& DAM CONSTRUCTION
Design of pervious pressure tunnels
By A. J. Schleiss
Reprinted from WATER POWER & DAM CONSTRUCTION May 1986
l1li
I article presents the summarized results of a study carried out by the author on the into account seepage forces and secondary permeability in lining and rock. The traditional tunnel statics" which assume lining and rock as impervious, is discussed. New pervious pressure tunnels are recommended.
UUUIILUUIII
f
the
that as long as the compared with the rock is in lining, accordmctv lining transmits only mechanical nounoarv is permeable mass. If the Fig. 2) the pressure the outside of the lining. In the and its reach depends on internal pressure in the rock. A single rock mass element Fig. 2) is loaded on all sides by a resulting force (the seepage nrpee11,'p dimension, like a body quantrtatrve seepage pressures on pressure tunnels Until now, these papers have given no inducement ing new design procedures for pervious pressure tunnels. of the rock Fractures and pores, which cause the mass, are deformed by the forces. rock mass around the pressure tunnel will be ' .... r» ..""·"",,'r1 internal pressure. This change in permeability, in turn, the seepage flow and therefore, the seepage forces. This mechanical-hydraulic coupling is shown schematically in Fig. 3. Due to the latest scientific findings, in rock mechanics and above all in rock hydraulics, such coupled can be solved theoretically", Inspite of this, coupling is rarely considered in pressure tunnel 11Y\''''''''~'''1f''n"
""""UH'P'U
",..,,,,,,I,-t,,~C}t-,,",,,,, continues to be used, because ofthe uncertainties in describing in situ behaviour. !\/lr,.. "".,",u,"'" there are no adequately realistic criteria for n""rrnp",lhlp pressure tunnels. This article presents a new design which takes into account the pressures and permeability influence of fracture deformation on rock masses.
Problematic nature of pressure tunnels U/-,',<4n,<");;,, water on an impervious nenaves as a load; uniformly distributed and perpendicular the surface (Fig. la). If the medium is permeable, water infiltrates cavities (pores, cracks, fissures, and and again exerts a surface pressure (Fig. 1b). In this case water pressure is gradually reduced by friction within the cavities. The force applied to the structure is transmitted along lines of seepage as a body force I. A permeable medium is therefore not loaded by the purely mechanical effect of water the boundary loading, but by hydraulic body forces. forces can be illustrated for a concrete-lined pressure tunnel (Fig. The traditional' 'pressure tunnel statics" theory is based on
U"?'U'r1'O
The porous, thick-walled cylinder Differential equation for the radial The state of stress due to internal water pressure in a porous cylinder, with homogeneous, isotropic material behaviour, is radially symmetrical. In contrast to the impervious cylinder, the elementary volume of the permeable cylinder is loaded not only by radial and tangential stresses but also by pore pressure (Fig. 4). The efficiency of this water pressure can be reduced by factor y/(O ~ Y/ ~ 1). In the past the factor Y/ was often equated with the superficial porosity of the material". In the meantime, experimental work has shown that Y/ is equal to unity, if rupture conditions are considered" 8. 9 The same is valid in highly I1 r1n'pr>t",An
1: Effect of
for the
pressure on an impervious and pervious rock hydraulics
R
/ 7](p
+
(per cent)
elementary volume combined with the plain strain relations derived from Hooke's law gives the following relation for the unknown radial displacement u(r): ~
+ du 1 + d2u
2
r
dr r
dr2
y/dp (1+v)(1-2v) dr E(1-v)
... (1)
The differential equation contains another unknown, namely the gradient of pore pressure in the cylinder, which, in the radially symmetrical case, is proportional to the hydraulic gradient. According to the permeability properties of the thickwalled cylinder, one can imagine different pore pressure distributions (Table 1). If the cylinder has a high porosity, theory would, potentially, lead to a logarithmic pore pressure distribution. If the permeability is caused by a great number of radial cracks, the pressure distribution would be linear, assuming laminar flow in the cracks. In a cylindrical zone of highly fractured rock mass, pore pressure distribution would be somewhere between these two assumptions. Deformations and stresses. Inserting one of the pressure gradients from Table I in Eq. 1, gives the well known inhomogeneous Euler differential equation. The general solution is found by superposition ofthe homogeneous with a particular solution. Eliminating the integration constants by the boundary (1-YJ)P a, deformaconditions Or(n) = (1-YJ)Pi and Or(ra) tions and stresses can be calculated. For the logarithmic and linear pore pressure distribution the corresponding relations are summarized in Table 11. If the cylinder is impervious, ie, YJ = 0, the relations in Table 11 are reduced to the well-known Lame formula. Fig. 5 shows the radial stress distribution in a thick-walled cylinder for logarithmic and linear pore pressure. If the pore pressure is fully effective (YJ = 1), the radial stresses lie entirely Table I Distribution of pore pressure, pressure gradient and water losses for thick-walled porous and cracked cylinders.
permeability properties of cylinder
radial cracked cylinder
pore pressure distribution per)
p
pressure gradientdp/dr
dp dr
water losses q
q
porous cylinder p
pa-pi ra-ri
(potential flow)
dp dr q
Pi In(ra/r)+Pa In(r/ri) 1n(ra/ri) pa-pi r'ln(r a/ fi) (Pi-Pa)2rc' K g'ln(r a/ri)
Qw'
(potential flow)
. - impervious cylinder (ry = 0) pervious cylinder (ry 0.9 and rl = 1.0) linear pressure gradient --Iogarithmical pressure gradient Fig. 5: Radial stresses in a thick-walled cylinder under internal and external pressure.
in the tension zone (see Fig. 5). On the other side, in an impervious cylinder (YJ = 0) only radial compressive stresses are possible. The linear pore pressure distribution causes greater radial stresses (tension) than the logarithmic distribution. With YJ< 1 the radial tensile stresses are somewhat reduced compared with YJ = I and displaced in the compressive zone towards the internal boundary of the cylinder. The maximum of the radial tensile stress is located throughout in the inner third of the cylinder, but not at the inner surface as in the impervious cylinder (compressive stress). In Fig. 6, it can be seen that the tangential tensile stresses in the permeable cylinder are always higher than in the impervious cylinder. Again the linear pore pressure distribution results in higher tangential tensile stresses. The same can be noted for the longitudinal stresses, whereby contrary to the impervious cylinder, considerable tensile stresses occur in the longitudinal direction. Influence of mechanical boundary loads. Mechanical boundary loads pF(ri) and pF(r a) in addition to the water pressure only change the boundary conditions. Therefore the resulting stresses and deformations can be calculated for the impervious cylinder. The effect of mechanical loads at the internal and external surface can be considered in Table 11 by adding the following expressions:
internal n .. o c:",,, .. ,, tunnel function of the ratio boundary stress is would develop greater the lining trnckness higher is the tensile tends to from the is the rock mass. similar to cooling of the varying the ratio in tensile stress between lining and rock if ~ . The boundary stress lining-rock is not influenced by the ratio if Kc/K r > l. The natural tensile strength of the oounoarv 11n"n ....._ ..r'f>v normally very small. High tensile transmitted to the if the lining is nrestressed
r /r.
... (4)
... (5) Computation of pressure tunnels Around a pressure tunnel there are zones with different mechanical and permeability properties (lining, grouted zone, cracked and fractured zones, etc). Ifthe zones are approximately radially and have a homogeneous isotropic material behaviour, they could be modelled with thick-walled cylinders. Furthermore, it is assumed that fractured roC;k masses behave
lining
(per
rock zone.
fair,
Fig. 8: Concrete-lined pressure tunnel. Mechanical boundarystress between lining and rock versus lining thickness for different ratio of elasticity modulus.
the at the inner (due strength of concrete. The the tensile only stresses and the pressure gradient seepage flow is The head at the outside ofthe cracked lining (pa) or rock mass is the continuity condition. Normally, the head in the cracked zones (ie, unreinforced linings) is neatunbte and therefore nearly the whole internal pressure becomes effective the boundary of the uncracked zones (Pi ::::; p, ::::; Pf). The total width of the cracks in the lining has to correspond with the tangential displacement of the rock mass. This relation allows the estimation of crack width (Eq, (2a)
= u(ra)2rr/n
. . . (7)
New cracks develop only in unfractured (bulky) rock mass. Otherwise the existing fractures are opened radially so much that they loose any contact. In such cracked or fractured zones the relation between width and tangential displacement is given by (2a)
u(rf)x2rr/n
+
linear pore pressure gradient
... (8)
Contrary to concrete linings (Eq. 7), the fractures in rock masses are enlarged only by a part of the tangential displacement. This reduction, due to the elastic compression strain, is considered in Eq. 8 with the factor x, which is approximately 1 Er/Eh. Normally the rock modulus Eh (measured from hand-piece) exceeds considerable the rock mass modulus Er and therefore x is near 1. The cracked or fractured zones in rock will expand as far as the natural compressive rock stresses surrounding the tunnel are greater than the stresses from the internal pressure. If the natural rock stresses cannot restrict the cracked zone, the bearing f"H'V""1h, of the rock masses is exceeded. number of cracks weak zones and in an unreinforced concrete transition floor-wall.
+------"-
It can be reduces the in linings and could therefore reduce cracks. Grouting is most effective in rock masses (see Fig. 11 right). economical depth of grouting is about 1 to 2 radius of the tunnel. The influence of grouting quality is more important than the grouting depth (see Fig. 11). Therefore it is better to lessen the distance between the boreholes than to lengthen the boreholes.
medium rock
~ .. £,,'tin~
(per cent)
r-r-:-
400 1--...
-,--
..- ----------+-
-,
--.....
1
Mechanical-hydraulic coupling
Influence of fracture deformation on secondary permeability. Due to internal. water pressure around the -nmnel, the width of the fracture will increase. In the following it is assumed that the natural rock stresses exceed the stresses due to internal pressure. Thus the fractures are able to transmit restricted tensile stresses, otherwise a "cracked" zone (see "Influence of cracks") would be formed. The coupling between stresses or deformations and rock mass permeability is based on the following statement: In a fixed circle around the tunnel the total sum of change in fracture width is equal to the tangential displacement due to internal water pressure. Assuming laminar, parallel flow in the fractures this relationship is: K(r)
Km +
--=---'-'---
3( 1
6Vw ttr 7i;
... (9)
1 )2
+ 7h
With the above equation the change in permeability in fractured rock masses is proportional to the cube of the radial displacement u(r) and inversely proportional to the square of fracture density (l/b, + l/b2)2. Eq. 9 is only valid for high fracture density (see earlier section).
Thick-walled cylinder with stress-dependant permeability. Assuming Darcy flow, the pressure gradient dp/dr in the cylinder with variable permeability K(r) is derived from the continuity condition:
1 dp dr
- (Pi
pal r,
f
. . . (10)
Naturally the coupled calculation is only sensible if the permeability properties of the rock masses are known adequately. Design criteria for pervious pressure tunnels For designing pervious pressure tunnels, three criteria are important: • Avoiding cracks in the lining; • Limiting water losses; • Ensuring the bearing capacity of the rock masses. Which one of these criteria governs the design in a particular case depends on the respective boundary conditions.
Avoiding cracks in lining. Cracks in the lining caused by internal water pressure can be prevented either by reducing forces in lining or by increasing resistance of the lining. Measures for reducing forces in lining are: increasing thickness of lining , rock grouting, drainage and controlled commencement of operation. These measures are successful only at relatively small internal pressure head (Pi < 20 bar) and good rock quality (Ei/E, < 3). At higher pressure heads cracks can be avoided only by increasing lining resistance, which is possible by prestressing. Different prestressing techniques have been developed: prestressing by rock grouting'? or gap
dr K(r).r
r. This pressure gradient, combined with Eq. 9 and inserted in Eq. 1, gives a differential equation which can be solved only with a numerical method, for example with the finite difference method. Fig. 12 shows the calculated distribution of radial displacements in a thick-walled cylinder; the permeability is dominated from two fracture sets. The mechanical-hydraulic coupling results in an increase of radial displacement compared with the cylinder with constant permeability. The difference is obvious above all towards the outside.
Pressure tunnel with stress-dependant permeability. As mentioned before, in the radial-symmetrical case, pressure tunnels can be modelled with several thick-walled cylinders. Detailed studies of the main parameters have shown that the effect of mechanical-hydraulic coupling is insignificant in calculating pure mechanical data such as stresses and deformations of pressure tunnels. However for the quantitative description of water losses and reach of seepage flow pattern it is essential to use a stress-dependant permeability (see Fig. 13).
E
E
radius r (m)
ra
Fig. 12: Distribution of radial displacement in a thick-walled, fractured cylinder. Comparison of deformation-dependent with constant permeability. (Assumed parameters: E/Pi 1000, U 0.2, Ko = 10- 6 m/s, "L1/bi 10,y/ x 1.0)
Conclusions The consideration mathematical rennement
Laboratory of
Ee/Er
3.0;
and active losses. of a pressure tunnel is dictated and depth under water table as well as long as the safety of the nr~'CC'lT'''' tunnel is not the allowable amount of water determined only by economic considerations. Besides limiting water losses should r!1c,,,,,t,,,,,,,'r<:>t',nn of some (anhydrite, rI'\'I'lrwl1t"" etc) or washing out of joint-fillings">'. In addition, water losses reaching the natural ground surface have occasionally in the past produced landslides-l-":". Water losses can be limited avoiding cracks in the concrete lining (see above) and to some extent by reinforcing as well as rock grouting. If these measures are not effective, special sealings such as plastic sheeting or thin steel tubes have to be used 27 ,13,2o . Bearing capacity ofrock masses. If the stresses in rock due to internal water pressure are greater than the natural stresses, the tunnel will probably fail because the expansion of the "cracked" rock zone cannot be limited. Considering a single fracture, for example in the tunnel roof, this hydro-fracturing is often compared with the effect of a hydraulic press": 13. Assuming that natural (primary) stresses in rock masses are influenced mainly by overburden, the criteria of bearing capacity is normally reduced to a criteria of minimal overburden28,29,3o . But such rule-of-thumb criteria are bound up with the local geological situation and therefore are not generally applicable. Of course the natural stresses in rock masses can be influenced locating the tunnel deep enough capacity of 'rI'\1'\""rUl'Ar.c tunnels with of influence around the tunnel
1973. 6. BROWN, E. T., AND BRAY ,J. W., "Rock-support interaction calculations for pressure shafts and tunnels. " ISRM Symposium Aachen , Vol. 2; 1982. ZIENKIEWICZ, O. "Stress of hydraulic structures including pore 1963. pressure effects." "Some interpretation of pore fluid 8. ROBINSON, L. H. , Rock Mechanics, Berkeley, effects in rock failure." 11th California, USA; 1969. 9. SERAFIM, J. L., "Influence of joint water in the stability of structures in rock drainage measure." Proceeding ISRM-Symposium: Percolation through fissured rock, Stuttgart T4 (General Report) 1972. 10. WITTKE W., Felsmechanik, "Grundlagen fur wirtschaftliches Bauen im Fels.' Springer-Verlag, Berlin, Heidelberg, New York, Tokyo; 1984. 11. G. , "Wirkung des Kluftwasserdruckes auf einen Felskorper. Felsbau 2 No. 1984. 12. SCHLEISS, A., "Bemessung von Druckstollen. Teil I: Literatur, Grundlagen, Felshydraulik insbesondere Sickerstromungen durch Auskleidung und Fels." Mitteilung der Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie an der ETH Zurich, Switzerland, No. 78; 1985. 13. SEEBER, G., "Power conduits for high-head plants." Water Power & Dam Construction, June 1985; July 1985. 14. BouvARD, M., "Les fuites des galeries en charge en terrain sec. Role du revetement, des injections, du terrain." In Houille Blanche, No. 4; 1975. 15. BOUVARD, M., AND NIQUET, J., "Ecoulernent transitoires dans les massifs autour d'une galerie en charge." In Houille Blanche, No. 3; 1980. 16. PONIMATKIN, P. U. "Calculation of pressure tunnels with a concrete lining and grouting considering the formation of cracks in the lining and rock. " Hvdrotechnical Construction, No. 3; March 1977. IVANOVIC, K., 0., AND 17. KUNJUNDZIC, storage hydroelectric 18. KIESER, A., "Druckstollenbau. 19. H., 'Vorspanninjektion No.
Scandinavia. " in Druckstollen.' und tngenieurueotogte . Supplement IV; 1968. Auskleidung von Druckstollen und Druckschachten." Verlag Berlin W. Germany; 1926. 26. J., "Wassersprengung und Sprengwasser." Geologie Bauwesen, Vol. 22, No. 2; 1956. WENZEL, K., "Neue Moglichkeiten zur wasserdichten Stollenauskleidung." Nos. 26,27,28; 1963. 28. BROCH, E., "The development of unlined pressure shafts and tunnels in Norway." ISRM Symposium Aachen, Vol. 2; 1982. BROCH, E., "Unlined high tunnels in areas phy." Water Power & Dam November; ing 30. SHARP, J. C., AND GONANO, C. P., concrete lined pressure tunnels of Drakensberg. " IS1W-Symposium Vol. 2; 1982. 31. JAEGER, C., "Present trends in the design of pressure tunnels and shafts for underground hydroelectric power stations. " Paper No. 5978, Proceedings Institution Civil Engineers, Part I, Vol. 4, March; 1955. 32. KASTNER, H., "Statik des Tunnel- und Stollenbaues." Springer-Verlag Berlin/Gottingen/Heidelberg, W. Germany 1962.
