Structural Geology
Lecture 18
Effective Stress
(Pore Pressure within the Earth)

During the previous lectures crack propagation was discussed in terms of stress risers.  The example of a stress riser was the point-to-point contact between grains.  Stress risers also occur around pores in rocks.  An example of a stress riser around a pore may be illustrated by examining the magnification of tectonic stress at the edge of a circular hole in an elastic medium (i.e. a rock).  Figure 18-1 shows that the tectonic stress s1 is magnified three times at the edge of the hole whose tangent parallels the orientation of the tectonic stress.  At 90° around the hole from the stress concentration the tectonic stress causes a tensile stress whose magnitude is equal to - s1.   

If this hole serves as a model for a pore in a rock, it is apparent that the position of easiest cracking is at the point of greatest tensile stress.  This stress distribution favors crack propagation in the plane of the maximum compressive stress and normal the least compressive stress ( s3).

(Fig. 18-1)

So far we have discussed the concept of stress and one of its effects (e.g. microcracking and then brittle fracture in shear) on the deformation of rocks within the upper crust.  The effect of stress within the crust can be modified by several mechanisms and processes.  One of the most profound "modifiers" of stress is pore water.  As was indicated in the discussion of the Griffith failure criterion rocks contain cracks and pores.  Most of these voids within rocks are interconnected and, hence, subject to penetration by fluids migrating through the crust.  It is quite rare to find a completely dry rock within the crust.

The fluids that penetrate pore space exert a "back pressure" on the pores.  This internal pressure tends to force the pores open, in contrast to the lithostatic stress of the surrounding rock which acts to close the pores.  Investigations of rock properties must account for the effect of this "back pressure" which modifies the behavior expected for a dry rock.  The effect of this back pressure is shown in the fluid filled crack of Figure 18-2.

(Fig. 18-2)

  Karl Terzaghi was a pioneer in the field of soil mechanics.  Much of what he proposed for the behavior of water saturated soils can be applied to the mechanics of rocks with fluid-filled pores.  His most important concept is known as the effective stress principle.  The effective stress within a soil or rock is equal to the total stress minus the pore pressure.   The effective stress principle is as follows.  Across any plane at element A within a rock, there are acting a total stress s and a pore water presssure pp.  Total stress ( s ) can be visualized as the weight of a water-saturated column  rock.   Two components of that weight are the rock with empty pores and the weight of the water that fills the pores.  We define effective stress as

Distinguishing the total vertical stress sv and the total horizontal stress sh, we have

and

I will define the coeffieient of lateral stress (L) based on effective stresses

	

Lets now consider the profile of various stresses and pressures with depth.   To do this we must define two constants:  the density of water rw and the total density of the rock rt which includes a component for water in the pore spaces.   In the following equations z is the depth of burial or depth below the surface of the earth.   Total vertical stress is

Hydrostatic water pressure is

Effective vertical stress is then

 
Now consider the horizontal stress generated from the vertical load.  To do this we assume plane strain.  This means that imaginary vertical planes within the rock are not allowed to move regardless of the vertical stress.  This is like putting sand into a glass jar and pushing on top of the sand.  Even though we can gradually push harder, the walls of the glass jar do not expand.  How is a vertical load transfered to the horizontal direction.  If we squeeze a rubber ball in a vertical direction it will want to expand in a horizontal direction.  However, if we do not allow the ball to expand in the horizontal direction by clamping the ball,  the clamps will  have to sustain increasing larger horizontal stresses with the additional vertical stresses.  This response of an elastic body by an expansion perpendicular to a compression is known as the Poisson effect.  The coeffieient of lateral stress (L) as defined above is a measure of the magnitude of the Poisson effect.  For our example where the only horizontal stress is due to a vertical stress then

                                                                
Now the total horizontal stress is

	

This is an illustration of how pore water pressure effects horizontal stress within the crust.

The final concept to be introduced in this lecture is the effect of pores on strength of rocks.  A rock without pores is considerable stronger.  This stands to reason because pore space is unsupported within the rock.  With the presence of unsupported space the intact rock between the pores has to take larger loads to sustain an overall applied stress.  This important effect also has a direct bearing on the effective stress law which is more properly written

Terzaghi suggested that a  =  1 based on experiments in soils.  His interpretation was that grain boundaries and grain contacts both have an effective porosity of one, which is to say, water at a pressure pp filled all space between grain contacts.  Others including Geertsma have suggested that 

	a  =  1  -  (K/K0)

where K and K0 are the effective and grain bulk moduli.  Nur and Byerlee have a very interesting derivation of a.  They assume an isotropic aggregate of solid material with connected pores.  The outside of the solid is subject to confining pressure Pc and the pores are subject to pressure pp.  Both of these pressures have an effect on the volumetric strain of the aggregate Q.  But what we will do is consider the effect of these pressures by dividing the application of the confining pressure into two steps.  The first application of pressure involves balancing pp and pc, so that pp'  =  pc.  The second application of pressure involves the remaining confining pressure  pc''  =  pc  -  pp.  Volumetric strain Q of a dry aggregate caused by the addition  of a confining pressure is

	Q  =  b.pc

where b =  1/K is the effective compressibility of the dry aggregate.  Now the volumetric strain of the aggregate Q'' due soley to the confining pressue pc'' is

	Q''  =  b.pc''  =  b.pc  -  b.pp.    

The volumetric strain Q' caused by the equal addition of pressure around the rock body and inside the pores is of interest.   The solution is to assume a solid in which the pores are filled with a solid.  Then an external confining pressure pc' causes an equal hydrostatic compression pc' in the solid.  The volumetric strain of the solid Qs is then

	Qs  =  bs.pc'   
                                                                
where bs is the compressibility of the solid devoid of any cavities.  In this analysis we can remove the solid from the pore and replace it with a fluid at pp'  =  pc'.  In doing so the deformation of the solid remains unchanged.  Consequently

	Q'  =  Qs  = bs.pc'  
Adding the equations to find the total strain

	Q  =  Q'  +  Q''  =  b.pc  -  ( b  -  bs)pp. 
                             
This equation can be written in terms of the bulk moduli where K is the bulk modulus of the dry aggregate and Ks is the intrinsic bulk modulus of the solid itself

	Q  =  1/K.pc  -  [1/K  -  1/Ks].pp                                        

Because the volmetric strain can be written
 
	Q  =  1/K.  pc''
                                                                       
the expression for the effective pressure can now be written

	pc''  =  pc   -  [1  -  (K/Ks)].pp  
                                     
and
	a  =  1  -  (K/Ks).

As discussed in the previous lecture, extension fractures are by far more common in the crust of the earth.  The explanation is simple:  the state of stress within the crust of the earth is such that the differential stress  (s1 - s3) is rarely large enough to fracture intact rock.  Many smallish appearing faults are nothing more then reactivated extension fractures.  Reactivated means that slip occured parallel to the extension fracture after it propagated normal to the least principal stress (s3).  Slip in shear requires that the stress field be reoriented so that the extension fracture is no longer sitting in a principal stress plane.

(Fig. 18-3)

If differential stress within the crust of the earth is rarely large enough to cause shear fractures, how are most fractures formed?  This question is particularly interesting because tensile stresses are required for the generation of extension fractures.  Yet, how are tensile stresses generated when lithostatic stress which is compressive become increasingly large with depth in the crust.  Remember that the vertical stress (sv) is proportional to the density (r) of the rock and the depth of burial (h).

	sv  =  r.g.h                                                                    
where g is the acceleration due to gravity.  In the absence of tectonic stresses the horizontal stress (sh) increases with depth as a fraction of the vertical stress.

	sh  =  u/(1 - u ). sv  =  u /(1 - u).r.g.h            

Figure 18-3 shows extreme examples of the development of sh with depth of burial in a delta environment.  The difference depends on whether water can leak or drain out of the rock during burial.  Clay, the forerunner of shale has such a low permeability that its leak rate is too slow to keep up with burial.  Therefore we call a clay "undrained".  For an undrained clay water supports the matrix.  A material such as water with no shear strength has a u = 0.5.  If a sandstone is allowed to drain, it has a u = 0.25.  There is a dramatic difference in the overburden induced sh between these two situations. However, there are four mechanisms by which these large compressive stresses can be negated.  These mechanisms either contribute large tensile stresses or act to create negative effective stresses within the crust of the earth.  These mechanisms include:  1.)  increasing pore pressure (negative effective stress);  2.)  thermal cooling (tensile stress);   3.)  bending of rock (tensile stress);  and 4.)  Poisson contraction (tensile stress).  Often more than one mechanism is active at the same time within the crust of the earth.

PORE PRESSURE  In a previous lecture we have already discussed the equations involved in pore presssures.  The effective stress concept was presented.  A simple model for effective stress involves pore water in a crack.  The lithospheric stress which may be either sv or sh depending on the orientation of the stress acting to close the crack.  These stresses are counterbalanced by water pressure within the crack which acts to open the crack (Fig. 18-4).  In a cross section through the rock if water has access to pores that fill the cross-sectional area then a change in pore pressure Dpp is effective in balancing the same change in lithospheric stress Ds.  Experimental evidence suggests that where cracks are in point contact, water cannot penetrate the point contacts to push the crack open  If this is the case then the pore fluid in a cross-sectional area does not fill the area and a Dpp does not act to offset Ds.   Here, then,  we would say that the effective stress changes less.  Hence, the general effective stress law is

	Dse  =  Ds  -  y. Dpp                                                    

where y  =  1.0 in the former case and y  <  1.0 in the latter case.  There is reason to believe that y is proportional to the ratio of pore space to total cross-sectional area as is shown in Figure 10-4.

(Fig. 18-4)

Evidence for abnormal pore pressures come from oil wells through out the world where the pore pressure is higher than expected for the weight of the water column at that level within a well.  Pore pressure is abnormal if

	Pp >  rH2Ogh  
                                                                  
Figure 10-5 shows data from the Gulf of Mexico where pore pressure is above hydrostatic pressure at depths greater than 7500 feet.  Rocks of ancient deltas contain shales that are undercompacted.  Undercompaction means that the pore space was not crushed by the weight of the overburden.  To prevent crushing the pore space must be supported by the pore water at abnormal pressures.  Joints have propagated in associated with the undercompacted shales.  These joints propagate only if the pore pressure in the rock has exceeded the inward compression of the tectonic stresses or the weight of the overburden.

Three mechanisms for generating abnormal pore pressures include compaction, aquathermal pressuring, and clay dehydration.  For pressuring by compaction the permeability of the rock must be low.  Captured pore water then prevents the pores from collapsing under the weight of the overburden.  The weight of the overburden is then transferred from the rock matrix to the water which increases in pressure.  For aquathermal pressuring the thermal expansivity of pore water is higher than the surrounding rock.  If the water is heated in closed pore space it will exert an ever increasing pressure against the walls of the pores.  At the temperature of the rock increases aquathermal pressuring increases at the rate of a few bars per °C.  A common clay in a sedimentary environment is a montmorillonite.  As the montmorillonite is heated it will dewater (lose water) during a phase transformation to illite.  This volume of water released during the transformation to illite equals about half the volume of the altered montmorillonite.  This process is called clay dehydration.

(Fig. 18-5)

THERMAL COOLING:  Under most circumstances materials including rock expand when heated and contract when cooled.  This property to expand when heated is called thermal expansivity.  The cooling of ice to develop water is one exception to the rule that materials expand when heated and this example involves a phase transformation.  A measure of expansivity is called the coefficient of thermal expansion (a).  Aquathermal pressuring is another example of the effect of thermal expansivity.  It denotes the change in length per unit length per degree of temperature change (DT).  Recall that a change in length per unit length is a strain.  A metal called Invar has an a = 0 ppm/°C.  Rock is commonly  a  =  5 x 10-6/°C  or 5 ppm/°C.  Linear strain caused by a temperature change is thus:

	e  =   dl/l  =  a.DT

If the rock is constrained and unable to expand or contract, a stress is developed in proportion to the rock's Young's modulus (E):

	s   =  E.e  =  E.a.DT  
                                                           
If DT = 100 °C and E = 0.5 x 106 b, the stress developed is:

	s  =  (0.5 x 106)( 5 x 10-6 )(102)  =  2.5 x 102 b 
                  
If the change in temperature was a cooling, this effects stress in rock by lowering the horizontal stress by 250 b.

(Fig. 10-6)

POISSON'S EFFECT:  As defined earlier, Poissons ratio (u) is the ratio of expansion in one direction of a rock caused by a contraction at right angles:

	u  = e2/e1.                                                                      

From this equation it is straight forward to calculate the change in horizontal stress (s2) accompanying the erosion of, say, 1 km of overburden.  If the density (r) of rock in the top km of the crust is 2.7, then the vertical stress (s1) caused by one km of overburden is then

	s1  =  r.g.h  =   (2.7 g/cm3)(980 cm/sec2)(105 cm)

	s1 ≈  270 x 106 dynes/cm2  =  270 bars                        

From Poisson's ratio

	u  =  e2/e1 =  Es2/Es1                                                       

If u  =  0.25, then

	s2  =   u. s1  =  67.5  b.                                                      

This calculation again assumes that the rock is constrained from moving in a horizontal direction (i.e. a plane strain model). 

Figure 10-6 illustrated two paths for horizontal stress with depth of burial.  Both paths lead to negative (tensile) effective stresses.  This stress condition is necessary for the formation of joints as described in the previous lecture.  Aquathermal pressuring leads to jointing by a hydraulic fracturing mechanism at considerable depth of burial.  Contraction by thermal cooling and the Poisson's effect leads to joint formation in near surface rock.

FOLDING OF ROCKS (i.e. bending of elastic beams)  Moments applied to the end of a beam would cause the beam of flex or bend with a constant radiius of curvature.  Rock layers can be treated as beams and respond in the same fashion to bending moments.  Engineers recognized that if an elastic beam were bent by moments applied at the ends of the beam, a neutral fiber would develop.  The neutral fiber is the plane within the beam where neither tensile nor compressive stresses would develop.   Below the neutral fiber the beam or rock layer is compressed.  Above the neutral fiber the beam or rock layer is extended, and may, in fact, extend far enough to overcome the horizontal stresses due to overburden.  Stresses in folds will be discussed in more detail in the lectures on folding.