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Pore pressure effects on interface behavior
Meclaanlcs of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All fights reserved.
449
Pore Pressure Effects on Interface Behavior R.O. Davis Department of Civil Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
1. THE EFFECTIVE STRESS PRINCIPLE The fact that pore fluid pressures are important in understanding and predicting the behavior of many geologic materials has been known at least since the late nineteenth century. A paper by Clarke 1 in 1904 presented correlations of monthly rainfall with displacement of a large slow-moving landslide at Portland, Oregon. Clarke's work showed clearly that it was the pressure in the pore water, rather than its weight or any lubricating effect, that caused the motion of the landslide to accelerate or to slow down. Although Clarke did not have the analytical tools to fully understand the Portland landslide, his paper was a remarkably well documented case history of pore pressure effects. There are a great many similar case histories in the engineering and geologic literature in which the reduction of frictional strength on a potential surface of sliding due to pore fluid pressures has resulted in failure. Sometimes the failure is manifest in only small displacements accompanied by minor damage. Other times catastrophic results have ensued. The Malpasset dam which failed in 1959 in southern France took the lives of 300 people. The dam failed when a massive slip occurred on a pre-existing fault surface which lay beneath the dam foundation. This failure occurred despite the fact the dam had been in operation for more than four years. The failure was directly linked to pore pressures developing on the fault due to abnormally heavy rainfall during the three days preceding failure. The link between pore pressure and frictional strength of geologic materials lies in understanding the principle of effective stress discovered by Karl Terzaghi 2. Terzaghi realized that whenever a pore fluid is present, the stress it supports will contribute to the overall stress state in the material. On any geologic interface there will be areas of rock or soil particles in direct contact and other areas where no contact occurs but where pore fluid separates material on either side of the interface. Let us consider some area A of the interface. At the points of contact between soil particles or rock asperities, there will exist some average contact stress crc acting normal to the interface. If we let A c denote the area of contact, then the normal force supported by the soil particles or the rock will be ocAc. The remaining area, A-Ac, will support the hydrostatic pore fluid stress which we will denote by u. Thus the normal force carried by the pore fluid is u(A-Ac). The combination of these two normal forces must equilibrate the applied normal stress o acting over the whole area A. We have o A = or162 + u ( A -
450 or
a =a/+u
1---~
(1)
where o / is the effective stress given by o/_
~ A
(2)
We see from this expression that the effective stress represents the ratio of normal force supported by the solid particle contacts or rock asperities per unit area of the interface. In many practical situations, the ratio AriA in equation (1) will be small in comparison with 1. Because of this, (1) is commonly approximated by o
=
o ~ +u
(3)
This is the form of the effective stress principle put forward by Terzaghi in 1925. The total stress o is partitioned into the effective stress o ~ and the pore pressure u. The more exact form of the effective stress principle given in (1) was first noted by Skempton 3 in 1960. It should be noted that eq. (1) or the approximate eq. (3) are appropriate for dealing with geologic interfaces where frictional strength and slip are matters of primary interest. The frictional strength will depend upon o / and will be independent of u, a fact experimentally verified many times. Other forms for the effective stress principle may be required for other types of problems 4 but these will not concern us here. We can generalize the effective stress principle for three-dimensional stress states. Equation (3) becomes '
Oij "-- Oij + U6ij
(4)
where oii is the Cauchy total stress tensor, o/..tj is the effective stress tensor and 6~i is the Kronecker delta symbol. We see that eq. (4) decomposes the stress state into a hydrostatic part with magnitude u and an effective stress part supported by the rock or soil skeleton. It should also be noted that even though we employ the notation of continuum mechanics, we must still consider finite sized areas and volumes. The usual limit procedures cannot be applied to granular materials or rock interfaces. Thus we define stress as the ratio of force divided by a finite area large enough to encompass a representative number of grains or a representative region of rock surface. The interfaces we wish to consider here are surfaces on which slip may occur in a geologic mass. With regard to soils the interface may be a well defined slip surface or it may be a shear zone where shearing deformation is concentrated in a narrow band. In rock we are concerned with joints or faults and these also may be well defined surfaces or may be narrow gouge filled shear bands. In either case the resistance to slip or shear deformation will be characterized by the Coulomb friction law which states the maximum shear stress ~: which may be supported is a linear function of the effective stress o / . r = c +/~ o /
(5)
Here c is the cohesion term, independent of normal stress, and p is a coefficient of friction. According to (5), the actual shear stress on the interface must always be less than or equal to ~:. If the actual stress is equal to r, then slip will occur.
451 Equation (5) is an idealization of the complex processes involved in slipping on a geologic shear surface. According to (5) no deformation occurs until the actual shear stress equals ~:, and deformation will cease immediately if the stress drops below ~:. In fact the slip mechanism is much more complicated than this and deformation may accompany lesser stresses. Nevertheless, (5) is a useful representation of strength which may be fruitfully applied in most practical problems. For high normal stress (e.g. crY> 200 MPa) the coefficients in eq. (5) become relatively independent of the material being tested. In 1978 Byerlee 5 compiled test data for a wide range of rock types and found generally good agreement that p = 0.6. For normal stresses lower than 200 MPa, the data becomes more scattered. For modest stress levels in soils (on the order of 100's of kPa) the value of p may take on values as small as 0.1 in clays to values in excess of 1.0 in sands and gravels. The cohesion c may also change as a function of confining stress. At low stresses on a well developed shear surface, c will generally be zero. At high stress levels c will normally have some small positive value. The strength parameter/J may also depend upon the rate of shearing. Rate effects have been observed in both soils 6'7 and rocks 8'9. Increasing the rate of shearing may lead to weakening or strengthening depending upon the material involved. The importance of pore fluid stress is clear in eq. (5). Increasing or decreasing u, while holding the total stress o constant, may have a dramatic effect on the stability of a fault or landslide. Changes in u may occur either because of outside influences such as infiltration in a landslide during heavy rainfall, or because of the mechanical behavior of the material itself. Practically all geologic materials exhibit some volume change when subjected to shear. Both compaction (volume decrease) and dilatation (volume increase) have been observed in different materials. If the void space in the material is filled with pore fluid, any tendency toward volume change will be accompanied by changes in pore fluid pressure. These pore pressures are required to drive the flow of pore fluid which must accompany the volume change. Once sufficient flow has occurred the pore pressures may return to their normal values, although situations arise in which flow and pore pressure dissipation encourage further shearing deformation with its accompanying compaction or dilatation. We will investigate these effects in more detail below. In order to quantify changes in pore pressure which result from either compaction or dilatation, Skempton 1~ introduced two pore pressure coefficients called A and B. These relate the change of pore pressure to changes of deviatoric and mean stress in undrained deformation. Let p represent total mean stress and let q be the deviatoric stress 1 p = 3O'kk
,
1 [3 Crkk)2]1 q = -~ OijOij - (
(6)
Then A and B are given by A=__
du
aq
, B=--
Ou
ap
(7)
The coefficient B depends primarily on whether the void space is fully saturated or not. For full saturation with de-aired water, B will theoretically be exactly equal to 1.0, and careful experiments confirm this. If undissolved air is trapped in the pores, B will be less than 1.0. The range of possible values for A is quite broad. Negative values correspond to dilating materials and positive values to compacting materials. If A equals zero, then
452 the material maintains constant volume during shearing. The value of A will not in general remain constant during any loading procedure. Depending on loading details, some geologic materials may first compact, then dilate, and finally deform at constant volume in a typical test. Nevertheless A can be a useful measure of the overall pore pressure behavior expected for a particular material. In the remainder of this chapter we will consider a number of situations where pore pressures may have a significant effect on the behavior of a geomaterial interface. In the next section, the question of induced seismicity due to pore pressure is examined. This is followed by a section describing how pre- and post-earthquake pore pressures may affect the earthquake process. Section 4 briefly discusses frictional heating of pore fluids with implication for landslide stability. Finally, in section 5, recent experimental evidence of dynamic fluctuation of pore pressures in shearing materials is described.
2. PORE PRESSURE INDUCED SEISMICITY The Denver earthquakes of 1966 first suggested that pore pressures could induce seismic activity 11. Injection of fluid waste in a disposal well was linked to earthquake occurrence, and seismic activity ceased when injection was discontinued. This phenomena was confirmed in an elaborate and carefully documented experiment carried out at Rangely, Colorado between 1971 and 197312. The Rangely experiment utilized existing wells at the Rangely oilfield to increase pore pressures in the region of an active fault. Since the overall fault dimensions were small, there appeared to be no possibility of producing a damaging earthquake, and indeed the largest earthquake induced by the experiment had magnitude 3.1. Using hydraulic fracturing techniques the in-situ stress field near the fault was measured, and it was found that the shear and normal stresses acting on the fault were approximately 7 and 34 MPa respectively. Taking c - 0 and/~-0.81 in eq. (5), the value of effective stress required to produce failure with 1:-7 MPa was found to be about 8.6 MPa. This suggested that the critical value of u required to induce earthquakes was ur
- ~'=34
- 8.6=25.4 MPa
The experiment consisted of carefully raising the pore pressure by water injection to a value greater than ur and then lowering it below ur while monitoring seismic activity on the fault. As expected a strong correlation between number of earthquakes and pore pressure near the fault was discovered, and the value of critical pore pressure was experimentally confirmed. It is noteworthy that the value of ur was predicted before the complete cycle of increasing and decreasing pore pressure was carried out. The Rangely experiment provided conclusive evidence for the applicability of the effective stress principle to frictional stability of faults. Many laboratory experiments had shown the principle to be valid for small scale samples, but the Rangety data extended the experimental dimensions by four to five orders of magnitude and removed any lingering doubt that pore pressures and effective stress were not controlling factors. Increased seismicity associated with injection from wells is not the only form of induced earthquake activity. The most common occurrences of induced seismicity are associated with reservoir construction. There have been numerous instances of increased numbers of earthquakes resulting from reservoir filling 13. These may simply result from increasing
453 levels of deviatoric stress in the rock near the reservoir due to the additional weight of water, or may be due to pore pressure effects combined with increased stress levels. Two pore pressure effects are present in reservoir filling. The most obvious effect is that water from the reservoir may infiltrate into the ground below and directly affect pore pressures in the adjacent rocks. A second, more immediate effect is the possibility of enhanced pore pressures due to the weight of the reservoir. We can gain an intuitive understanding of both effects by considering the idealized situation depicted in Figure 1. In the figure the reservoir is idealized as having constant depth h and extending indefinitely in all horizontal directions. Uniaxial strain conditions exist and the increase in vertical and horizontal components of total stress due to reservoir filling will be
Figure 1. Reservoir induced seismicity
'
&~
v ]v.h
A~
(8)
l-v
Here v~ represents the value of Poisson's ratio which is applicable for undrained loading of the rock. The pore pressure increase Au will be given by eqs. (6) and (7). Using (8) we have 1
Ap = 3 ( A O , + 2 k , OH)
_
_
,
1[ 1 + v ]?w h
~q = ~o
=
- Ao H
1-2v.)
y~h
(9)
)] 1~,~h -v,
(10)
I -v
Thus the pore pressure increase is Au=BAp
+AAq=
-1B 3 (l+v)
+A(1-2v
If the degree of saturation of the rock is near total, then B will be close to 1.0. In most instances the value of A will be negative but its magnitude will be small. If we assume A(1-2v~) is near zero and set B equal to 1.0, then
454
,u
=
_
3 i-v
?w h
(11)
If v~ were 0.4 for example, we see that Au = 0.78Ywh. Thus reservoir filling will immediately increase the pore pressure in the surrounding rock. How this increase may affect the stability of a fault depends primarily on the fault surface orientation. If the fault surface is horizontal, the effective stress increases AOZv= A o , -
Au = -
2[ 1 - 2 v
] u yw h 1-v u
3
If the fault surface is vertical, the effective stress it supports decreases by this amount.
AOH:AOH AU:
-
3 ....1
-
v u
/
Y
wh
Thus vertical fault surfaces become less stable and horizontal surfaces more stable. Surfaces with intermediate orientation suffer this change Ao~ = [ 1 - 2 v ) ( 1 ) 1-vu c~
Ywh
where 1~ denotes the angle between the fault surface and the horizontal. We see that cos"11/v3 - 55 degrees marks the maximum value of 0 for which the effective stress will be increased. These effects occur immediately upon loading. As time passes two things may occur. First, the additional or excess pore pressure beneath the reservoir may dissipate as flow occurs into the surrounding rock. Second, infiltration from the reservoir may link the existing ground water with the reservoir water. The result of these two processes will be a return of the effective stress conditions to the situation which existed before reservoir filling. Both pore pressure and total stress will be increased, but effective stress will be unchanged, implying unaltered stability of existing faults. This too is an idealized situation. Infiltration may often not connect the reservoir to existing ground water due to the presence of impermeable materials. The dissipation of excess pore pressures will occur regardless, in which case the long term equilibrium pore pressure may be unchanged from pre-reservoir conditions. In that instance the reservoir induced changes in effective stress will be equal to the changes in total stress and stability may be enhanced. In general, the most likely time of occurrence for reservoir induced earthquakes is shortly after reservoir filling. The stress increase caused by the added weight of water will, in fully saturated rock, be initially largely carried by the pore fluid. As flow occurs away from the loaded region the stress increase will gradually be transferred from the pore fluid to the rock. This process is familiar in soil mechanics where it is called consolidation, and it was the discovery of the effective stress principle which led Terzaghi to an understanding of the consolidation problem. The same understanding is necessary for the reservoir induced seismicity problem.
455
3. PORE PRESSURE EFFECTS ASSOCIATED WITH EARTHQUAKES Changes in pore pressures accompany earthquakes and may affect the aftershocks which follow. Pore pressure changes may also precede an earthquake and, in the 1970's, much interest was generated by the possibility that precursor changes in pore pressure could be used as a prediction tool. The dilatancy-diffusion theory was put forward as an explanation of changes in the ratio of dilatational and shear wave velocity observed before several earthquakes. The basic idea which underlies the dilatancy-diffusion theory is simple 14as. Shearing deformation in the rock adjacent to a fault will generally be accompanied by dilatation which in turn will cause a decrease in pore pressure. The mechanism is embodied in eq. (7)1 whenever the pore pressure coefficientA is negative. Lowered pore pressures near the fault will result in increased effective stress and greater strength. The phenomena is called dilatancy hardening. Strength increases will only be temporary however, since flow will occur into the region due to the negative pore pressure gradient. Diffusion of pore fluid into the region increases the pore pressure, decreasing the effective stress, and decreasing the strength. In theory then, the increased shear stress which might have caused an earthquake, instead causes the fault to become temporarily stronger. The increased strength cannot persist, and after some time the pore pressure drops and rupture occurs. The amount of time needed for all this to occur depends on the level of dilatation, the volume of rock involved and the permeability in the region of the fault. One particular aspect of the dilatancy-diffusion theory made it especially attractive from the standpoint of earthquake prediction, namely that the whole process could, in theory, be monitored. It was speculated that dilatancy would manifest itself by changing the velocity of seismic waves which propagate through the affected rock. Early observations suggested that the ratio of speeds of dilatational and shear waves, Vp/Vs, decreased by as much as ten percent in response to dilatation. As dilatation developed, the Vp/Vs ratio decreased. Then as diffusion brought additional pore fluid into the focal region and pore pressures increased, the ratio Vp/V~also increased, supposedly returning to its normal value of about 1.75, roughly at the time the earthquake would occur. The optimistic view taken in the 1970's held that observation of the vp/v~ ratio was one key to earthquake prediction. Unfortunately, subsequent events do not totally conform with the theory 16. Careful measurements of seismic wave velocities were carried out annually in Japan using explosive generated waves in the Izu region. Three large earthquakes occurred in the region in 1974, 1978 and 1980 during the time the velocity measurements were being made. No significant changes in vp/v~ were observed, and the dilatancy-diffusion theory has since lost much of its credibility in regard to earthquake prediction 17. Failure of the dilatancy-diffusion theory as an earthquake prediction tool does not imply the phenomena of dilatant hardening accompanied by pore fluid flow may never occur. There is considerable geological evidence suggesting just the opposite. Sibson TM has summarized a number of cases of hydrothermal vein systems associated with faults which could have arisen from fracture dilatancy and pore fluid flow. Post rupture flow and redistribution of pore fluid stress has also been advanced to explain both the temporal and spatial distribution of aftershocks associated with large earthquakes. A simplified model connecting pore fluid motion with aftershock occurrence was put forward by Nur and Booker 19. Their model considered pore pressure variations
456 set up by a two-dimensional edge dislocation in an infinite elastic space. Regions of compression and dilation lie on either side of the dislocation and corresponding pore pressure changes are expected to be found near the ends of a fault immediately following rupture. Flow then occurs from the region of higher pressure to the region of lower pressure. Nur and Booker hypothesized that occurrence of aftershocks was directly related to the rate of increase of pore pressure in the dilatant quadrant. Their simple model explained clustering of aftershocks near fault ends as well as the absence of aftershocks following deep earthquakes or small earthquakes. A more sophisticated model for this process has since been advanced by Li, et al.20.
4. FRICTIONAL HEATING EFFECTS Whenever slip occurs on a geological interface, energy will be dissipated. If we let r denote shear stress and v slip velocity, then at any instant the rate of dissipation per unit area of interface is the product rv. Dissipated energy takes the form of heat which may be transported away from the interface by a combination of convection and conduction. Heat conduction occurs through rock or the solid particle matrix in soil and also through pore fluids. Convection is manifest in flow of pore fluids away from the interface. If steady state slip is occurring, a state of thermal equilibrium will exist. The frictional heat generated will be balanced by conduction alone as convection can occur only as a transient process. No excess pore pressure will accompany steady state slip. In contrast to the steady state, transient conditions give rise to a rich variety of possible responses depending upon the details of loading and the materials involved. Any increase in either r or v will result in an increase in temperature at the interface surface. In general, for a fully saturated geomaterial, increasing temperature will result in increasing pore pressure since the coefficient of thermal expansion for most common minerals is about an order of magnitude smaller than that for water 21. Increasing pore pressure of course implies decreasing effective stress and decreasing strength. This slip-weakening effect may have important ramifications for overall stability of a fault or landslide. The simplest situation of interest occurs when we assume the interface is of infinite extent with uniform conditions everywhere. In this case heat transport occurs perpendicular to the interface and only one spatial dimension is required. Letting x be the spatial coordinate normal to the interface, the field equations appropriate to the problem are 22
mfi- anO=-aV dx V=_
(12)
k du "tw Ox
6 = 6 . 020 aX 2
(13/ PwCw O(OV) pC
O~X
where u = u(x,t) = pore pressure 0 = 0(x,t) = pore fluid temperature V = V(x,t) -- pore fluid velocity
(14)
457 n -- porosity rn = compressibility of solid matrix = (1 + v)(1 - 2v) E(1 -
a k 6 p law c %
= = = = = = =
coefficient of thermal expansion of pore fluid coefficient of permeability of solid matrix thermal diffusivity of solid-pore fluid mixture mass density of solid-pore fluid mixture mass density of pore fluid specific heat capacity of solid-pore fluid mixture specific heat capacity of pore fluid.
Here eq. (12) represents conservation of pore fluid mass, eq. (13) is Darcy's law, and eq. (14) represents conservation of pore fluid energy. These three equations can be solved, given appropriate boundary conditions and a constitutive equation for the shear stress 1:. The boundary conditions for this problem depend on whether we are concerned with earthquakes or landslides. In the latter case, there will be an isothermal boundary (the ground surface) not too distant from the interface. In the former case we may assume the interface is contained within an infinite medium and require the solution to remain bounded as x -, oo. In either case, the boundary conditions at the interface are atx=0"
V=0
and
- K a0 ax
=
t:v
(15)
Here K denotes the coefficient of thermal conductivity of the solid-pore fluid mixture. The no-flow boundary condition (15)1 implies a symmetric pore pressure gradient on either side of the interface. Finally, we require a constitutive equation relating shear and normal effective stress on the slipping interface. The Coulomb equation (5) is appropriate, and would normally be applied with c equal to zero. The coefficient of friction/~ may be taken to be a function of the slip velocity v as discussed in Section 16'7'8'9. Consideration of frictional heating of pore fluids in regard to earthquakes were first treated by Lachenbruch 23. Without solving the field equations, he considered critical combinations of the material parameters in special cases. He concluded that if the permeability was sufficiently large, thermal effects would not be important; but, in sufficiently impermeable rock, high pore pressures could be thermally generated and these could affect the dynamics of faulting. Lachenbruch's work was pursued further by Mase and Smith 24. They solved the governing equations for the special case of a fault slipping with constant velocity. They identified a range of material parameters within which frictional heating affected the strength of the fault. Again permeability was the most interesting parameter. In cases where permeability was sufficiently high, any excess pore pressures which might be generated by frictional heating were quickly dissipated by flow away from the fault. For low permeability rock however, high pore pressures could be generated by their model. Effects of frictional heating in landslide behavior have been considered by several investigators. Habib 25 carried out a simplified analysis, omitting the effects of fluid flow, and concerned primarily with the possibility of vaporization of pore fluid. Gogue126 was also concerned with the possibility of vaporization, particularly in regard to very large rockslides which appear to exhibit behavior compatible with near zero friction coefficients.
458 The Vaiont rockslide 27 is a case which has generated particular interest in the rock mechanics community. Voight and Faust 2a'29 carried out numerical calculations for the Vaiont slide which incorporated conduction and convection of heat. Their findings suggested that vaporization of pore fluid was unlikely, but pore pressures could nevertheless be significantly enhanced by frictional heating, leading to loss of frictional strength and high slide velocities. Similar conclusions were given by Anderson 3~ Frictional heating of pore fluids has also been implicated as a possible mechanism for the gradual acceleration and loss of stability observed in many creeping landslides 22. In materials of low permeability, small changes in creep velocity may result in slightly increased pore pressures, and this effect may compound in time leading to complete loss of stability. The effect may have been observed in the East Abbotsford landslide 31. Frictional heating can also result in stick-slip behavior for a simple elastic slider such as illustrated in Figure 2. The slider is pulled by an elastic spring which is connected to a load point moving with constant velocity vo. Frictional dissipation and pore fluid heating occur at the slider-base interface affecting frictional resistance. For certain combinations of material parameters, the slider exhibits stationary periods interspersed with rapid jumps forward 32. The full role of pore fluid heating in interface behavior is not yet clear. While theoretical calculations suggest its importance in certain situations, these are specialized cases dependent on full saturation and (especially) low permeability. The exceptional mobility of some large rock slides remains both the impetus and the only experimental evidence for the theory. Figure 2. Elastic slider 5. DYNAMIC PORE PRESSURE FLUCTUATIONS An alternative explanation for landslide mobility has been advanced by Iverson and LaHusen 33. They hypothesize that while the overall motion of a landslide may appear steady, internally there exist isolated regions of high or low pore pressure induced by local compaction or dilatation of the solid matrix. At any point on the sliding surface, local compaction and dilatation would be expected to occur in some roughly periodic fashion. The rate at which changes occur would presumably depend upon the velocity of sliding v and some characteristic dimension A. The dimension A might be the mean particle size or might be larger if groups of particles are moving together as a relatively rigid mass. The characteristic time (period) associated with the dilatation-compaction fluctmtions would be A/v. Pore pressures will develop locally in response to the dilatation-compaction fluctuations. Any excess pore pressure, whether negative or positive, will result in flow, either toward or away from the region affected. Flow results in dissipation of pore pressure depending upon the permeability and compressibility of the solid matrix and the distance to regions of lower pressure. The characteristic time for pore pressure dissipation may be taken as
459 A2/c where A is the characteristic particle (or particle group) dimension, and c is the coefficient of consolidation, defined by k c--
Ywm
where k and m are the permeability and compressibility of the solid matrix [defined previously following eq. (4)] and y, is the unit weight of the pore fluid. The coefficient of consolidation c is a familiar parameter in soil mechanics. We now have two characteristic times: A/v, the period of fluctuation of compression and dilatation, and A2/c, the time for dissipation of excess pore pressure. The ratio of these defines a dimensionless number R34 A/v c t~2/c vA whose magnitude characterizes the tendency of a landslide to develop sustained excess pore pressures. For large values of R the dissipation time is short in comparison with the period of pore pressure generation, and sustained pore pressures are unlikely. Conversely, for small R, the pore pressure dissipates slowly and sustained pressures are likely. Iverson and LaHusen 33 carried out pore pressure measurements in controlled shearing of a carefully constructed array of fibreglass rods. For values of R between 10 and 50 they found pore pressure fluctuations on the slip surface characterized by plateaus of high pressure (slightly higher than the initial static pressure) separated by deep troughs of low pressure. During the high pressure plateaus the pore pressure was sufficient to reduce the effective stress to zero. The low pressure troughs were of shorter duration but during them the pore pressure was sufficiently reduced so that the mean pressure over the plateautrough cycle was equal to the initial hydrostatic pressure. They also performed measurements in a large scale simulated landslide which showed quite large pore pressure fluctuations for an R value of approximately 0.3. If significant local pore pressure fluctuations do accompany sliding on geomaterial interfaces, many regions of high and low excess pore pressure might exist simultaneously on the sliding surface. The regions of high pressure would be weakened while the low pressure regions would be hardened. The edges of the high pressure region might be expected to exhibit stress concentrations tending toward dilatation, while flow from high to low pressure regions would decrease strength in areas of dilatency hardening. Migration of pore pressure from one region to another might presumably result in an overall pattern of motion which is apparently uniform. 6. SUMMARY Pore pressures tend to complicate the picture of geomaterial interface strength. They may be generated by the material itself through compaction or dilatation or possibly frictional heating, and they will dissipate only as rapidly as the material will allow. They introduce the dimension of time to strength considerations which might otherwise exhibit no time dependence, and their effects on stability of faults or landslides may be catastrophic. An understanding of the effective stress principle together with the phenomena of compaction and dilatation and the laws governing flow through porous media is required for any consideration of strength of geomaterial interfaces where pore fluids are involved.
460 REFERENCES
1. Clarke, D.D., (1904) A phenomenal landslide, Tram. ASCE, vol. 53 (paper 984) pp. 332-397. 2. Terzaghi, K., (1925) Erdbaumechanik, Franz Deuticke, Wien, 399 p. 3. Skempton, A.W. (1960) Correspondence, Geotechnique, vol. 10, No. 4, p. 186. 4. Robin, P-Y.F. (1973) Note on effective pressure, Jour. Geophys. Res. vol. 78, No. 14, pp. 2434-2437. 5. Byerlee, J.D., (1978) Friction of rocks, Pure AppL Geophys., vol. 116, pp. 615-626. 6. Hvorslev, M.J. (1960) Physical components of the shear strength of saturated clays, Proc. Res. Conf. on Shear Strength of Cohesive Soils, ASCE, Boulder, pp. 169-273. 7. Salt, G. (1985) Aspects of landslide mobility, Proc. llth Int. Conf. Soil Mechanics and Foundation Engineetqng, Part 3/A/4, pp. 1167-1172. 8. Dieterich, J.H. (1979) Modelling rock friction: 1 Experimental results and constitutive relations, Jour. Geophys. Res., vol. 84, pp. 2161-2168. 9. Tullis, T.E. (1988). Rock friction constitutive behavior from laboratory experiments and its implication for earthquake prediction field monitoring program, Pure AppL Geophys., vol. 126, pp. 555-588. 10. Skempton, A.W. (1954) The pore pressure coefficients A and B, Geotechnique, vol. 4, pp. 143-147. 11. Healy, J.A., Rubey, W.W., Griggs, D.T., and Raleigh, C.B. (1968) The Denver earthquakes, Science, vol. 161, pp. 1301-1310. 12. Raleigh, C.B., Healy, J.H., and Bedehoeft, J.D. (1976) An experiment in earthquake control at Rangely, Colorado, Science, vol. 191, pp. 1230-1237. 13. O'Reilly, W. and Rastogi, B.K. (eds) (1986) Induced seismicity, Phys. Earth Planetary Interiors, vol. 44, pp. 73-199. 14. Nur, A. (1972) Dilatency, pore fluids, and premonitory variations in tJtp travel times, Bull. SeismoL Soc. Am., vol. 62, pp. 1217-1222. 15. Whitcomb, J.H., Garmony, J.D., and Anderson, D.L., (1973) Earthquake prediction: Variation of seismic velocities before the San Fernando earthquake, Science, vol. 180, pp. 632-641. 16. Mogi, K. (1985) Earthquake Prediction, Academic Press, Tokyo, 355 p. 17. Scholz, C.H. (1990) The Mechanics of Earthquakes and Faulting, Cambridge University Press, Cambridge, 439 p. 18. Sibson, R.H. (1981) Fluid flow accompanying faulting: Field evidence and models, in Earthquake Prediction, an International Review, ed. D. Simpson and P. Richards, Am. Geophys. Union, Washington, pp. 593-603. 19. Nur, A. and Booker, J.R. (1972) Aftershocks caused by pore fluid flow? Science, vol. 175, pp. 885-887. 20. Li, V.C., Seale, S.H., and Cao, T. (1987) Postseismic stress and pore pressure readjustment and aftershock distributions, Tectonophysics, vol. 144, pp. 37-54. 21. Campanella, R.G., and Mitchell, J.K. (1968) Influence of temperature variations on soil behavior, Jour. Soil Mech. Found. D&. ASCE, vol. 94, pp. 709-734. 22. Davis, R.O., Smith, N.R., and Salt, G. (1990) Pore fluid frictional heating and stability of creeping landslides, Int. Jour. Num. AnaL Methods Geomech., vol. 14, pp. 427-443.
461 23. Lachenbruch, A.H. (1980) Frictional heating, fluid pressure, and the resistance to fault motion, Jour. Geophys. Res., vol. 85(B11), pp. 6097-6112. 24. Mase, C.W., and Smith, L. (1985), Pore-fluid pressures and frictional heating on a fault surface, Pure AppL Geophys., vol. 122, pp. 583-607. 25. Habib, P. (1975) Production of gaseous pore pressure during rockslides, Rock Mech., vol. 7, pp. 193-197. 26. Goguel, J (1978) Scale dependent rock slide mechanisms, with emphasis on the role of pore fluid vaporization, in B. Voight (ed.), Rockslicles and Avalanches 1: Natural Phenomena, Elsevier, pp. 693-705. 27. MiJller, L. (1968), New considerations on the Vaiont slide, Rock Mech. Engrg. GeoL, vol. 6, pp. 1-91. 28. Voight, B. and Faust, C. (1982) Frictional heat and strength loss in some rapid landslides, Geotechnique, vol. 32, pp. 43-54. 29. Voight, B and Faust, C. (1992) Frictional heat and strength loss in some rapid landslides: error correction and affirmation of mechanism for Vaiont landslide, Geotechnique, vol. 42, pp. 641-643. 30. Anderson, D.L. (1980), An earthquake induced heat mechanism to explain the loss of strength of large rock or earth slides, Proc. Int. Conf. Eng. for Protection from Natural Disasters, Bangkok, pp. 569-580. 31. Smith, N.R. and Salt, G. (1988) Predicting landslide mobility: an application to the East Abbotsford Landslide, Proc. 5th New Zealand Geomech. Conf., Sydney, pp. 567-572. 32. Davis, R.O., and Mullenger, G. (1992) Frictional sliding in the presence of thermally induced pore pressures, in C.S. Desai, et aL (ed.) Constitutive Laws for Engineering Materials, ASME Press, New York, pp. 549-552. 33. Iverson, R.M., and LaHusen, R.G. (1989) Dynamic pore-pressure fluctuations in rapidly shearing granular materials, Science, vol. 246, pp. 796-799. 34. Rudnicki, J.W. (1984) Effects of dilatent hardening on the development of concentrated shear deformation in fissured rock masses, Jour. Geophys. Res., vol. 89, pp. 9259-9270.
449
Pore Pressure Effects on Interface Behavior R.O. Davis Department of Civil Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
1. THE EFFECTIVE STRESS PRINCIPLE The fact that pore fluid pressures are important in understanding and predicting the behavior of many geologic materials has been known at least since the late nineteenth century. A paper by Clarke 1 in 1904 presented correlations of monthly rainfall with displacement of a large slow-moving landslide at Portland, Oregon. Clarke's work showed clearly that it was the pressure in the pore water, rather than its weight or any lubricating effect, that caused the motion of the landslide to accelerate or to slow down. Although Clarke did not have the analytical tools to fully understand the Portland landslide, his paper was a remarkably well documented case history of pore pressure effects. There are a great many similar case histories in the engineering and geologic literature in which the reduction of frictional strength on a potential surface of sliding due to pore fluid pressures has resulted in failure. Sometimes the failure is manifest in only small displacements accompanied by minor damage. Other times catastrophic results have ensued. The Malpasset dam which failed in 1959 in southern France took the lives of 300 people. The dam failed when a massive slip occurred on a pre-existing fault surface which lay beneath the dam foundation. This failure occurred despite the fact the dam had been in operation for more than four years. The failure was directly linked to pore pressures developing on the fault due to abnormally heavy rainfall during the three days preceding failure. The link between pore pressure and frictional strength of geologic materials lies in understanding the principle of effective stress discovered by Karl Terzaghi 2. Terzaghi realized that whenever a pore fluid is present, the stress it supports will contribute to the overall stress state in the material. On any geologic interface there will be areas of rock or soil particles in direct contact and other areas where no contact occurs but where pore fluid separates material on either side of the interface. Let us consider some area A of the interface. At the points of contact between soil particles or rock asperities, there will exist some average contact stress crc acting normal to the interface. If we let A c denote the area of contact, then the normal force supported by the soil particles or the rock will be ocAc. The remaining area, A-Ac, will support the hydrostatic pore fluid stress which we will denote by u. Thus the normal force carried by the pore fluid is u(A-Ac). The combination of these two normal forces must equilibrate the applied normal stress o acting over the whole area A. We have o A = or162 + u ( A -
450 or
a =a/+u
1---~
(1)
where o / is the effective stress given by o/_
~ A
(2)
We see from this expression that the effective stress represents the ratio of normal force supported by the solid particle contacts or rock asperities per unit area of the interface. In many practical situations, the ratio AriA in equation (1) will be small in comparison with 1. Because of this, (1) is commonly approximated by o
=
o ~ +u
(3)
This is the form of the effective stress principle put forward by Terzaghi in 1925. The total stress o is partitioned into the effective stress o ~ and the pore pressure u. The more exact form of the effective stress principle given in (1) was first noted by Skempton 3 in 1960. It should be noted that eq. (1) or the approximate eq. (3) are appropriate for dealing with geologic interfaces where frictional strength and slip are matters of primary interest. The frictional strength will depend upon o / and will be independent of u, a fact experimentally verified many times. Other forms for the effective stress principle may be required for other types of problems 4 but these will not concern us here. We can generalize the effective stress principle for three-dimensional stress states. Equation (3) becomes '
Oij "-- Oij + U6ij
(4)
where oii is the Cauchy total stress tensor, o/..tj is the effective stress tensor and 6~i is the Kronecker delta symbol. We see that eq. (4) decomposes the stress state into a hydrostatic part with magnitude u and an effective stress part supported by the rock or soil skeleton. It should also be noted that even though we employ the notation of continuum mechanics, we must still consider finite sized areas and volumes. The usual limit procedures cannot be applied to granular materials or rock interfaces. Thus we define stress as the ratio of force divided by a finite area large enough to encompass a representative number of grains or a representative region of rock surface. The interfaces we wish to consider here are surfaces on which slip may occur in a geologic mass. With regard to soils the interface may be a well defined slip surface or it may be a shear zone where shearing deformation is concentrated in a narrow band. In rock we are concerned with joints or faults and these also may be well defined surfaces or may be narrow gouge filled shear bands. In either case the resistance to slip or shear deformation will be characterized by the Coulomb friction law which states the maximum shear stress ~: which may be supported is a linear function of the effective stress o / . r = c +/~ o /
(5)
Here c is the cohesion term, independent of normal stress, and p is a coefficient of friction. According to (5), the actual shear stress on the interface must always be less than or equal to ~:. If the actual stress is equal to r, then slip will occur.
451 Equation (5) is an idealization of the complex processes involved in slipping on a geologic shear surface. According to (5) no deformation occurs until the actual shear stress equals ~:, and deformation will cease immediately if the stress drops below ~:. In fact the slip mechanism is much more complicated than this and deformation may accompany lesser stresses. Nevertheless, (5) is a useful representation of strength which may be fruitfully applied in most practical problems. For high normal stress (e.g. crY> 200 MPa) the coefficients in eq. (5) become relatively independent of the material being tested. In 1978 Byerlee 5 compiled test data for a wide range of rock types and found generally good agreement that p = 0.6. For normal stresses lower than 200 MPa, the data becomes more scattered. For modest stress levels in soils (on the order of 100's of kPa) the value of p may take on values as small as 0.1 in clays to values in excess of 1.0 in sands and gravels. The cohesion c may also change as a function of confining stress. At low stresses on a well developed shear surface, c will generally be zero. At high stress levels c will normally have some small positive value. The strength parameter/J may also depend upon the rate of shearing. Rate effects have been observed in both soils 6'7 and rocks 8'9. Increasing the rate of shearing may lead to weakening or strengthening depending upon the material involved. The importance of pore fluid stress is clear in eq. (5). Increasing or decreasing u, while holding the total stress o constant, may have a dramatic effect on the stability of a fault or landslide. Changes in u may occur either because of outside influences such as infiltration in a landslide during heavy rainfall, or because of the mechanical behavior of the material itself. Practically all geologic materials exhibit some volume change when subjected to shear. Both compaction (volume decrease) and dilatation (volume increase) have been observed in different materials. If the void space in the material is filled with pore fluid, any tendency toward volume change will be accompanied by changes in pore fluid pressure. These pore pressures are required to drive the flow of pore fluid which must accompany the volume change. Once sufficient flow has occurred the pore pressures may return to their normal values, although situations arise in which flow and pore pressure dissipation encourage further shearing deformation with its accompanying compaction or dilatation. We will investigate these effects in more detail below. In order to quantify changes in pore pressure which result from either compaction or dilatation, Skempton 1~ introduced two pore pressure coefficients called A and B. These relate the change of pore pressure to changes of deviatoric and mean stress in undrained deformation. Let p represent total mean stress and let q be the deviatoric stress 1 p = 3O'kk
,
1 [3 Crkk)2]1 q = -~ OijOij - (
(6)
Then A and B are given by A=__
du
aq
, B=--
Ou
ap
(7)
The coefficient B depends primarily on whether the void space is fully saturated or not. For full saturation with de-aired water, B will theoretically be exactly equal to 1.0, and careful experiments confirm this. If undissolved air is trapped in the pores, B will be less than 1.0. The range of possible values for A is quite broad. Negative values correspond to dilating materials and positive values to compacting materials. If A equals zero, then
452 the material maintains constant volume during shearing. The value of A will not in general remain constant during any loading procedure. Depending on loading details, some geologic materials may first compact, then dilate, and finally deform at constant volume in a typical test. Nevertheless A can be a useful measure of the overall pore pressure behavior expected for a particular material. In the remainder of this chapter we will consider a number of situations where pore pressures may have a significant effect on the behavior of a geomaterial interface. In the next section, the question of induced seismicity due to pore pressure is examined. This is followed by a section describing how pre- and post-earthquake pore pressures may affect the earthquake process. Section 4 briefly discusses frictional heating of pore fluids with implication for landslide stability. Finally, in section 5, recent experimental evidence of dynamic fluctuation of pore pressures in shearing materials is described.
2. PORE PRESSURE INDUCED SEISMICITY The Denver earthquakes of 1966 first suggested that pore pressures could induce seismic activity 11. Injection of fluid waste in a disposal well was linked to earthquake occurrence, and seismic activity ceased when injection was discontinued. This phenomena was confirmed in an elaborate and carefully documented experiment carried out at Rangely, Colorado between 1971 and 197312. The Rangely experiment utilized existing wells at the Rangely oilfield to increase pore pressures in the region of an active fault. Since the overall fault dimensions were small, there appeared to be no possibility of producing a damaging earthquake, and indeed the largest earthquake induced by the experiment had magnitude 3.1. Using hydraulic fracturing techniques the in-situ stress field near the fault was measured, and it was found that the shear and normal stresses acting on the fault were approximately 7 and 34 MPa respectively. Taking c - 0 and/~-0.81 in eq. (5), the value of effective stress required to produce failure with 1:-7 MPa was found to be about 8.6 MPa. This suggested that the critical value of u required to induce earthquakes was ur
- ~'=34
- 8.6=25.4 MPa
The experiment consisted of carefully raising the pore pressure by water injection to a value greater than ur and then lowering it below ur while monitoring seismic activity on the fault. As expected a strong correlation between number of earthquakes and pore pressure near the fault was discovered, and the value of critical pore pressure was experimentally confirmed. It is noteworthy that the value of ur was predicted before the complete cycle of increasing and decreasing pore pressure was carried out. The Rangely experiment provided conclusive evidence for the applicability of the effective stress principle to frictional stability of faults. Many laboratory experiments had shown the principle to be valid for small scale samples, but the Rangety data extended the experimental dimensions by four to five orders of magnitude and removed any lingering doubt that pore pressures and effective stress were not controlling factors. Increased seismicity associated with injection from wells is not the only form of induced earthquake activity. The most common occurrences of induced seismicity are associated with reservoir construction. There have been numerous instances of increased numbers of earthquakes resulting from reservoir filling 13. These may simply result from increasing
453 levels of deviatoric stress in the rock near the reservoir due to the additional weight of water, or may be due to pore pressure effects combined with increased stress levels. Two pore pressure effects are present in reservoir filling. The most obvious effect is that water from the reservoir may infiltrate into the ground below and directly affect pore pressures in the adjacent rocks. A second, more immediate effect is the possibility of enhanced pore pressures due to the weight of the reservoir. We can gain an intuitive understanding of both effects by considering the idealized situation depicted in Figure 1. In the figure the reservoir is idealized as having constant depth h and extending indefinitely in all horizontal directions. Uniaxial strain conditions exist and the increase in vertical and horizontal components of total stress due to reservoir filling will be
Figure 1. Reservoir induced seismicity
'
&~
v ]v.h
A~
(8)
l-v
Here v~ represents the value of Poisson's ratio which is applicable for undrained loading of the rock. The pore pressure increase Au will be given by eqs. (6) and (7). Using (8) we have 1
Ap = 3 ( A O , + 2 k , OH)
_
_
,
1[ 1 + v ]?w h
~q = ~o
=
- Ao H
1-2v.)
y~h
(9)
)] 1~,~h -v,
(10)
I -v
Thus the pore pressure increase is Au=BAp
+AAq=
-1B 3 (l+v)
+A(1-2v
If the degree of saturation of the rock is near total, then B will be close to 1.0. In most instances the value of A will be negative but its magnitude will be small. If we assume A(1-2v~) is near zero and set B equal to 1.0, then
454
,u
=
_
3 i-v
?w h
(11)
If v~ were 0.4 for example, we see that Au = 0.78Ywh. Thus reservoir filling will immediately increase the pore pressure in the surrounding rock. How this increase may affect the stability of a fault depends primarily on the fault surface orientation. If the fault surface is horizontal, the effective stress increases AOZv= A o , -
Au = -
2[ 1 - 2 v
] u yw h 1-v u
3
If the fault surface is vertical, the effective stress it supports decreases by this amount.
AOH:AOH AU:
-
3 ....1
-
v u
/
Y
wh
Thus vertical fault surfaces become less stable and horizontal surfaces more stable. Surfaces with intermediate orientation suffer this change Ao~ = [ 1 - 2 v ) ( 1 ) 1-vu c~
Ywh
where 1~ denotes the angle between the fault surface and the horizontal. We see that cos"11/v3 - 55 degrees marks the maximum value of 0 for which the effective stress will be increased. These effects occur immediately upon loading. As time passes two things may occur. First, the additional or excess pore pressure beneath the reservoir may dissipate as flow occurs into the surrounding rock. Second, infiltration from the reservoir may link the existing ground water with the reservoir water. The result of these two processes will be a return of the effective stress conditions to the situation which existed before reservoir filling. Both pore pressure and total stress will be increased, but effective stress will be unchanged, implying unaltered stability of existing faults. This too is an idealized situation. Infiltration may often not connect the reservoir to existing ground water due to the presence of impermeable materials. The dissipation of excess pore pressures will occur regardless, in which case the long term equilibrium pore pressure may be unchanged from pre-reservoir conditions. In that instance the reservoir induced changes in effective stress will be equal to the changes in total stress and stability may be enhanced. In general, the most likely time of occurrence for reservoir induced earthquakes is shortly after reservoir filling. The stress increase caused by the added weight of water will, in fully saturated rock, be initially largely carried by the pore fluid. As flow occurs away from the loaded region the stress increase will gradually be transferred from the pore fluid to the rock. This process is familiar in soil mechanics where it is called consolidation, and it was the discovery of the effective stress principle which led Terzaghi to an understanding of the consolidation problem. The same understanding is necessary for the reservoir induced seismicity problem.
455
3. PORE PRESSURE EFFECTS ASSOCIATED WITH EARTHQUAKES Changes in pore pressures accompany earthquakes and may affect the aftershocks which follow. Pore pressure changes may also precede an earthquake and, in the 1970's, much interest was generated by the possibility that precursor changes in pore pressure could be used as a prediction tool. The dilatancy-diffusion theory was put forward as an explanation of changes in the ratio of dilatational and shear wave velocity observed before several earthquakes. The basic idea which underlies the dilatancy-diffusion theory is simple 14as. Shearing deformation in the rock adjacent to a fault will generally be accompanied by dilatation which in turn will cause a decrease in pore pressure. The mechanism is embodied in eq. (7)1 whenever the pore pressure coefficientA is negative. Lowered pore pressures near the fault will result in increased effective stress and greater strength. The phenomena is called dilatancy hardening. Strength increases will only be temporary however, since flow will occur into the region due to the negative pore pressure gradient. Diffusion of pore fluid into the region increases the pore pressure, decreasing the effective stress, and decreasing the strength. In theory then, the increased shear stress which might have caused an earthquake, instead causes the fault to become temporarily stronger. The increased strength cannot persist, and after some time the pore pressure drops and rupture occurs. The amount of time needed for all this to occur depends on the level of dilatation, the volume of rock involved and the permeability in the region of the fault. One particular aspect of the dilatancy-diffusion theory made it especially attractive from the standpoint of earthquake prediction, namely that the whole process could, in theory, be monitored. It was speculated that dilatancy would manifest itself by changing the velocity of seismic waves which propagate through the affected rock. Early observations suggested that the ratio of speeds of dilatational and shear waves, Vp/Vs, decreased by as much as ten percent in response to dilatation. As dilatation developed, the Vp/Vs ratio decreased. Then as diffusion brought additional pore fluid into the focal region and pore pressures increased, the ratio Vp/V~also increased, supposedly returning to its normal value of about 1.75, roughly at the time the earthquake would occur. The optimistic view taken in the 1970's held that observation of the vp/v~ ratio was one key to earthquake prediction. Unfortunately, subsequent events do not totally conform with the theory 16. Careful measurements of seismic wave velocities were carried out annually in Japan using explosive generated waves in the Izu region. Three large earthquakes occurred in the region in 1974, 1978 and 1980 during the time the velocity measurements were being made. No significant changes in vp/v~ were observed, and the dilatancy-diffusion theory has since lost much of its credibility in regard to earthquake prediction 17. Failure of the dilatancy-diffusion theory as an earthquake prediction tool does not imply the phenomena of dilatant hardening accompanied by pore fluid flow may never occur. There is considerable geological evidence suggesting just the opposite. Sibson TM has summarized a number of cases of hydrothermal vein systems associated with faults which could have arisen from fracture dilatancy and pore fluid flow. Post rupture flow and redistribution of pore fluid stress has also been advanced to explain both the temporal and spatial distribution of aftershocks associated with large earthquakes. A simplified model connecting pore fluid motion with aftershock occurrence was put forward by Nur and Booker 19. Their model considered pore pressure variations
456 set up by a two-dimensional edge dislocation in an infinite elastic space. Regions of compression and dilation lie on either side of the dislocation and corresponding pore pressure changes are expected to be found near the ends of a fault immediately following rupture. Flow then occurs from the region of higher pressure to the region of lower pressure. Nur and Booker hypothesized that occurrence of aftershocks was directly related to the rate of increase of pore pressure in the dilatant quadrant. Their simple model explained clustering of aftershocks near fault ends as well as the absence of aftershocks following deep earthquakes or small earthquakes. A more sophisticated model for this process has since been advanced by Li, et al.20.
4. FRICTIONAL HEATING EFFECTS Whenever slip occurs on a geological interface, energy will be dissipated. If we let r denote shear stress and v slip velocity, then at any instant the rate of dissipation per unit area of interface is the product rv. Dissipated energy takes the form of heat which may be transported away from the interface by a combination of convection and conduction. Heat conduction occurs through rock or the solid particle matrix in soil and also through pore fluids. Convection is manifest in flow of pore fluids away from the interface. If steady state slip is occurring, a state of thermal equilibrium will exist. The frictional heat generated will be balanced by conduction alone as convection can occur only as a transient process. No excess pore pressure will accompany steady state slip. In contrast to the steady state, transient conditions give rise to a rich variety of possible responses depending upon the details of loading and the materials involved. Any increase in either r or v will result in an increase in temperature at the interface surface. In general, for a fully saturated geomaterial, increasing temperature will result in increasing pore pressure since the coefficient of thermal expansion for most common minerals is about an order of magnitude smaller than that for water 21. Increasing pore pressure of course implies decreasing effective stress and decreasing strength. This slip-weakening effect may have important ramifications for overall stability of a fault or landslide. The simplest situation of interest occurs when we assume the interface is of infinite extent with uniform conditions everywhere. In this case heat transport occurs perpendicular to the interface and only one spatial dimension is required. Letting x be the spatial coordinate normal to the interface, the field equations appropriate to the problem are 22
mfi- anO=-aV dx V=_
(12)
k du "tw Ox
6 = 6 . 020 aX 2
(13/ PwCw O(OV) pC
O~X
where u = u(x,t) = pore pressure 0 = 0(x,t) = pore fluid temperature V = V(x,t) -- pore fluid velocity
(14)
457 n -- porosity rn = compressibility of solid matrix = (1 + v)(1 - 2v) E(1 -
a k 6 p law c %
= = = = = = =
coefficient of thermal expansion of pore fluid coefficient of permeability of solid matrix thermal diffusivity of solid-pore fluid mixture mass density of solid-pore fluid mixture mass density of pore fluid specific heat capacity of solid-pore fluid mixture specific heat capacity of pore fluid.
Here eq. (12) represents conservation of pore fluid mass, eq. (13) is Darcy's law, and eq. (14) represents conservation of pore fluid energy. These three equations can be solved, given appropriate boundary conditions and a constitutive equation for the shear stress 1:. The boundary conditions for this problem depend on whether we are concerned with earthquakes or landslides. In the latter case, there will be an isothermal boundary (the ground surface) not too distant from the interface. In the former case we may assume the interface is contained within an infinite medium and require the solution to remain bounded as x -, oo. In either case, the boundary conditions at the interface are atx=0"
V=0
and
- K a0 ax
=
t:v
(15)
Here K denotes the coefficient of thermal conductivity of the solid-pore fluid mixture. The no-flow boundary condition (15)1 implies a symmetric pore pressure gradient on either side of the interface. Finally, we require a constitutive equation relating shear and normal effective stress on the slipping interface. The Coulomb equation (5) is appropriate, and would normally be applied with c equal to zero. The coefficient of friction/~ may be taken to be a function of the slip velocity v as discussed in Section 16'7'8'9. Consideration of frictional heating of pore fluids in regard to earthquakes were first treated by Lachenbruch 23. Without solving the field equations, he considered critical combinations of the material parameters in special cases. He concluded that if the permeability was sufficiently large, thermal effects would not be important; but, in sufficiently impermeable rock, high pore pressures could be thermally generated and these could affect the dynamics of faulting. Lachenbruch's work was pursued further by Mase and Smith 24. They solved the governing equations for the special case of a fault slipping with constant velocity. They identified a range of material parameters within which frictional heating affected the strength of the fault. Again permeability was the most interesting parameter. In cases where permeability was sufficiently high, any excess pore pressures which might be generated by frictional heating were quickly dissipated by flow away from the fault. For low permeability rock however, high pore pressures could be generated by their model. Effects of frictional heating in landslide behavior have been considered by several investigators. Habib 25 carried out a simplified analysis, omitting the effects of fluid flow, and concerned primarily with the possibility of vaporization of pore fluid. Gogue126 was also concerned with the possibility of vaporization, particularly in regard to very large rockslides which appear to exhibit behavior compatible with near zero friction coefficients.
458 The Vaiont rockslide 27 is a case which has generated particular interest in the rock mechanics community. Voight and Faust 2a'29 carried out numerical calculations for the Vaiont slide which incorporated conduction and convection of heat. Their findings suggested that vaporization of pore fluid was unlikely, but pore pressures could nevertheless be significantly enhanced by frictional heating, leading to loss of frictional strength and high slide velocities. Similar conclusions were given by Anderson 3~ Frictional heating of pore fluids has also been implicated as a possible mechanism for the gradual acceleration and loss of stability observed in many creeping landslides 22. In materials of low permeability, small changes in creep velocity may result in slightly increased pore pressures, and this effect may compound in time leading to complete loss of stability. The effect may have been observed in the East Abbotsford landslide 31. Frictional heating can also result in stick-slip behavior for a simple elastic slider such as illustrated in Figure 2. The slider is pulled by an elastic spring which is connected to a load point moving with constant velocity vo. Frictional dissipation and pore fluid heating occur at the slider-base interface affecting frictional resistance. For certain combinations of material parameters, the slider exhibits stationary periods interspersed with rapid jumps forward 32. The full role of pore fluid heating in interface behavior is not yet clear. While theoretical calculations suggest its importance in certain situations, these are specialized cases dependent on full saturation and (especially) low permeability. The exceptional mobility of some large rock slides remains both the impetus and the only experimental evidence for the theory. Figure 2. Elastic slider 5. DYNAMIC PORE PRESSURE FLUCTUATIONS An alternative explanation for landslide mobility has been advanced by Iverson and LaHusen 33. They hypothesize that while the overall motion of a landslide may appear steady, internally there exist isolated regions of high or low pore pressure induced by local compaction or dilatation of the solid matrix. At any point on the sliding surface, local compaction and dilatation would be expected to occur in some roughly periodic fashion. The rate at which changes occur would presumably depend upon the velocity of sliding v and some characteristic dimension A. The dimension A might be the mean particle size or might be larger if groups of particles are moving together as a relatively rigid mass. The characteristic time (period) associated with the dilatation-compaction fluctmtions would be A/v. Pore pressures will develop locally in response to the dilatation-compaction fluctuations. Any excess pore pressure, whether negative or positive, will result in flow, either toward or away from the region affected. Flow results in dissipation of pore pressure depending upon the permeability and compressibility of the solid matrix and the distance to regions of lower pressure. The characteristic time for pore pressure dissipation may be taken as
459 A2/c where A is the characteristic particle (or particle group) dimension, and c is the coefficient of consolidation, defined by k c--
Ywm
where k and m are the permeability and compressibility of the solid matrix [defined previously following eq. (4)] and y, is the unit weight of the pore fluid. The coefficient of consolidation c is a familiar parameter in soil mechanics. We now have two characteristic times: A/v, the period of fluctuation of compression and dilatation, and A2/c, the time for dissipation of excess pore pressure. The ratio of these defines a dimensionless number R34 A/v c t~2/c vA whose magnitude characterizes the tendency of a landslide to develop sustained excess pore pressures. For large values of R the dissipation time is short in comparison with the period of pore pressure generation, and sustained pore pressures are unlikely. Conversely, for small R, the pore pressure dissipates slowly and sustained pressures are likely. Iverson and LaHusen 33 carried out pore pressure measurements in controlled shearing of a carefully constructed array of fibreglass rods. For values of R between 10 and 50 they found pore pressure fluctuations on the slip surface characterized by plateaus of high pressure (slightly higher than the initial static pressure) separated by deep troughs of low pressure. During the high pressure plateaus the pore pressure was sufficient to reduce the effective stress to zero. The low pressure troughs were of shorter duration but during them the pore pressure was sufficiently reduced so that the mean pressure over the plateautrough cycle was equal to the initial hydrostatic pressure. They also performed measurements in a large scale simulated landslide which showed quite large pore pressure fluctuations for an R value of approximately 0.3. If significant local pore pressure fluctuations do accompany sliding on geomaterial interfaces, many regions of high and low excess pore pressure might exist simultaneously on the sliding surface. The regions of high pressure would be weakened while the low pressure regions would be hardened. The edges of the high pressure region might be expected to exhibit stress concentrations tending toward dilatation, while flow from high to low pressure regions would decrease strength in areas of dilatency hardening. Migration of pore pressure from one region to another might presumably result in an overall pattern of motion which is apparently uniform. 6. SUMMARY Pore pressures tend to complicate the picture of geomaterial interface strength. They may be generated by the material itself through compaction or dilatation or possibly frictional heating, and they will dissipate only as rapidly as the material will allow. They introduce the dimension of time to strength considerations which might otherwise exhibit no time dependence, and their effects on stability of faults or landslides may be catastrophic. An understanding of the effective stress principle together with the phenomena of compaction and dilatation and the laws governing flow through porous media is required for any consideration of strength of geomaterial interfaces where pore fluids are involved.
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