Rock Mech. Rock Engng. (1997) 30 (3), 121-127 Rock Mechanics and Rock Engineering 9 Springer-Verlag 1997 Printed in Austria

Effect of Tunnel Depth on Modulus of Deformation of Rock Mass

By

M. Verman 1, B. Singh 2, M. N. Viladkar 2, and J. L. Jethwa 3

l Advanced Technology and Engineering Services, AIMIL Ltd., New Delhi, India 2 Department of Civil Engineering, University of Roorkee, Roorkee, India

3 Central Mining Research Institute Regional Centre, Nagpur, India

Summary Deformability of rock mass significantly influences its behaviour and is, tberelore, an important consideration for the design of underground openings. The modulus of deformation of rock mass is, however, normally obtained from expensive and time-consuming uniaxial jacking tests, whose results often have a large scatter. An empirical correlation has, therefore, been proposed for a quick and inexpensive preliminary estimation of the modulus of deformation of rock mass on the basis of field instrumentation carried out in tunnels in India.

1. Empirical Correlation for Modulus of Deformation

1.1 Back Analysis of Instrumentation Results

Field instrumentation was carried out in several Indian tunnels (Table 1) to obtain the data of support pressures and tunnel closures at various tunnel sections exhibiting elastic ground behaviour. The data were then back-analysed to obtain the modulus of deformation of rock mass using the following expression applicable for a circular opening driven through a homogeneous, isotropic and linearly elastic rock mass under hydrostatic stress field:

Ua/a = (1 + v)(po -p i ) /Ed

where,

u a = radial tunnel closure, a = radius of tunnel opening, v = Poissoffs ratio of rock mass, Ed = modulus of deformation of rock mass, Po -= in-situ stress magnitude ( = rH, for hydrostatic stress field), r = unit weight of rock mass,

(1)

122 M. Verman et al.

Table 1. An overview of tunnels

Tunnel Depth of P u r p o s e Instruments used Method of instrumented excavation sections Tunnel closure Support pressure (m)

Tehri Tunnels 225-295 Hydropower Tape extensometer Load cell Drill and blast Lower Periyar 197 Hydropower Tape extensometer Load cell Drill and blast

Tunnel Maneri-Uttarkashi 225-467 Hydropower Tape extensometer Load cell Drill and blast

Tunnel Maneri Stage-II 100-200 Hydropower Tape extensometer Load cell Drill and blast

Tunnel Bagur-Navile 50 Irrigation Tape extensometer Load cell Drill and blast

Tunnel

H = height of overburden, and Pi = support pressure (short-term).

The value of v was taken as 0.25. Assuming a hydrostatic stress field (the hydrostatic solution is being used as an average even though all of the data may not have been obtained from a hydrostatic stress field), Po was considered equal to 7H and its values for different tunnel sections were accordingly obtained. The back-analysed values of the modulus of deformation are plotted against RMR (Rock Mass Rating), proposed by Bieniawski (1973) in Fig. 1, from which the following correlation may be obtained for modulus of deformation of dry rock masses, Ed:

E a = 10(RMR-Z0)/38GPa (2)

(correlation coefficient = 91%). Mehrotra (1992) also obtained nearly the same correlation from uniaxial jacking

tests on dry rock masses. Thus, one may use Eq. (2) with confidence in poor rock conditions also.

As shown in Table 1, all the tunnel sections considered for the present study were excavated using the drill-and-blast method of excavation. The damage tendency of blasting may be different for different rock mass qualities which could be reflected by the field data. Therefore, to correlate Ed (obtained from the field data; Section 1.1) with RMR, only the post-blasting RMR was used. Thus, if the field data (used to obtain Ed) reflects the damage tendency of blasting, so does the RMR. Moreover, the damage tendency of blasting is not likely to differ significantly in various rock mass qualities as the critical blasting parameters, such as blast pattern, charge, type of explosive etc., are also changed according to the rock mass quality.

1.2 Effect of Depth on Modulus of Deformation of Rock Mass

The back-analysed values of the modulus of deformation indicated its dependence on the height of overburden. To account for the effect of the height of overburden (or

Effect of Tunnel Depth on Modulus of Deformation 123

2o

E 8 N

11 3~

a - Tehr i Tunne ls b - L o w e r Per iyar Tunne l c - M a n e r i - U t t a r k a s h i Tunne l d - Mane r i S t a g e - I I Tunnel e - B a g u r - N a v i l e Tunnel

d

a

4~ RMR

e u

~1 ( R M R - 2 0 ) / 3 8 Ed = 10 , (Corr. C o e f f . = 9 1 % )

Fig. 1. Correlation between RMR and modulus of deformation of rock mass

depth of tunnel), a correction factor, f, was introduced in Eq. (2) in the following manner:

E d = f . 10(RMR-20)/38GPa. (3)

From Eq. (3),

Ed f = ~0(RMR_:0)/38 . (4)

The correction factor, f, is plotted against the height of overburden, H, in Fig. 2, from which the following correlation has been obtained:

f = 0.3 H a, (5)

where o~ = 0.16 to 0.3, and H > 50m. Equation (3) may, therefore be written as:

E d = 0.3 H a . 10 (RMR 2~ (6)

where H is in meters.

1.3 Effect of Stress-dependent Modulus of Deformation Equation (6) has been arrived at on the basis of the linear elastic solution given by Eq. (1). Since the attempt here is to derive a depth-dependent correlation for the modulus of deformation, it would have been more appropriate to use a stress-dependent solution in place of the linear elastic solution given by Eq. 1. This is because if the modulus of deformation is proportional to the depth, then it is proportional to the stress also. A stress-dependent solution will, however, affect the absolute value in Eqs. (5) and (6) but not the main argument. A stress-dependent solution would normally require

124

z

M. Verman et al.

r -

Pz 3

r , .

o

P~

(RMR-20 ) /38 0.3 f = Ed / lO Upper bound, f = 0.3H

i" i l l

I

J

I I

0.16 . Lower bound, f = 0.3H

r 400 500

, I t

H,rm ) Height of Overburden,

Fig. 2. Correction factor for effect of depth on modulus of deformation of rock mass

a numerical method. The calculations, which have been performed by Referee (1995), are summarised below:

Using an average value of unit weight of rock mass, ~ = 0.0255 MPa/m,

Po = 1.02 MPa at H = 40 m Po = 2.04 Mpa at H = 80 m Po = 5.1 MPa at H = 200 m Po = 10.2MPa at H = 400m Po = 12.75 MPa at H ----- 500m.

Referring to the subsequent Section 2, the following corresponding values of RMR and o~ have been considered: RMR = 31 and ~ = 0.3 Substituting these values in Eq. (6),

E d = 584 (39.2 po) ~ MPa (7)

For the above values of the in situ stress, Eq. (1) has been used to arrive at the linear elastic displacement (as a ratio of tunnel radius), le. This is compared with the stress- dependent displacement (as a ratio of tunnel radius), sd, for various tunnel depths in Table 2.

The term E d . . . . (rings) indicates that the solution is calculated iteratively over several rings of radii. In this case, it has been calculated over 38 rings of radii as follows:

1 - 1.000 2 - 1.100 3 - 1.200 4 - 1.300 5 - 1.400 6 - 1,500 7 - 1.600 8 - 1.700 9 - 1.800 10- 1.900 11- 2.000 12- 2.100

13 - 2.200 14- 2.300 15- 2.400 16- 2.500 17- 2.600 18- 2,700 19- 2.800 2 0 - 2.900 2 1 - 3.000 2 2 - 3.250 2 3 - 3.500 2 4 - 3,750

Effect of Tunnel Depth on Modulus of Deformation

Table 2. Results of linear elastic and stress-dependent solutions

125

H (m) 40 80 200 400 500 Po (MPa) 1 2 5 10 12.75

E~ . . . . 874-1770 1027-2170 1309-2850 1601-3528 1708-3771 (rings) (MPa)

sd 0.94 1.56 2.99 4.95 5.79 Ej . . . . 1770 2170 2850 3528 3771

(Elast) (MPa)

le 0.71 1.15 2.20 3.62 4.23 le/sd 0.75 0.74 0.74 0.73 0.73

2 5 - 4.000 2 6 - 4.250 2 7 - 4.500 2 8 - 4.750 2 9 - 5.000 3 0 - 6.000 31 - 7.000 3 2 - 8.000 3 3 - 10.000 3 4 - 12.000 35 - 15.000 3 6 - 20.000 37 - 25.000 38 - 30.000

The range of values of Ed . . . . (rings) given is for the value 1.05 and 27.5 (i.e., at the centre of the inner and outer rings). These depend on the radial stress, cr r, and are given by E d = 584 (39.2crr)~ MPa.

It may be seen from Table 2 that the displacement obtained using a linear elastic solution is nearly 0.75 times the displacement obtained using a stress-dependent solution for various values of the in-situ stress. This means that the introduction of a correction factor, f ' , in Eq. (1) in the following manner, will take care of the stress- dependency of the modulus of deformation:

Ua/a = (1 4- v)(po --Pi)/f' "Ed (8)

where

Ed = Modulus of deformation of rock mass at a given depth far away from the opening, f l = A/B, A = displacement obtained from a linear elastic solution, and B = displacement obtained from a stress-dependent solution.

Therefore, f~ = 0.75. Thus, f ' 9 E d represents the stress-dependent modulus of deformation.

Equation (6) may, therefore, be rewritten as,

or,

f " Ed = 0.3 H a . 10 (RMR-20)/38 GPa

E d = (0 .3 / f ' ) . H a - 10 (aMR-2~ GPa

Substituting, f ' = 0.75,

E d = 0.4 H a - 10 (RMR-;~ GPa. (9)

126 M. Verman eta|.

2. Discussion on Effect of Depth on Modulus of Deformation

The case-histories, considered to arrive at Eq. (6), pertain to poor to good rock mass quality (RMR ----- 31 to 68). It is quite likely that for rock masses having a higher RMR value than 68 (i.e., good to very good rock mass), the value of c~ is lower than 0.16, and for rock masses with a lesser RMR value than 31 (i.e., very poor to poor rock mass), the value of c~ is greater than 0.3. This argument originates from a growing evidence, mainly based on the laboratory experiments (Kulhawy, 1975; Santarelli and Brown, 1987; Brown et al., 1989; Duncan Fama and Brown, 1989), to suggest that

a) the elasticity modulus increases with the confining pressure and has a relationship similar to Eq. (5), and

b) this pressure dependency, reflected in the value of c~, of the modulus of elasticity is more pronounced in the weaker rock materials and is almost absent in strong, brittle rock materials. Kulhawy (1975), for instance, proposed the following expression for modulus of elasticity of rock material, E r, after examining a wide range of data:

Er ---- E0~ , (10)

where 0.3 is the minor principal stress and E 0 is the Young's modulus measured in a uniaxial compression test (g3 = 0). The value of a varies between 0 and 1.

A similar expression was obtained by Santarelli and Brown (1987) on the basis of triaxial compression tests on hollow cylinders of Carboniferous sandstone. This is given by:

E r = 15.08 0 "0'195 GPa (11)

where ~3 is in MPa. Other investigators have also obtained similar or slightly different expressions. The

increase with confining pressure of the modulus of elasticity is, therefore, a well documented phenomenon, largely based on the laboratory tests. It is interesting, therefore, to observe the occurrence of this phenomenon in the field as indicated by Eq. (9), which is based on the actual support pressure and tunnel closure measurements in tunnels.

The form of Eq. (9) is such that Ee ~ 0 as H---* 0, making the correlation inapplicable to situations where the height of overburden is less than say 50 m. Such situations, however, are irrelevant in the context of the underground openings.

3. Conclusions

Based on the results of the field instrumentation carried out in several Indian tunnels, an empirical correlation has been proposed for prediction of the deformation modulus of dry rock masses. The correlation indicates that:

1) The deformation modulus of the rock mass increases with increase in RMR and the tunnel depth.

2) This depth dependency of the deformation modulus is likely to be more pronounced in weaker rock masses and almost absent in strong, brittle rock masses, due to the effect of the confining pressure.

Effect of Tunnel Depth on Modulus of Deformation I27

References

Bieniawski, Z. T. (1973): Engineering classification of jointed rock masses. Trans. S. Afr. Inst. Civ. Eng. 15(12), 335-344.

Brown, E. T., Bray, J. W., Santarelli, F. J. (1989): Influence of stress-dependent elastic moduli on stress and strains around axisymmetric boreholes. Rock Mech. Rock Engng. 22, 189-203.

Duncan Fama, M. E., Brown, E. T. (1989): Influence of stress dependent elastic moduli on plane strain solution for boreholes. In: Proc., Int. Symposium on Rock at Great Depth, Vol. 2, Pau, France, 819-826.

Kulhawy, F. H. (1975): Stress deformation properties of rock and rock discontinuities. Engng. Geol. 9, 327-350.

Mehrotra, V. K. (1992): Estimation of engineering parameters of rock mass. Ph.D. Thesis Department of Civil Engineering, University of Roorkee, India.

Referee (1995): Personal communication, University of Roorkee, India.

Santarelli, F. J., Brown, E. T. (1987): Performance of deep wellbores in rock with a confining pressure-dependent elastic modulus. In: Proc., 6th Congress Int. Society for Rock Mechanics, Vol. 2, Montreal, 1217-1222.

Authors ' address: Dr. Manoj Verman, Advanced Technology and Engineering Services, AIMIL Ltd., Naimex House, A-8, Mohan Cooperative Industrial Estate, Mathura Road, New Delhi, 110 044 India.