface. This method can beextended to solve problems involving several homogeneous,isotropic and linear elastic subregions. However, this means thatwe will have to solve the boundary integral equations with a largenumber of coefficients. To save the requirement on centralmemory for boundary element simulation, a simultaneousequation solution method developed by Beer and Watson can beadopted [14]. Since in this study only two homogeneoussubregions were represented, the procedure of Beer and Watsonwas not adopted.

2.3. Check of the inhomogeneous modelling code

Liu et al. [13] carried out numerical calculation using their BEMcode developed for inhomogeneous modelling. They took theinfinite model including concentric double cylindrical surfaces

whose lengths are finite. As shown in Fig. 3, the inner boundarycontour with radius a is the free surface boundary subject to theuniform pressure and the outer boundary contour with radius b isthe interface between the subregion R1 and R2, where R1 is thefinite region between the free surface and the interface, and R2 isthe infinite region outside of the interface although the long-itudinal length of those contours are L (not infinite). Theycompared the numerical results with the analytical solution [15]and found that the numerical and analytical solutions are almostthe same even in the case of L/(2b) ¼ 2. Please refer to [13] forfurther information about material parameters and solutions indetail.

ARTICLE IN PRESS

Fig. 5. Location of MIU in Tono district. The rectangle indicates the area to be

simulated.

G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147142

3. Application to the stress field determination for undergroundresearch laboratory

3.1. Introduction of Mizunami underground research laboratory

(MIU) and Tono district

One aim of the geological disposal policy in Japan is theestablishment of URLs. The URLs are distinguished from a disposalfacility, as outlined in [16]. Research on the deep geologicalenvironment will provide the scientific and technological basis forR&D on geological disposal of high-level radioactive waste. TheAEC also stipulated that research at the underground laboratoriescontribute to Japan’s scientific research on the geologicalenvironment. URL projects of the Japan atomic energy agency(JAEA) are directed towards improving the reliability of geologicaldisposal technologies and developing advanced safety assessmentmethodologies.

As a part of the projects, the MIU, located in Mizunami city ofGifu Prefecture in central Japan, is under construction for researchworks 1000 m underground (Fig. 4). The construction includestwo vertical shafts excavated 1000 m deep underground andhorizontal tunnels at depths of 500 and 1000 m. First stageexcavation of the vertical shafts to 300 m depth has already beencarried out. The laboratory shall be utilized as a base for research

500m

500m

82m

Fig. 4. Diagram of the Mizunami underground research laboratory (MIU).

of rock mechanics and groundwater flow relating to deepgeological repositories.

Fig. 5 shows the location of MIU in Tono district. Local hillsreach elevations of 200–300 m. Two main geologic units exist atMIU, a basement of Toki granite and the overlying Akeyoformation of mudstone and sandstone, which has a maximumthickness of �150 m [17]. The Tsukiyoshi fault is a reverse faultwith a strike of N80W, a dip of 60 and an estimated throw ofabout 30 m located in Tono district.

3.2. Local stress measurements using conventional hydraulic

fracturing

Rock stress measurements have been performed at 247 pointsunder 9 different locations within Tono district by variousmeasurement techniques, including the hydraulic fracturing (HF)[17], deformation rate analysis (DRA), and acoustic emission (AE).However, only the in situ stresses measured by HF are introducedinto the left column of Eq. (5). Fig. 6 shows the locations of theboreholes for stress measurement at locations 1–9 relative to thelocation of the MIU. Table 1 shows the in situ stresses measuredby HF at 101 points in different locations at various depths. Notethat stress convention in this paper is that tensile stresses areconsidered positive and compressive stresses negative.

Sano et al. [18] reviewed methods of measuring stress anduncertainties in HF. The shut-in pressure, ps, is widely consideredto equal the minimum principal stress Sh in the plane perpendi-cular to the borehole. In conventional HF analyses, the maximumprincipal stress (SH) is determined through

SH ¼ 3Sh � pr � pp (11)

where pr is the reopening pressure and pp is the pore pressure.Eq. (11) assumes that a HF closes completely before reopening. Inorder, for Eq. (11) to apply, the ‘‘flow rate should be sufficientlyhigh to prevent fracturing fluid percolation into the closedfracture before the actual mechanical fracture reopening’’ [19].However, the residual aperture of the induced fracture cannot benegligible for rocks [20–23]. Numerical simulations indicate thatthe pressurized fluid should permeate easily into a HF, even for aninitial aperture of 3mm and the aperture should be enlargedby the penetration of fluid [24]. Eq. (11) should be modified by

ARTICLE IN PRESS

Table 1 (continued )

Measurement

locations

Depth (m) s1 (MPa) s2 (MPa) Degree of s1

(clockwise,

G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147 143

substituting pr into pp [24,25] as

2pr ¼ 3Sh � SH (12)

Fig. 6. Map of measurement locations within the area of simulation shown in Fig.

4. The y-axis points north and the x-axis points east. Point 10 shows MIU.

Table 1In situ measured stresses for the Tono district

Measurement

locations

Depth (m) s1 (MPa) s2 (MPa) Degree of s1

(clockwise,

from north)

1 233.6 �7.26 �12.16 �46.69

235.5 �6.97 �11.85 �71.46

238.5 �8.45 �15.78 �70.02

249.9 �5.80 �9.89 �25.78

251.8 �5.71 �9.47 �14.32

253.0 �5.70 �9.42 �23.27

258.9 �5.97 �9.66 �39.95

260.9 �5.55 �8.85 �27.12

263.9 �5.95 �9.45 �23.14

2 52.3 �1.00 �1.40 45.20

64.5 �1.70 �3.30 65.20

77.0 �1.40 �2.40 35.40

102.0 �1.90 �3.20 26.60

122.0 �6.20 �12.50 �89.60

141.0 �7.50 �11.40 �72.80

157.8 �6.60 �11.40 �46.00

170.3 �3.40 �4.80 �42.60

184.2 �10.10 �20.20 �51.30

192.5 �4.50 �8.30 �61.00

3 39.0 �1.00 �2.20 32.30

44.0 �1.50 �1.90 �36.70

75.0 �1.80 �3.80 �13.80

79.0 �1.70 �3.30 �20.40

87.0 �1.40 �1.80 �38.80

105.0 �2.10 �3.70 22.30

127.0 �2.10 �4.00 �28.10

162.0 �3.60 �5.30 32.50

188.5 �3.20 �5.40 �50.50

199.0 �3.90 �6.30 �18.90

205.0 �4.50 �6.50 �58.50

4 49.0 �2.90 �5.10 0.40

199.0 �6.80 �14.10 3.80

249.0 �9.10 �20.30 10.70

309.0 �3.80 �4.80 �40.20

351.0 �8.80 �17.50 �26.80

from north)

404.0 �10.40 �19.20 �32.50

498.5 �13.60 �28.00 �55.90

564.0 �14.10 �29.70 �50.90

600.0 �15.80 �25.30 �39.90

651.0 �16.10 �29.20 �36.10

700.0 �12.90 �20.80 �41.20

790.0 �15.70 �22.80 �41.40

850.0 �18.40 �28.30 �45.40

900.0 �25.50 �48.50 �53.30

941.0 �23.40 �42.70 �27.50

991.0 �18.30 �27.80 �71.30

5 233.6 �7.26 �12.16 �46.69

235.5 �6.97 �11.85 �71.46

238.5 �8.45 �15.78 �70.02

249.9 �5.80 �9.89 �25.78

251.8 �5.71 �9.47 �14.32

253.0 �5.70 �9.42 �23.27

258.9 �5.97 �9.66 �39.95

260.9 �5.55 �8.85 �27.12

263.9 �5.95 �9.45 �23.14

6 138.2 �7.40 �15.60 7.90

158.0 �6.50 �14.20 �10.00

187.3 �6.90 �15.40 �39.60

254.0 �6.60 �13.00 �33.00

294.7 �4.20 �5.80 �60.50

301.5 �4.00 �3.30 �57.50

356.4 �9.60 �16.90 �57.70

413.4 �13.70 �24.30 �4.50

452.0 �12.70 �25.20 �19.80

491.0 �15.10 �32.30 �54.30

555.0 �11.70 �19.80 �39.70

604.0 �9.80 �14.60 �44.10

651.0 �15.40 �25.80 �57.70

682.0 �13.20 �23.20 �68.90

698.5 �13.50 �23.90 �20.60

733.7 �14.00 �24.40 �37.30

761.3 �12.50 �20.20 �52.70

811.3 �16.20 �26.50 �36.40

837.7 �15.30 �27.30 �45.70

878.1 �15.50 �27.30 �26.10

7 122.0 �5.30 �11.40 �80.80

266.0 �8.40 �17.20 �13.80

338.0 �10.50 �20.60 �35.90

509.0 �16.30 �35.20 �30.90

589.0 �12.10 �21.90 �42.40

847.0 �11.70 �16.00 �17.70

858.0 �8.80 �9.30 �23.90

946.0 �12.10 �14.70 �42.00

988.0 �13.10 �16.20 �28.90

8 47.0 �1.22 �2.24 �82.20

62.5 �2.06 �3.82 �87.10

98.0 �1.67 �2.74 �19.80

100.7 �2.34 �4.26 19.70

149.4 �2.62 �4.38 21.30

192.2 �5.48 �12.47 10.90

194.3 �3.06 �5.36 2.20

9 32.0 �0.90 �1.00 �38.60

40.0 �1.50 �2.20 42.10

73.0 �1.80 �2.50 �19.10

80.0 �1.90 �2.60 34.50

84.5 �2.50 �4.00 �27.60

95.4 �2.00 �2.00 �16.30

171.0 �4.70 �9.30 �22.90

179.1 �5.40 �10.20 �24.30

182.0 �4.40 �8.10 �81.80

185.0 �5.10 �10.10 �34.30

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Table 2Input material properties for BEM modelling

Materials Akeyo

formation

Toki

granite

Tsukiyoshi

fault

Young’s modulus, E (GPa) 2.21 50.09

Poisson’s ratio, n 0.30 0.35

Density, r (g/cm3) 1.84 2.56

Normal stiffness, kn (MPa/m) 50.0

Shear stiffness, ks (MPa/m) 50.0

G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147144

Based on experimental results for a granitic rock mass, Pineet al. [26] considered reopening pressure as a measure ofminimum horizontal stress, namely

2pr ¼ Sh (13)

Ito and Hayashi [27] numerically showed that the conventionalpr could be equal to ps by considering that the pressure fluidpermeates the fracture deeply before reopening. Rutqvist et al.[28] suggested that Eqs. (11)–(13) might be true for fractures withextremely small apertures, medium apertures and sufficientlylarge apertures, respectively.

In general, numerical simulations [25,27,28] suggest that waterpermeates HFs at a lower pressure than conventional reopeningpressure, and that pr is affected by flow rate. The magnitude of pr

should also be affected by water volume in the pressurizingsystem [24]. When the constant flow rate is so small that thepressure gradient in the fracture is negligible, temporal variationof pressure are given by [23]

dp=dt ¼ Q=½dVc=dpÞ þ C� (14)

where Q, Vc and C are the flow rates, volume change due tofracture opening and compliance of the system, respectively. Sincethe conventional pr could only be detected when the waterpermeated a fracture several times longer than the boreholeradius, this result explained why the conventional pr is almostequal to ps. Ito et al. also proposed the use of a high-stiffness HFsystem instead of a compliant conventional HF system.

As it is not easy to make a HF system sufficiently rigid formeasurements in a deep borehole, the authors used the conven-tional HF procedure for local stress measurements described inthis paper. However, in the measurements at the MIU, the flowmeter was installed at the position of the straddle packer and theflow rate at that position was controlled in the fracture reopeningprocedure, while the flow meter is put at the ground surface in theconventional HF procedure. The measurements at the MIU werecarried out recently, whereas the measurements at other locationswere carried out before 21st century.

3.3. Estimation of field stress state by 3D BEM

The BEM model for the Tono district simulates an area 3.2 kmlong and 2.4 km wide. The BEM element (Fig. 7) generally is anisosceles-right triangle with 200 m�200 m in both directions,while the elements that cover most field-measured locations arefinely discretized (100 m�100 m) to improve the accuracy ofstress calculations. Interface elements simulate the boundarybetween different geologic units, here the Akeyo formation andthe Toki granite. The subregion between the surface and the

Fig. 7. Grid of triangular elements for the BEM modelling. The volume modelled has a p

the surface is shown by a dotted line.

interface in Fig. 7 represents the Akeyo formation, while thesubregion under the interface represents the Toki granite. TheTsukiyoshi fault is modelled with displacement discontinuitymethod (DDM) joint elements [9]. Elastic constants and densitiesof the formations were determined by laboratory tests of corespecimens taken from various locations at various depths.However, the averaged values shown in Table 2 are adopted innumerical modelling.

Model parameters are based on a combination of fieldmeasurements and results from numerical simulations. The meaninclination of Tsukiyoshi Fault is 56.31. The mean verticaldimension of the fault that is obtained from boring data is23.8 m and thus, the mean thickness of the fault, t, is 13.2 m. TheP-wave velocities in the fault have been found from logging datato be 65.3–77.4% (mean ratio, rp ¼ 0.717) of that of the sound rock.Assuming that the normal stiffness kn and shear stiffness ks aregiven by

kn ¼EF

t; ks ¼

EF

2ð1þ nÞt and EF ¼ r2pEcore (15)

where Ecore is Young’s modulus measured from core specimens,then, kn ¼ 86 MPa/m, ks ¼ 33 MPa/m for the fault in the Akeyoformation and kn ¼ 1950 MPa/m, ks ¼ 722 MPa/m for the fault inthe Toki granite. However, normal and shear stiffness of the DDMjoint elements are both assumed to be 50 MPa/m, because ourFEM modelling yields stress distribution in the Toki granite thatare gross inconsistent with the measured values, unless stiff-nesses of �50 MPA are used [19].

In Eq. (5), six coefficients are unknown since ax, ay and az areknown, whereas 101 stress measurements exist. Only verticalstresses were determined from DRA and AE, and the measuredvalues were compared with overburden pressure, rgh. HF can onlydetermine the stress states in the plane perpendicular to thevertical holes (see Table 1 and Fig. 8). Hence, data of sz, tyz and tzx

components are not in the left column of Eq. (5). Furthermore,there are no sudden changes in the stresses determined fromEq. (2), although sudden changes near the ground surface or

lan view area of 3.2 km�2.4 km and a depth of 1.2 km. The projection of MIU onto

ARTICLE IN PRESS

-1200

-1000

-800

-600

-400

-200

0

-60-40-200

measured

calculated

-1200

-1000

-800

-600

-400

-200

0

-60-40-200

measured

calculated

σ1 (MPa) σ2 (MPa) σv (MPa)

-1200

-1000

-800

-600

-400

-200

0

-60-40-200

measured

calculated

Ele

vatio

n, m

Fig. 8. In situ stresses versus depth for the Tono district.

200100

0-100-200-300-400-500-600-700-800-900

200100

0-100-200-300-400-500-600-700-800-900

0 -10 -20 -30 -40 0 -10 -20 -30 -40 0 -10 -20 -30 -40

Ele

vatio

n, m

measuredcalculated

measuredcalculated

measuredcalculated

σ1 (MPa) σ2 (MPa) σv (MPa)200

0

-200

-400

-600

-800

-1000

Fig. 9. The local stress distributions calculated by FLAC3D forward analysis compared with the measurement results at location 6 of Fig. 6.

Table 3Far-field stress state determined by BEM3D

Sx (MPa) �1.99

Sy (MPa) �3.45

Sxy (MPa) 0.755

kx 0.778

ky 0.674

kxy �0.168

ax (Akeyo) 0.429

ay (Akeyo) 0.429

az (Akeyo) 1

ax (Toki) 0.538

ay (Toki) 0.538

az (Toki) 1

a (m) 904.6

b (m) 812.1

G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147 145

subsurface are in the calculated stresses through fast Lagrangiananalysis of continua in three dimensions (FLAC3D) (Fig. 9).

Based on the measurement results and material propertiesadopted above, we first used BEM method to calculate the matrixof induced state of stresses in Eq. (5). Second, we used a simplexmethod [29] to determine the two constants a and b in Eq. (2),with a ¼ 904.6 m and b ¼ 812.1 m. A least-squares approach ischosen to control the distribution of error and find the bestapproximate solution for this over-determined problem [29,30].Table 3 shows the calculated results for far-field stress state. Thesolution for the far-field stresses with the non-linear assumptionof the initial state of stresses is affected by local effects such asnon-linearity and large strains. Furthermore, the fault stiffnessadopted from the FEM modelling results might be a key factor that

would contribute to error in the BEM modelling. Finally, there stillexists the possibility that one of the principal stresses is notvertical.

Fig. 8 shows the far-field principal stresses, calculated usingthe BEM results and the coefficients for Eqs. (2) and (5), as afunction of elevation. The vertical stress sv has a stress gradientof 0.026 MPa/m. The calculated stresses are calculated based onin situ measured stresses shown in Table 1. In most cases, theabsolute values of measured horizontal principal stresses, s1 ands2, exceed those of the vertical stresses at shallow depths. Theabove results show that the stress state measured varied withdepth from thrust fault type (|s1|4|s2|4|sv|) to strike-slip faulttype (|s1|4|sv|4|s2|).

3.4. Estimation of field stress state by 3D FDM with boundary

conditions determined by the above BEM analysis

To illustrate how the above far-field stress coefficients canbe adopted as input boundary conditions, a 3D FDM namedFLAC3D [31] is applied to simulate the same area of 3D BEManalysis. FLAC3D can solve a wide range of complex problemsin mechanics, and has some advantages over a BEM approach incertain situations, such as modelling of the non-linear behaviourof materials and large-strain deformation analysis. The regionmodelled is 3.2 km long, 2.4 km wide and 1.1 km deep. The modelwas composed of 280,160 elements and 301,704 gridpoints withan interface to simulate the Tsukiyoshi fault. The Akeyo formationand Toki granite are accounted for in the simulation; theirproperties are shown in Table 2.

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G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147146

A stress boundary determined by the above-calculated coeffi-cients of far-field stresses is applied to the FLAC3D model. InFLAC3D forces or stresses may be easily applied to any boundaryby some inherent commands. The boundaries should be placedsufficiently far away from the area of interest such that thebehaviour in that area is not greatly affected, with constraints onmemory and computer time being considered. In this study, aspace about 300 m between the boundaries and the areas ofinterest is adopted (see Fig. 6). Fig. 9 shows the stresses calculatedwith FLAC3D at location 6 (in Fig. 6) with the measured stresses.We can tell that calculations with FLAC3D provide a better fit atshallow depths than at great depth, which could be a resultof more input data, i.e., in situ measurement results, at shallowdepths (see Table 1). The least-squares procedure adopted to solveEq. (5) for the unknown coefficients reflects the influence of fieldmeasurements at shallow depths better than those at greatdepths. The estimated stresses distribution has the same patternas the measured stresses, i.e., the absolute value of the verticalstress is least at shallow depths and increases with depth.

All the stress measurements at the locations 1–9 were carriedout before 21st century by the conventional HF method and thoseHF data were used for stress field estimation at location 10, whichis the location of the MIU. We assumed that locations 1–10 wereunder the influence of the same far-field stress state, i.e., theestimation results of the above BEM modelling. Therefore, theunknown stress state at location 10 could be estimated basedon the FLAC3D model with boundary conditions determined bythe above BEM analysis. Recently, the stress measurements at theMIU were carried out by the improved HF method, whichprovided us a chance to verify our estimation of the stress stateat location 10.

Fig. 10 shows stresses estimated using FLAC3D compared withthe in situ stresses measured by HF at MIU at depths as great as1000 m. The horizontal dashed lines in Fig. 10 show where theMIU borehole intersects three small faults that were not modelledin our numerical simulations. When we compare the estimated

0

-200

-400

-600

-800

-1000

-12000 -10 -20 -30 -40

Ele

vatio

n, m

σ1 (MPa)

measuredfault

Best-fit line

FLAC3Destimation

Fig. 10. In situ stresses measured by HF at depths as great as 1000 m at MIU (location

dashed lines show where the MIU borehole intersects three small faults.

stresses with the measured ones, good matches can be found forboth the maximum and minimum principal stresses as shown inFig. 10, which means that it is practical to estimate the stress stateat MIU (location 10) based upon the in situ stress measurementresults at locations 1–9. Therefore, the method proposed hereprovides a practical way to estimate the stress state based onmeasured results with reasonable pre-determined boundaryconditions by BEM analysis. Since the FLAC3D model is onlyabout 1.1 km in depth, the stress state estimated at greater depth,i.e., more than 1000 m, deviates from the measurement resultsdue to the influence of the bottom boundary. A model with greaterdepth at least 1200 m will be necessary for accurate estimation ofstress state at depth near 1000 m. A best-fit line for the in situmeasurement data was added in this figure to compare with theFLAC3D estimated results. The FLAC3D estimation fits themeasurement data better than the best-fit line at shallow depth.

Thus, by adopting the stress boundary determined by non-linear far-field stress coefficients calculated from BEM modelling,the in situ state of stress can be calculated by FDM simulationconsidering tectonic stresses, inhomogeneity and discontinuitiesof rock mass. With the understanding of this initial state of stress,we can easily study other aspects that we are interested in alongwith the excavation of MIU, although more detailed fieldformation should be taken into modelling for better estimationof the local stresses.

4. Conclusion

A new proposal for the determination of the far-field stressesbased on stress measurement results is developed in this paper.The method considered the tectonic stresses, topography andinhomogeneity of the rock mass. The in situ measurement stress isdecomposed into tectonic and gravitational components, and sixindependent far-field stress coefficients are achieved by backanalysis using newly developed 3D BEM simulation incorporating

0

-200

-400

-600

-800

-1000

-12000 -10 -20 -30 -40

σ2 (MPa)

measuredfault

FLAC3Destimation

Best-fit line

10 in Fig. 6) compared with the stresses estimated using FLAC3D. The horizontal

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G. Li et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 138–147 147

inhomogeneity for determination of gravitational stresses dis-tribution.

The 3D BEM and FDM applications to the MIU site have shownthat the estimated stresses agree well with the tendency of the insitu measurement data. Therefore, it is practical to estimate thestress state in an unknown location based upon the in situmeasurement data in the close locations by inverse analysis. It islikely that the tectonic stresses have to be considered for a betterestimation of underground stress distribution in tectonicallyactive region, such as in Japanese islands. The method describedhere provides boundary conditions for commercially availableFEM or FDM programs by considering the influence of tectonicstresses based on stress measurements.

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