Determination of the stress field in a mountainous granite rock mass

B. Figueiredo a,n, F.H. Cornet b, L. Lamas a, J. Muralha a

a Portuguese Laboratory for Civil Engineering (LNEC), Lisbon, Portugalb Institut de Physique du Globe de Strasbourg (IPG-S-CNRS), Strasbourg, France

a r t i c l e i n f o

Article history:Received 17 June 2013Received in revised form20 July 2014Accepted 28 July 2014Available online 19 September 2014

Keywords:In situ stress measurementsRegional stress fieldElastic rock massShear stress relaxationNatural fracture network

a b s t r a c t

The design of an underground hydroelectric power scheme in northern Portugal has required thecharacterisation of the local stress field. Nineteen hydraulic tests have been conducted in two, 500 mdeep, vertical boreholes. In addition twelve overcoring tests together with twelve flat jack tests havebeen performed from an existing adit located some 1.7 km away from the location of the hydraulic tests.Results have been integrated into a stress model that takes into account both topography and tectonicseffects. Most of the data are consistent with a linearly elastic, gravity loaded model, provided a very softgeomaterial is considered. This implies that the stress field in this granite rock mass is controlled bygravity alone and shear stress relaxation along faults and fractures but is unaffected by present-daytectonics.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The re-powering scheme of the Paradela hydroelectric infra-structure developed on the Cávado River in Northern Portugalinvolves a new 10 km long power conduit and a powerhousecomplex located about halfway in the conduit and 500 m belowground level. It includes a new powerhouse cavern, a valveschamber and a large surge chamber with several adits. The localgeological formation is mostly granite (Fig. 1).

The design of this new scheme requires a sound understandingof the regional stress field and this has prompted an in situ stressmeasurement campaign. Results have raised important questionson the relative influence of topography and regional tectonics aswell as on that of the rock mass rheological characteristics. Suchissues have been frequently addressed in the literature [1,2] andwe describe here after results derived from a stress determinationstrategy somewhat similar to that proposed for the InternationalSociety of Rock Mechanics (ISRM) [3].

After identifying the objective of the stress determinationcampaign, we present results obtained with three different meth-ods, namely hydraulic tests in 500 m deep boreholes as well asovercoring and flat jack tests conducted at different locations.Then we introduce the numerical model that has been developedfor integrating the various measurements in order to identify anoptimum solution. This model helps to discriminate the respective

contributions of gravity and tectonics and provides means todetermine the long-term rock mass rheological behaviour thatbest fits observations.

2. Stress determination campaign

2.1. Objectives and design of the campaign

The design of the underground excavations planned for hostingthe new powerhouse as well as the valves and large surge systemimplies a complete 3D characterisation of the stress field at thelocation of the excavations. But the design of the 10 km longpressure tunnel requires only a sound evaluation of the minimumprincipal stress magnitude all along the tunnel as well as anestimate of its extreme local variations.

On site, two 500 m deep vertical boreholes (PD19 and PD23)were available in the immediate vicinity of the planned under-ground powerhouse. Further a horizontal adit located some 1.7 kmfrom the vertical boreholes, with a 2.4�2.0 m2 rectangular crosssection, was excavated some 50 years ago and provided theopportunity for further stress measurements (Figs. 1–3).

It was decided to run a combination of hydraulic fracturing (HF)tests together with hydraulic tests on pre-existing fractures (HTPF)in both vertical boreholes in order to constrain the direction andmagnitude for the three principal stress components at thelocation of the excavation. Further the objective was also toprovide data on the spatial variability of these quantities alongthe vertical direction.

Contents lists available at ScienceDirect

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

International Journal ofRock Mechanics & Mining Sciences

http://dx.doi.org/10.1016/j.ijrmms.2014.07.0171365-1609/& 2014 Elsevier Ltd. All rights reserved.

n Correspondence to: Uppsala University, Villavägen 16, Uppsala, Sweden.Tel.: þ46 739735500.

E-mail address: bruno.figueiredo@geo.uu.se (B. Figueiredo).

International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48

Validity of these results for other locations would be estab-lished next, through further testing conducted in the adit. Hence itwas decided to run overcoring and flat jack tests in the adit in

order to combine various measuring techniques for this stressevaluation, as recommended by the ISRM [3].

Locations of the various boreholes are shown on Figs. 2 and 3.The 60 m deep boreholes PD1 and PD2 are 150 m apart and havebeen drilled specifically for overcoring measurements (Fig. 3). Thethree locations SFJ1, SFJ2 and SFJ3 of flat jack tests are also shownon Fig. 3.

2.2. Hydraulic tests

Hydraulic tests involved two different techniques, hydraulicfracturing (HF) and hydraulic testing of pre-existing fractures(HTPF), following the procedures described by Haimson andCornet [4].

For HF, a portion of a borehole free of pre-existing fractures isisolated with a straddle inflatable packer. The pressure is progres-sively raised in the isolated interval till a hydraulic fracturedevelops at the so called breakdown pressure Pb. Then the fractureis extended till it reaches zones outside the domain of influence ofthe borehole. When the fracture stops, the hydraulic injectionsystem is kept shut so as to monitor the subsequent pressuredecay for some time (a few minutes). This testing period is calledthe shut-in period. At the end of shut-in, the system is shortly bledoff (a few seconds) and the subsequent pressure build-up isobserved for a few minutes. This part of the test is called flow-back and ends the first testing cycle. This testing cycle is repro-duced at least twice and sometimes more, when results showsome drifting from one cycle to the next (Fig. 4).

The pressure at which the fracture closes is called the instan-taneous shut-in pressure Ps. It is equal to the normal stresscomponent acting on the fracture plane, i.e. the natural minimumprincipal stress component when the hydraulic fracture haspropagated far enough from the borehole.

The HTPF procedure is very similar to the HF procedure exceptthat the isolated borehole portion is supposed to include onesingle pre-existing fracture. The pressure is raised sufficiently

Fig. 1. Layout of the Paradela II hydroelectric repowering scheme (courtesy of Energy of Portugal-EDP).

Fig. 2. Vertical cross-section A–A' along the pressure tunnel showing the relative location of the adit with respect to the 500 m deep vertical boreholes PD19 and PD23.

Fig. 3. Vertical cross-section along the adit axis (above) and three dimensionalscheme of the adit (below) showing the location of overcoring and flat jack tests.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–4838

slowly to insure a uniform pressure distribution within thefracture close to the borehole. When the injection pressure reachesthe normal stress supported by the fracture, the fracture opens.This quasi-static fracture opening operation alters all the compo-nents of the stress field in the vicinity of the fracture except thestress component normal to the fracture plane. Once the fracturehas been opened up to distances far enough from the well, theinstantaneous shut-in pressure of HTPF's yields an estimate on thenormal stress component that was supported before testing by thefracture plane away from the borehole.

The hydraulic testing equipment used for these measurementsincludes an inflatable straddle packer system together with anelectrical imaging device (Cornet et al. [5]). The system is used firstto log the borehole so as to identify the optimum location of testintervals (homogeneous rock formation for HF, single isolatedfractures for HTPF). Then hydraulic tests are conducted and anelectrical image of the tested interval is produced just after testing.Comparison of pre- and post-fracturing images provides therequired information for determining the geometry of the fracturethat has just been tested. The uncertainty on depths measure-ments is close to 0.5 m.

The stress normal to a tested fracture plane is directly related tothe shut-in pressure measured during each test cycle. Aamodt andKuriyagawa's method [6] was applied for identifying the max-imum borehole pressure for which the fracture is completelyclosed, i.e. the highest pressure for which flow obeys Darcy's law.This pressure value is an underestimate of the normal stresssupported by the fracture at closure. Hayashi and Haimson'smethod [7] was applied for determining the pressure at the endof fracture propagation, just prior to fracture closing. This providesan overestimate of the normal stress supported by the fracture. Foreach test sequence (three or more shut-in readings), the minimumvalue obtained with the Aamodt and Kuriyagawa method and themaximum value obtained with Hayashi and Haimson's methodwere used to define the 99% confidence interval for the normalstress estimate. If the error on the normal stress componentestimate is assumed to obey a normal distribution, then the extentof the 99% confidence interval is equal to six standard deviations.

The geometry and orientation of the tested fracture planes havebeen determined through a comparison between the orientedelectrical images obtained before and after hydraulic tests. Thefracture planes are recognised as sinusoids on electrical imaginglogs. Two sinusoidal curves, that completely cover the extent ofthe identified fracture plane, have been drawn to identify extremevalues for the azimuth and inclination of the normal to thefracture planes. From the scatter described by the two sinusoidal

curves, a 99% confidence interval is defined for the two angles. Inborehole PD23, a technical difficulty prevented the determinationof the tool orientation during logging. This difficulty was overcomeby running a properly oriented high-resolution acoustic televiewer(HRAT) log. A comparison with the electrical log provided theproper orientation of all the hydraulically tested fractures for thisborehole.

Tables 1 and 2 present a summary of the hydraulic test results.In these tables, z is the depth of tests, ϕ is the azimuth of thenormal to the fracture plane with respect to the North (positiveeastward), θ is the inclination angle of the normal to the fractureplane with respect to the vertical direction, and σn is the normalstress measurement for the corresponding depth interval. Thestandard deviations associated with z, ϕ, θ and σn are δz, δϕ, δθand δσn, respectively. In Table 2, the orientations marked with n

were only obtained with the HRAT log.The tables show that eight out of the nineteen tests are

ambiguous because more than one fracture plane is observed inthe test interval. For test number 1 in borehole PD23, the normal

Fig. 4. Typical curve for the interval pressure record versus time during a hydraulicfracturing test. Pb is the breakdown pressure and Ps is the instantaneous shut-inpressure.

Table 1Summary of the hydraulic test results obtained from borehole PD19.

Test Type Depth Azimuth Dip Normal stress

z (m) δz (m) ϕ (1) δϕ (1) θ (1) δθ (1) σn (MPa) δσn (MPa)

1 HF 471.8 0.5 108 4 87 2 10.3 0.42 HF 455.5 0.5 18 7 90 2 9.0 0.23 HF 450.1 0.5 126 5 90 2 7.8 0.2

HTPF 450.4 0.5 281 6 60 14 HF 442.1 0.5 133 5 90 2 9.0 0.2

HTPF 442.2 0.5 303 5 42 25 HTPF 436.3 0.5 108 6 32 2 8.9 0.26 HF 414.9 0.5 133 4 90 2 7.1 0.17 HTPF 393.4 0.5 270 4 66 2 7.3 0.3

393.9 0.5 284 3 79 2394.1 0.5 277 5 52 2394.3 0.5 50 5 38 2

8 HTPF 379.3 0.5 88 3 57 2 5.6 0.29 HF 335.6 0.5 119 3 61 2 6.7 0.110 HTPF 293.1 0.5 14 2 44 3 7.5 0.311 HF 279.8 0.5 22 4 90 2 5.8 0.112 HF 164.6 0.5 320 7 86 2 2.6 0.3

Table 2Summary of the hydraulic test results obtained from borehole PD23.

Test Type Depth Azimuth Dip Normal stress

z (m) δz (m) ϕ (1) δϕ (1) θ (1) δθ (1) σn (MPa) δσn (MPa)

1 HF 490.7 0.5 199 5 90 2 20.2 0.52 HF 421.8 0.5 19 5 90 2 9.9 0.2

HTPF 420.9 33 5 42 23 HF 402.4 0.5 348 5 68 2 8.9 0.2

HTPF 402.5 252 5 34 34 HTPF 377.4n 0.5 215n 5n 81n 3n 9.7 0.2

377.8n 106n 5n 12n 3n

377.8n 46n 5n 20n 3n

378.4 2 5 35 2377.9 137 5 22 2377.9 59 5 24 2377.7 358 10 37 2

5 HF 364.7n 0.5 325n 5n 81n 3n 7.0 0.2HTPF 364.6 252 5 35 2

6 HF 356.8 0.5 269 4 90 2 5.9 0.1HTPF 357.0n 33n 5n 6n 3n

7 HF 176.6 0.5 243 3 77 2 3.2 0.2

n orientation obtained from the re-orientation of the electrical logs by using theHRAT log.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48 39

stress magnitude is significantly larger than all the other values.However, the measurement is reproducible and satisfies all theprerequisite conditions of validity. A detailed examination of boththe electrical and the HRAT images shows a clear inclined fracturebelow the lower packer. In comparison to other post-fractureimages, no other fracture is observed within the tested interval.This result strongly suggests that flow occurred within the inclinedfracture below the packer. In such conditions, the pressure appliedby the packer is also applied on some parts of the inclined fracturesuch that its opening remains very limited in azimuth. The stressanalysis for such fractures has been discussed in [8], who showedthat the shut-in pressure in such configurations does not yield thefar-field normal stress value. This result has not been included inthe data base for the stress determination.

For test number 8 conducted in borehole PD19, a stronglyaltered zone was observed in the electrical imaging log, approxi-mately 20 m above the test location. This zone dips steeply and isnot very far from the tested fracture. It affects most probably thestress field measured with test 8.

Finally, we note that tests number 7 in borehole PD19 and testnumber 4 in borehole PD23 resulted respectively in four and sevendifferent fractures observed within the tested interval. These twotests do not bring any significant constrain to the solution. Theyhave not been considered for the determination of the optimumsolution.

Fig. 5 shows a plot of the normal stress magnitudes as obtainedwith HF and HTPF as a function of depth as well as the 99%confidence limit associated with the measurements.

Fig. 6 shows a plot of the azimuth of the normal to the fractureplanes as obtained with HF tests only, as a function of depth aswell as the 99% confidence limit associated with the measure-ments. In these tests, the fracture planes are parallel to theborehole axis which demonstrates that, at these locations, theborehole is parallel to a principal stress direction and that theminimum principal stress is oriented perpendicularly to the bore-hole axis. Two different groups of data are noted. The solid linerepresents the azimuth of fractures for which the measurednormal stress is close to the expected minimum stress magnitudeat the same depth. The dashed line represents a perpendicular setof azimuths. Results shown in Figs. 5 and 6 reveal that the normalstress magnitudes measured for sub-vertical fractures that arenearly perpendicular to each other are yet very similar. This resultsuggests that differences between the maximum and minimumprincipal horizontal stress magnitudes are likely fairly small. Thisis particularly noticeable for tests 2, 3 and 4 in PD19. The threetests are only 10 m apart, yet the fracture of tests 2 is nearly

normal to that of test 3 whilst shut-ins for tests 2, 3 and 4 are verysimilar.

2.3. Overcoring tests

The overcoring method is based on the stress relief principle. Ityields the complete stress state at the corresponding locationprovided linear elasticity applies. For these measurements theStress Tensor Tube (STT), initially developed by Rocha and Silvério[9], has been used for it provides all tensor components through asingle overcoring operation.

The STT strain measurement device is a hollow epoxy resincylinder with an outer diameter of 35 mm, an approximate lengthof 20 cm, and a thickness of 2 mm (Fig. 7). The cell has tenelectrical resistance strain gauges embedded in positions normalto the faces of a regular icosahedron, which enables sampling ofthe displacement field in all corresponding directions [10]. The cellincludes a metal capsule that houses the data acquisition unit aswell as a thermocouple. Readings of all strain gauges and of thetemperature are conducted at fixed time intervals (60 s) and thenare stored in the local memory.

A test consists of the following operations: (1) drilling of a140 mm diameter borehole to the depth of interest; (2) drilling aconcentric 37 mm diameter borehole from the bottom of the largediameter hole in which the STT cell is inserted and glued againstthe walls; and (3) resuming the drilling of the large diameter holeto a depth compatible with a complete stress relief around the cell.

After overcoring, the rock core together with the STT isrecovered. Fig. 8 provides an example of the variation of strainswith time at the location of the ten strain gauges during anovercoring test. The dashed line represents the variation of thetemperature with time. Strain readings are taken before and afterovercoring when the temperature is stabilised. The differencebetween these values corresponds to the strains that result fromthe overcoring stress relief.

Elastic constants of the rock core have been determinedthrough pressure tests conducted on the cores with a biaxialchamber in which a radial hydraulic pressure p is applied. Threeloading and unloading cycles were performed. The first cyclereached a maximum pressure of 2 MPa. For the other two cycles,a maximum pressure of 6 MPa was applied. The deformation atthe location of the ten strain gauges that resulted from the appliedpressure has been measured.

The model used for interpreting the STT tests results assumes thatthe rock is homogeneous, linearly elastic and isotropic; that thelength to diameter ratio for the cell is high; and that the stiffness ofthe hollow cylinder is significantly less than the stiffness of the rock.Fig. 5. Variation of the normal stress σn magnitudes as a function of depth.

Fig. 6. Variation with depth of the azimuth ϕ of the normal to the fracture planesobtained during hydraulic fracturing tests.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–4840

In this model, given the arrangement of the strain gauges, the strainεi at the location of strain-gauge i (i¼1,..,10) is linearly related to thecomponents of the in situ stresses σj (j¼1,..,6):

εi ¼ aijσj; ð1Þ

where aij are the components of a matrix with ten rows and sixcolumns that depends on the elastic parameters (elastic modulus Eand Poisson's ratio ν) of the overcored core [10].

With the measured ten strains that result from a biaxial loading(σ1¼σ2¼p and σ3¼σ4¼σ5¼σ6¼0), a least squares method isapplied to solve the matrix Eq. (1) and determine the elasticconstants. By considering the set of six biaxial tests obtained ineach borehole, the following mean values (E and ν) and associatedstandard deviations (δE and δν) for the elastic constants have beendetermined: borehole PD1: E¼60.8 GPa, δE¼12.9 GPa, ν¼0.30,δν¼0.05; borehole PD2: E¼57.7 GPa, δE¼9.5 GPa, ν¼0.35,δν¼0.11.

The standard deviations of the elastic modulus are approxi-mately 21% and 16% of the mean, whereas the standard deviationsfor Poisson's ratio are approximately 17% and 31% of the mean forboreholes PD1 and PD2, respectively. This analysis outlines asignificant dispersion, which may be compared to results fromuniaxial compression tests conducted on cores: E¼44.6 GPa,δE¼9.0 GPa, ν¼0.25, δν¼0.06.

For these tests, the standard deviations of the elastic modulusand Poisson's ratio are approximately 20% and 24% of the mean,respectively. Hence Poisson's ratio values as determined frombiaxial testing are found to be considerably larger than thoseobtained with uniaxial tests. This may be attributed to thedevelopment of microcracks normal to the axes of the rocksamples collected during overcoring. Indeed, it is well known thatunder specific loading conditions such microcracking leads todisking [11].

Once the strains resulting from overcoring have been measuredand the elastic constants have been determined, a least squaresmethod is applied for determining the six components of thestress tensor in a co-ordinate system associated with the STT cell.The magnitude and orientation of the principal stresses (σI, σII, σIII)obtained in each test are presented in Table 3. In this table, theorientations are described by two angles; the first angle is thedirection of the principal stress component with respect to the

North, and the second angle is the inclination with respect to ahorizontal plane.

The table shows that the maximum principal stress (σI) is sub-vertical and that the other two principal components (σII and σIII)are sub-horizontal and of similar magnitude. Hence the apparentdispersion for the direction of sub-horizontal principal stresscomponents is not significant. In borehole PD1, the sub-verticalcomponent measurements are significantly greater than thosemeasured in borehole PD2, a feature which cannot be explainedby the depth difference between the tests. It has been oftenobserved that overcoring tests lead to high stress values in thedirection parallel to the borehole axis. This has been explained bythe large deformations that occur in this direction as a result of theglue yield caused by the heat generated during the drillingoperation [12]. It may also result from an improper elastic modelbecause of the nonlinearity induced by the microcracking phe-nomenon already mentioned.

In borehole PD2, stress values obtained for test 5 at 580.5 mabove sea level are very questionable, considering the valuesobtained with neighbouring tests. This test result has not beenconsidered for further analysis.

2.4. Small flat jack tests

The small flat jack testing method [13,14] is based on the stressrelief principle, and implies a partial stress relief followed by stresscompensation. In this technique, two pairs of pins are placed onthe rock surface, and the initial distance between the pins ismeasured with digital transducers. Then, a 10 mm-thick slot is cutperpendicularly to the rock surface, using a 60 cm diameterdiamond disk, till a 27 cm depth is reached. Due to the partialstress relief, deformations in the direction normal to the slot occurand the distance between the pins decreases. Subsequently acircular flat jack, consisting of two thin metal plates weldedtogether, is inserted into the slot and pressurised until the distancebetween the pins is restored. During the test, the variation of therelative displacements between the two pairs of pins caused by

Fig. 7. STT cell and data acquisition unit.

Fig. 8. Strain versus time curves obtained during an overcoring test.

Table 3Principal stress (σI, σII, σIII) magnitudes and orientations obtained from interpretingthe overcoring tests.

Borehole Test Elevation (m) Principal stressmagnitudes (MPa)

Principal stressorientations (1)

σI σII σIII σI σII σIII

PD1 1 618.2 6.4 5.6 4.7 325/66 85/12 179/202 614.5 7.0 3.4 3.2 249/86 147/1 56/43 599.5 8.7 3.4 3.3 354/86 231/2 141/44 598.9 9.3 6.9 5.9 36/64 188/23 283/115 570.1 10.6 8.2 7.0 94/54 200/11 298/346 569.3 11.3 6.5 6.0 26/73 150/10 242/14

PD2 1 619.5 11.9 6.9 5.9 94/79 260/11 350/32 617.9 8.0 6.3 4.3 328/75 229/2 139/153 598.6 6.2 2.8 2.0 324/79 177/10 86/64 597.9 5.1 4.6 4.3 285/63 24/5 116/265 580.5 �1.9 �3.6 �4.0 267/77 69/12 160/46 579.2 6.0 4.2 3.4 289/39 147/45 36/20

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48 41

the applied pressure is recorded. Fig. 9 shows a plot of displace-ment versus pressure as obtained during a typical test. Generally,some non-elastic behaviour is observed and non-recoverabledisplacements are detected after unloading.

The pressure required to restore the initial position of the pinsis called the “cancellation pressure”, and it is assumed to be equalto the stress component normal to the slot plane. In this determi-nation, only the loading phase observed during the first cycle wasconsidered.

Results are presented in Table 4 where d is the distancebetween the test location and the adit's entrance, ϕ is the azimuthof the normal to the slot with respect to the North, θ is theinclination of the normal to the slot plane with respect to thevertical direction and σn is the normal stress component.

For tests number 1, 5 and 9, somewhat similar sub-verticalstress components have been measured, with values equal to8.9 MPa, 9.9 MPa and 9.9 MPa, respectively. But comparisonbetween the horizontal stress components parallel to the aditdirection and measured during test numbers 3, 7 and 11, showsthat the normal stress value measured during test number 3 issignificantly larger than that measured during the two other tests.Actually, for test number 3, because of a technical problem, it wasnot possible to run the measurement immediately after the slothad been cut. The rock was observed to have a creeping behaviour.The pressure versus displacement curve shows a significant slotclosure during the time interval that separates the end of thecut from the beginning of pressurisation. Consequently a high

cancellation pressure was required for restoring the initial positionof the pins within the same time span as for the other tests.

3. The optimum stress model

3.1. Data integration and solution identification

When in situ stress measurement data of various kinds havebeen acquired at diverse locations, a common procedure foridentifying the regional stress field is to develop a numericalmodel that fits best all observations [15–19]. Then this numericalmodel helps evaluate the stress tensor at the various locations ofinterest.

The aim is to determine the model that minimises differencesbetween a number of observations and predictions from themodel. This determination requires a model definition, a definitionof the misfit for describing the discrepancy between observed andpredicted values, and a normative measure of the misfit forquantifying the residuals for all observations [20].

3.2. Model's definition

We call model a set of unknown parameters that is looked forin order to fit observations. The objective is to define a simpleenough model so that the number of constraining data is largerthan the number of unknown parameters (degrees of freedom) forthe model. We consider the volume of interest as perfectlydefined, i.e. topography and all structural elements like the shapeof the adit are considered to be well known. This volume is filledwith an equivalent continuous geomaterial the mechanical char-acteristics of which are part of the model definition. The corre-sponding set of differential equations that describe its mechanicalbehaviour are solved with explicit finite differences using thesoftware FLAC3D [21].

The mesh constructed with the FLAC3D software (Figs. 1 and 10)is composed of 600,000 elements. It is finer above sea level, withcubic 25 m-sided elements. Below sea level, the elements are50 m�50 m�100 m. Boundary conditions imposed on the verticalboundaries are part of the model definition, as discussed here after,but a condition of no vertical displacement is imposed on thehorizontal basal boundary.

In order to determine the influence of volume size on the stressdetermination, more specifically the influence of the proximity ofvertical boundaries to the points of interest, various geometrieshave been considered when performing the first gravitationalanalysis (g¼9.81 m s�2). Results show that the 5 km long and3 km wide region shown in Fig. 1 is sufficiently large for obtaining

Fig. 9. Typical pressure versus displacement curves obtained during small flatjack tests.

Table 4Results from the small flat jack tests.

Location Test d (m) ϕ (1) θ (1) σn (MPa)

SFJ1 1 447 0 0 8.92 448 110 45 6.43 437 110 90 9.44 446 290 45 3.7

SFJ2 5 332 0 0 9.96 333 290 45 2.67 331 290 90 2.08 332.5 110 45 6.4

SFJ3 9 278 0 0 9.910 275 290 45 3.211 279 290 90 4.112 280 110 45 3.0

Fig. 10. Mesh of the FLAC3D model.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–4842

reliable estimates. Indeed, for larger regions, the maximum differ-ence between the computed three principal stress magnitudes atthe locations of the various tests gets smaller than 0.5 MPa. Withinthe domain of interest elevation varies between 315 m and 1030 mabove sea level. A 2.5 km extension below sea level has beenassigned in the vertical direction so that boundary conditions atthe base of the model do not affect the stress evaluations wheretopography is significant.

Various models have been considered. For the first model, theelastic constants that characterise the geomaterial are assumed tobe those measured on the cores (2 parameters model) and onlyvertical gravity loading is considered (no displacement normal tothe vertical boundaries). The adit is ignored and the data setincludes only HF and HTPF results. The second model is similar tothe first model except for a horizontal tectonic stress, which isintroduced so as to decrease the minimum computed misfit. Thethird model is similar to the first one but the Poisson's ratio for therock mass is taken as an unknown. Further the data set includesboth hydraulic tests and overcoring tests. In the fourth model, onlygravity loading is considered together with the optimum Poisson'sratio identified with model 3 but the adit is introduced so as todetermine the fit with data gathered close to the adit walls.

3.3. Definition and measurement of misfit

For the ith hydraulic normal stress measurement, the misfit ψiHF

may be expressed as

ψ iHF ¼

jσin;mes�σi

n;calcjδinþδif

; ð2Þ

where σin;mes and σi

n;calc are, respectively, the measured andcalculated normal stresses obtained at the location of the ithhydraulic test, δin is the uncertainty on the normal stress measure-ment and δif is the uncertainty on the normal stress estimateassociated with uncertainties on the orientation of the fractureplane selected for the ith test [22].

For ambiguous tests with two or more observed fractureplanes, the fracture plane has been selected to be that whichyields the smallest difference between measured and calculatednormal stress. Uncertainties on the normal stress measurementsand on the fracture plane orientation determinations correspondto the 99% confidence interval for the measurements. Uncertain-ties on the normal stress component due to the uncertainty on thefracture plane orientation determination have been estimatedusing the FLAC3D code. Values have been found to be of the sameorder of magnitude as those of uncertainties associated with theshut-in pressure measurements.

For overcoring data, the uncertainties are associated with thestrain measurement technique, the orientation of the strain gaugesin the three dimensional space and the determination of elasticproperties of the overcored cores. The misfit ψj

OC that has beenused is simplified and is defined as

ψ jOC ¼ jσj

mes�σjcalcj

δjσþδjbh; ð3Þ

where σjmes and σj

calc are respectively the measured and calculatedvalue for the jth principal stress components as obtained at thelocation of the kth overcoring test (j¼3k�2þp; p¼0,1,2); δjσ is theuncertainty associated with the stress measurement and δjbh is theuncertainty on the computed stress component because of uncer-tainties on the location of the test (mostly associated withorientation of the borehole axis).

Because overcoring tests were conducted within a small depthrange (approximately 60 m) in vertical boreholes, the uncertaintyassociated with the orientations of these boreholes was neglected.

Evaluation of the uncertainty on the stress measurements is basedon considerations on the dispersion of the measurements ratherthan on a rigorous error analysis procedure. We considered thevarious pairs of overcoring tests that have been conducted (i.e.tests conducted within 5 m from one other). For each pair of testswe determined the mean difference between principal stresscomponents and then we determined the mean value for all thepairs. This has led to a 1.5 MPa dispersion estimate, which is thevalue chosen for characterising the uncertainty on the variousprincipal stress components. With this approach, no concern isgiven to the systematic bias that may result from the linear elastichypothesis.

The hydraulic testing and overcoring methods are of differentnatures and concern different rock volumes. This has been takeninto consideration for the definition of the global misfit functionby introducing weighting parameters that take into account thevolume, or the area, involved by a given measurement for each ofthe method. Further each measurement has been weightedaccording to the relative value of its misfit with respect to theglobal value that may be used for characterising stress determina-tions conducted with hydraulic tests alone or overcoring testsalone. The general misfit ψq

HFOC for model Mq may be expressed as

ψ qHFOC ¼ ∑

M

i ¼ 1ωi

HFψ iHFþ ∑

N

k ¼ 1ωj

OCψ jOC ; ð4Þ

where M and N are, respectively, the total number of hydraulictests and of overcoring tests and the weight factors for hydraulicand overcoring data, ωi

HF and ωjOC, are given by

ωiHF ¼ AHF

AREV UψHF

i

ψHFmin

; ð5Þ

ωOCj ¼ VOC

VREV UψOC

j

ψOCmin

; ð6Þ

where AHF and AREV denote respectively the measurement area andthe area involved in the representative elementary volume of therock mass (REV) for hydraulic testing; VOC and VREV are thecorresponding notations associated with the overcoring technique(measurement volume and REV volume, respectively); and ψHF

minand ψOC

min are respectively the minimum values for the misfitsobtained when the stress determination is derived only fromhydraulic tests or only from overcoring tests, i.e. minimisationfor the sums of Eqs. (2) or (3), respectively.

The area involved by hydraulic testing depends on the area ofthe opened fracture. It depends on many parameters such as theinjected flow rate but also fluid losses through the walls of thefracture. It was set somewhat arbitrarily equal to 1 m2, given theinjection tests characteristics. The volume involved by overcoringmeasurements was set equal to the average volume of theresulting hollow rock cylinder. The REV for the rock mass wasset equal to 1 m3 (i.e., area 1 m2). Thus, the suggested global misfitgives more weight to hydraulic data than to overcoring data.

Two types of normative methods are commonly used to evaluatethe misfit between a model and a set of data: the l1-norm considersthe sum of the absolute values of the differences between observa-tions and predictions; whilst the l2-norm considers the sum of thesquares of the misfits. The l2-norm is generally chosen when alluncertainties obey a normal distribution [23]. But in the present casesome of the uncertainties are not Gaussian, such as for example thelinear elastic hypothesis for the overcoring measurements. Hence thel1-norm has been chosen.

Once a global minimum has been identified (minimum ofψHFOC), the limits of the 90% confidence level for a posteriori

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48 43

evaluations may be estimated by

ψHFOC90% ¼ 1:645 π=2�1

� �1=2 MþNð Þ1=2þMþNðMþNÞ�W

ψHFOCmin ; ð7Þ

whereW is the number of unknown parameters of the model usedto describe the regional stress field (Parker and Mcnutt [23]).

3.4. Solution with gravity loading alone and elastic parameters fromtests on core samples

A first stress calculation was conducted to evaluate the influ-ence of topography on stresses at the location of hydraulic tests, onthe assumption that the rock mass is linearly elastic. Its elasticcharacteristics have been assumed to be identical to those of therock cores tested in the laboratory (E¼45 GPa, ν¼0.25). The rockmass density was set equal to 2650 kg/m3. No optimisation wasundertaken. A comparison between measured and computedvalues is shown on Fig. 11.

For ambiguous tests with two observed fracture planes, com-puted values concern the plane for which the difference betweenobserved and computed value is the smallest. For most tests,differences between measured and calculated values are found tobe much larger than uncertainties on the measurements. So, thismodel has not been optimised and the possibility that tectonicstresses affect the massif has been investigated.

3.5. Testing the existence of tectonic stresses

The objective here was to explore the possibility of improvingthe fit between model and observations by introducing so calledtectonic stress components, i.e. horizontal stress components, onthe vertical boundaries of the model.

Several assumptions are made: (i) the rock mass exhibits alinear, isotropic and elastic behaviour; (ii) the total stress field maybe decomposed into gravity and tectonic components; (iii) thevertical component is equal to the weight of the overlying materialand only of gravitational origin; and (iv) with exception for thezones close to ground level where topography effects are impor-tant, the tectonic stresses are considered independent of depth.

The normal stress (σn,mes) measured on each tested fractureplane is decomposed into a component due to gravity (σn,grav) and

a component associated with tectonic (σn,tect) loading:

σn;mes ¼ σn;gravþσn;tect : ð8Þ

Unit normal (Sxx, Syy) and shear stress (Syy) components areintroduced in the model: uniform displacements are imposed onthe lateral boundaries of the model in order to yield unithorizontal stress components for elements in contact with thebasal boundary [24]. As a result, non-uniform and balanced stressdistributions are generated at the lateral boundaries that simulatethe influence of topography effects on the stress field.

The normal stress magnitudes σn,Sxx, σn,Syy and σn,Sxy at thelocation of each tested fracture plane due to unit tectonic stresscomponents Sxx, Syy and Sxy are computed. The normal stressmagnitudes due to tectonic loading (σn,tect) is estimated as a linearcombination of the response to unit tectonic stresses:

σn;tect ¼ Aσn;SxxþBσn;SyyþCσn;Sxy; ð9Þ

where coefficients A, B and C are unknown parameters thatcharacterise the tectonic stress field. Substituting Eq. (9) intoEq. (8) yields

σn;calc ¼ σn;gravþAσn;SxxþBσn;SyyþCσn;Sxy: ð10Þ

The right side of Eq. (10), σn,calc, is introduced in the misfitdefinition [2] for determining the coefficients A, B and C. Thefollowing values have been obtained:

A¼ 4:9;B¼ 5:1;C ¼ 0:0 ð11Þ

A comparison between measured and computed normal stress(σn) when combining tectonic and gravity loadings is shown inFig. 12. Differences between measured and computed values forhydraulic tests are found to be larger than uncertainties on themeasurements for approximately 60% of the tests. Further we notethat the values for A, B and C that best fit the results are such thatparameter C is null, whereas parameters A and B are similar. Thiscorresponds to horizontal compressive normal stresses of similarmagnitudes and to zero shear stresses.

This is in agreement with hydraulic data collected at thebottom of the boreholes which suggests that both horizontalprincipal stress magnitudes at this depth are very similar to eachother. But, this does not agree with focal plane solutions of localearthquakes that show an orientation for the regional tectonic

Fig. 11. Variation of the magnitudes of the normal stresses obtained by hydraulic testing (σn,mes) and with the FLAC3D model (σn,calc) run with gravity loading only inboreholes (a) PD19 and (b) PD23 as a function of depth, when the Poisson's ratio is taken equal to 0.25.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–4844

stress approximately N1351E [25,26] and we conclude that tec-tonic stresses are likely not effective at the depth of the tests.

3.6. Identification of the rock mass rheological characteristics

The analysis conducted in this section assumes a linearly elasticbehaviour for the rock mass. Because changing the elastic modulusdoes not induce changes in the stress field for a homogeneous rockmass, only an increase of the Poisson's ratio value is tested. TheFLAC3D model has been used in the inversion of the Poisson's ratiovalue to minimise the misfits defined by Eqs. (2)–(4). The resultspresented in Table 5 show that a Poisson's ratio value of 0.47provides the best fit between results obtained from both hydraulicand overcoring data and those computed with the model. Eq. (7)was used (W¼1) to calculate a 90% confidence interval for theminimum of the global misfit. The interval obtained in this waycorresponds to a Poisson's ratio value that ranges between 0.45and 0.49.

The profiles of the measured and calculated normal stresses(σn) due to gravity loading with a 0.47 Poisson's ratio are shown inFig. 13. The difference between measured and computed normalstress magnitudes is less than three standard deviations forapproximately 75% of the tests which is considered acceptablegiven the many simplifying assumptions implied by this model.

Comparison of measured and computed principal stress valuesat the location of overcoring tests in the PD1 and PD2 boreholesis shown in Fig. 14. Lines represent values computed with theFLAC3D model, and dots represent overcoring results. The positionabove sea level of the adit axis is also represented. At depth of thetests, the calculated maximum principal stress is parallel tothe boreholes axis and the other two components are sub-horizontal and similar magnitude. Further the FLAC3D modelresults also show that the stresses magnitude at location ofboreholes PD1 and PD2 are similar. This agrees with the resultsof the stress measurement results. In addition, approximately 80%of the measured and computed principal stresses are in satisfac-tory agreement (the difference between both values is less than1.5 MPa).

Hence, hydraulic and overcoring data may be explained by alinearly elastic rock mass under gravitational load provided thePoisson's ratio for the equivalent material is considerably largerthan that measured during short-term uniaxial compression tests.

3.7. Analysis of the test results obtained close to the adit

In this model exploration, we investigate how the linearlyelastic model defined here above fits the flat jack stress measure-ments and overcoring tests run the closest to the adit walls.

Firstly, a comparison of measured and computed normal stressvalues was made at the location of the small flat jack tests. Sincethe stresses provided by the small flat jack technique do notcorrespond to the far-field stress components because they areinfluenced by the existing adit, a three-dimensional numericalmodel of finite differences was developed using the code FLAC3D(Fig. 15) for the interpretation of the small flat jack test results.This model is a 30 m�30 m�5 m solid and includes the rectan-gular cross section of the adit. Note that the elastic modulus has noeffect on the solution. A Poisson's ratio value of 0.25 obtained fromuniaxial compression tests conducted on intact cores was con-sidered. Variations of the Poisson's ratio between 0.25 and 0.47resulted in a maximum variation of 0.5 MPa for the calculatednormal stresses. In addition to this uncertainty, an uncertainty of1 MPa on the normal stress measurements and an uncertainty of0.5 MPa on the calculated normal stresses due to the uncertaintyon the geometry of the adit and on the flat jack orientations wereassumed. This results in a 2.0 MPa for the maximum uncertaintyon the normal stress values. The model is used to calculate thenormal stresses at the location of the small flat jack tests byconsidering the far-field stress tensor components obtained withthe large-scale model shown in Fig. 10, providing a Poisson's ratioof 0.47 is taken for the rock mass. A comparison between thenormal stresses obtained this way and the stresses actuallymeasured with the small flat jacks is shown in Fig. 16. The figure

Fig. 12. Variation with depth of the normal stress magnitudes as measured by hydraulic testing (σn,mes) and computed with the FLAC3D model (σn,calc) for the combined effectof gravity and tectonics in boreholes (a) PD19 and (b) PD23.

Table 5Variation of the misfit value with the Poisson's ratio.

ν ψHF ψOC ψHFOC

0.25 56.7 43.9 295.00.35 37.1 38.5 126.30.45 13.2 35.3 16.40.46 11.8 35.1 13.20.47 10.9 34.9 11.50.48 12.2 34.3 14.20.49 12.6 34.6 14.9

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48 45

Fig. 13. Variation with depth of the normal stress magnitudes as measured by hydraulic testing (σn,mes) and computed with the FLAC3D model with ν¼0.47, consideringgravity effects only, in boreholes (a) PD19 and (b) PD23.

Fig. 14. Variation with elevation above sea level of the magnitude of the principal stresses (σI, σII, σIII) obtained by overcoring testing (OC) and with the FLAC3D model (FM)with ν¼0.47 considering gravity effects only.

Fig. 15. Three-dimensional model used for interpretation of the small flatjack tests.

Fig. 16. Comparison of the magnitudes of the normal stresses obtained by small flatjack technique (σn,mes) and with the FLAC3D model (σn,calc) with ν¼0.47 consideringgravity effects only.

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–4846

shows that for approximately 75% of the small flat jack tests, thediscrepancy between measured and calculated normal stresses issmaller than the uncertainty on the normal stresses. In testnumber 3, the discrepancy between the measured and calculatednormal stress value is understandable because of the alreadymentioned technical problem that occurred during the test.

Secondly, the stress field without the adit was calculated with themodel shown in Fig. 10, at the location of the shallow overcoring testsdone in boreholes PD1 and PD2. To analyse the influence of the adit,these stress components were used as boundary conditions in themodel shown in Fig. 15, and the stresses at the location of the shallowovercoring tests were calculated.

Comparison between measured and calculated principal stressesmagnitude and orientation is presented in Table 6. In this table, theorientations are described by two angles; the first angle is thedirection of the principal stress component with respect to theNorth, and the second angle is the inclination with respect to ahorizontal plane.

The table shows that the FLAC3D model results obtained at depthof the shallow overcoring tests done in borehole PD2, are not insatisfactory agreement with the stress measurements provided byovercoring (the difference between measured and calculated principalstress values is higher than 1.5 MPa). For these two tests, large sub-vertical stress components were measured which is also visible fromthe results of flat jack tests number 5 and 9. Results from FLAC3Dmodel and overcoring tests show that at depth of the tests done inboreholes PD1 and PD2, one of the principal stresses is sub-verticaland the other two components are sub-horizontal and of similarmagnitude. Due to this the dispersion observed on the direction ofsub-horizontal principal stresses is not significant. This analysis con-cludes that, with an elastic solution, the stress perturbation induced bythe adit at the location of the shallow overcoring tests is negligible,and it is not possible to explain the stresses measured by the twoshallow overcoring tests in borehole PD2 and flat jack tests number5 and 9. The source of this local stress heterogeneity is likely linked toa local fracture zone but this possibility has not been explored further.

4. Discussion

Results from the FLAC3D model show that most of the data areconsistent with a linearly elastic equivalent geomaterial, with proper-ties that correspond to a much softer material than suggested bylaboratory tests on cores. At this scale, the rock mass includes animportant natural fracture network (see Fig. 1, and images fromelectrical and HRAT logs) and the material filling these variousfractures and faults has been subjected to creep over time so thatvery small shear components are left along these planes of weakness.

Creep effects may be assessed by using a visco-elasto-plasticmodel. However, our objective here is not to simulate the timetransients of the deformation but rather to extrapolate results of local

stress measurements to locations of interest for the design of openingsrequired by the future hydroelectric infrastructures. Our analysisshows that choosing an equivalent linearly elastic material with verysoft elastic properties fulfils this need. Further, changing the elasticmodulus does not induce any change in the stress field for ahomogeneous rock mass, so that only an increase of the apparentPoisson's ratio value has revealed necessary. This equivalent linearlyelastic model has been found to fit approximately 75% of the in situstress measurements. It may be used for the design of the under-ground opening required by the hydroelectric repowering project.

Interestingly, this linearly elastic behaviour for the granite rockmass is markedly different from the short-term one considered forboth interpreting overcoring tests and simulating results from the flatjack tests, which depend on the stress concentration associated withthe adit. The adit is 50 years old so our results suggest that, for thistime scale, the elastic parameters as derived from tests on core yieldsatisfactory results except for the local stress measurements conductedfor tests 1 and 2 in borehole PD2 and nearby flat jack tests number5 and 9.

This suggests two possibilities for modelling this stress field. Wemay consider an elasto-viscous model with very high viscosity so thatit behaves elastically at the scale of 50 years but behaves like a fluid atthe scale of million years. Or we consider a linearly elastic model witha network of fracture sets with sufficiently diverse orientations andlittle long-term shear strength so that the long-term stress field is sub-hydrostatic away from topographic limits. The existence of local stressconcentrations, either positive or negative, as observed for very few ofthe hydraulic tests as well as for two overcoring and two flat jack testsare in qualitative agreement with models of shear motions [27]. Thissuggests that the second proposition is more appropriate than the firstone. And this has strong consequences for the design of the pressuretunnel. Indeed, the safe design of the unlined hydraulic pressuretunnel requires an absence of significant water leakage as would beobserved if some hydraulic jacking occurs along the tunnel. Thisrequires that the planned water pressure be smaller than the smallestminimum principal stress value encountered along the tunnel. As aconsequence, the design of the hydraulic pressure tunnel should bemade in terms of the minimum expected value for the minimumprincipal stress magnitude instead of the average value, implicit in thesolution presented in Section 3.4.

5. Conclusion

Several in situ stress measurements were conducted at theParadela II site for the design of an underground reinforcementpower scheme that includes a large powerhouse cavern and ahydraulic pressure tunnel. The measurements include nineteenhydraulic tests in two 500 m deep vertical boreholes, twelveovercoring tests in two 60 m deep vertical boreholes drilled from

Table 6Comparison of the magnitudes of the principal stresses (σI, σII, σIII) obtained by overcoring testing (OC) and with the FLAC3D model (FM) with ν¼0.47 considering gravityeffects only.

Borehole Test Elevation (m) Method Principal stress magnitudes (MPa) Principal stress orientations (1)

σI σII σIII σI σII σIII

PD1 1 619.5 OC 6.4 5.6 4.7 325/66 85/12 179/20FM 6.0 4.5 3.8 225/71 127/3 36/19

2 617.9 OC 7.0 3.4 3.2 249/86 147/1 56/4FM 6.1 4.6 3.9 225/70 127/3 36/20

PD2 1 618.2 OC 11.9 6.9 5.9 94/79 260/11 350/3FM 5.2 3.8 3.0 193/58 293/6 27/31

2 614.5 OC 8.0 6.3 4.3 328/75 229/2 139/15FM 5.2 3.8 3.1 193/60 293/6 26/29

B. Figueiredo et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 37–48 47

an existing adit and twelve small flat jack tests in the walls ofthe adit.

Analysis of hydraulic and overcoring data demonstrates thatone principal stress component is sub-vertical within most of thevolume that was tested. The other two components are sub-horizontal and of similar magnitude. Some local zones of hetero-geneity were encountered, which have been attributed to pre-existing inclined fractures, as observed on the electrical imaginglogs. Sub-equality between the two horizontal principal stresscomponents and the local heterogeneities explain the dispers-ion on orientations observed for horizontal principal stressesdirection.

75% of the measurements are consistent with a linearly elasticrock mass under gravity loading if the Poisson's ratio of theequivalent geomaterial is taken equal to 0.47, a value quite largerthan the value measured on cores and used for interpreting bothovercoring and flat jack tests.

It has been concluded that the observed large-scale stress fieldresults from the shear stress relaxation over a large number of pre-existing fractures and faults with very variable orientations ratherthan from the shear stress relaxation to be expected with a verylong-term viscous behaviour for this rock mass. Indeed only slipalong weakness planes is consistent with observed large-scalestress relaxations and local stress concentrations that are eitherpositive or negative.

Acknowledgements

This paper is a part of the PhD thesis by Bruno Figueiredo at theStrasbourg University. The work was funded by the PortugueseLaboratory for Civil Engineering (LNEC) and the Foundation forScience and Technology (FCT) PhD grant SFRH/BD/68322/2010.Authorisation by EDP-Energies of Portugal to publish the stressmeasurement results obtained at the Paradela II site areacknowledged.

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