Page 82 - Reservoir Geomechanics

66 Reservoir geomechanics

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

σ ij = S ij − δ ij P p (3.8)
which means that pore pressure influences the normal components of the stress tensor,
σ 11 , σ 22 , σ 33 and not the shear components σ 12 , σ 23 , σ 13 .Skempton’s coefficient, B,
is defined as the change in pore pressure in a rock,
P p , resulting from an applied
pressure, S 00 and is given by B =
P p /S 00 .
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

F T = F g
which, in terms of stress and area, can be expressed as

S ii A T = A c σ c + (A T − A c )P P
where A c is the contact area of the grain and σ c is the effective normal stress acting on
the grain contact. Introducing the parameter a = A c /A T , this is written as

S ii = aσ c + (1 − a)P P
Theintergranularstresscanbeobtainedbytakingthelimitwhereabecomesvanishingly
small

lim aσ c = σ g
a→0
such that the “effective” stress acting on individual grains, σ g ,isgivenby

σ g = S ii − (1 − a)P p = S ii − P P (3.9)