In a porous elastic solid saturated with a fluid, the theory of poroelasticity describes the constitutive behavior of rock. Much of poroelastic theory derives from the work of
Biot (1962). This is a subject dealt with extensively by other authors (e.g. Kuempel 1991; Wang 2000) and the following discussion is offered as a brief introduction to this topic.
The three principal assumptions associated with this theory are similar to those used for defining pore pressure in Chapter 2. First, there is an interconnected pore system uniformly saturated with fluid. Second, the total volume of the pore system is small compared to the volume of the rock as a whole. Third, we consider pressure in the pores, the total stress acting on the rock externally and the stresses acting on individual grains in terms of statistically averaged uniform values.
The concept of effective stress is based on the pioneering work in soil mechanics by Terzaghi (1923) who noted that the behavior of a soil (or a saturated rock) will be controlled by the effective stresses, the differences between externally applied stresses and internal pore pressure. The so-called “simple” or Terzaghi definition of effective stress is
σi j = Si j−δi jPp (3.8)
which means that pore pressure influences the normal components of the stress tensor, σ11,σ22,σ33 and not the shear componentsσ12,σ23,σ13. Skempton’s coefficient, B, is defined as the change in pore pressure in a rock,Pp, resulting from an applied pressure, S00and is given by B=Pp/S00.
In the context of the cartoon shown in Figure 3.5a, it is relatively straightforward to see that the stresses acting on individual grains result from the difference between the externally applied normal stresses and the internal fluid pressure. If one considers the force acting at a single grain contact, for example, all of the force acting on the grain is transmitted to the grain contact. Thus, the force balance is
FT=Fg
which, in terms of stress and area, can be expressed as Si iAT= Acσc+( AT−Ac)PP
where Acis the contact area of the grain andσcis the effective normal stress acting on the grain contact. Introducing the parameter a=Ac/AT, this is written as
Si i =aσc+(1−a)PP
The intergranular stress can be obtained by taking the limit where a becomes vanishingly small
a→0limaσc=σg
such that the “effective” stress acting on individual grains,σg, is given by
σg =Sii−(1−a)Pp =Sii −PP (3.9)
P o re pressure a c t i n g i n p o r e space
S tresses a c ti ng o n gra i ns S t r e s s = Fo r c e / A r e aTo t a l
S = F / AT
( A c t i n g o u t s i d e a n i m p e r me a b l e b o u n d a r y )
D R Y F O U N T A I N E B L E A U S A N D S T O N E f = 0 . 1 5
0.5 0.9
0.8
0.7
0.6 1
0.4
Alpha
50 40
30
10 20
0 60
Pressure (MPa) D RY OT TOWA S A N D f = 0 . 3 3
a .
c .
tc sc
t
AT
Ac Pp
b.
Figure 3.5. (a) Schematic illustration of a porous solid with external stress applied outside an impermeable boundary and pore pressure acting within the pores. (b) Considered at the grain scale, the force acting at the grain contact is a function of the difference between the applied force and the pore pressure. As Ac/ATgoes to zero, the stress acting on the grain contacts is given by the Terzaghi effective stress law (see text). (c) Laboratory measurements of the Biot coefficient,α, for a porous sand and well-cemented sandstone courtesy J. Dvorkin.
for very small contact areas. It is clear in Figure 3.5b that pore fluid pressure does not affect shear stress components, Si j.
Empirical data have shown that the effective stress law is a useful approximation which works well for a number of rock properties (such as intact rock strength and the frictional strength of faults as described in Chapter 4), but for other rock properties, the law needs modification. For example, Nur and Byerlee (1971) proposed an “exact”
effective stress law, which works well for volumetric strain. In their formulation
σi j = Si j−δi jαPp (3.10)
whereαis the Biot parameter α=1−Kb/Kg
and Kbis drained bulk modulus of the rock or aggregate and Kgis the bulk modulus of the rock’s individual solid grains. It is obvious that 0≤α≤1. For a nearly solid rock with no interconnected pores (such as quartzite),
φ→0limα=0
such that pore pressure has no influence on rock behavior. Conversely, for a highly porous, compliant formation (such as uncemented sands)
φ→0limα=1
and pore pressure has maximum influence. Figure 3.5c shows measured values of the Biot parameter for two materials: a compliant unconsolidated sand in whichαis high and a dry, well-cemented sandstone in whichα has intermediate values (J. Dvorkin, written communication). In both cases,αdecreases moderately with confining pressure.
Hofmann (2006) has recently compiled values forαfor a wide range of rocks.
Thus, to consider the effect of pore fluids on stress we can re-write equation (3.3) as follows
Si j =λδi jε00+2Gεi j −αδi jP0 (3.11)
such that the last term incorporates pore pressure effects.
The relation of compressive and tensile rock strength to effective stress will be dis- cussed briefly in Chapter 4. With respect to fluid transport, both Zoback and Byerlee (1975) and Walls and Nur (1979) have shown that permeabilities of sandstones con- taining clay minerals are more sensitive to pore pressure than confining pressure. This results in an effective stress law for permeability in which another empirical constant replaces that in equation (3.10). This constant is generally≥1 for sandstones (Zoback and Byerlee 1975) and appears to depend on clay content (Walls and Nur 1979). More recently, Kwon, Kronenberg et al. (2001) have shown that this effect breaks down in shales with extremely high clay content. For such situations, permeability seems to
depend on the simple form of the effective stress law (equation 3.8) because there is no stiff rock matrix to support externally applied stresses. Other types of effective stress laws describe the dependence of rock permeability on external “confining” pressure and internal pore pressure.