Page 99 - Reservoir Geomechanics

83 Basic constitutive laws

the b parameter, and smaller values of b represent greater viscosities. Thus, Wilmington
sand is more viscous than the sand from the Gulf of Mexico. The effective pressure
exponent d represents compliance, with smaller values being stiffer. Thus, Wilmington
sand is stiffer than the GOM sand. Note that stiffness is not related to d in a linear way,
because strain and stress are related via a power law.
We will revisit the subject of viscoplastic compaction in weak sand reservoirs in
Chapter 12 and relate this phenomenon to the porosity change accompanying com-
paction of the Wilmington reservoir in southern California and a field in the Gulf of
Mexico.

Thermoporoelasticity

Because thermoporoelastic theory considers the effects of both pore fluids and temper-
ature changes on the mechanical behavior of rock, it could be utilized as a generalized
theory that might be applied generally to geomechanical problems. For most of the
problems considered in this book, this application is not necessary as thermal effects
are of relatively minor importance. However, as will be noted in Chapters 6 and 7,
theromoporoelastic effects are sometimes important when considering wellbore failure
in compression and tension and we will consider it briefly in that context.
Fundamentally, thermoporoelastic theory allows one to consider the effect of tem-
perature changes on stress and strain. To consider the effect of temperature on stress,
equation (3.21)is the equivalent of equations (3.11) where the final term represents the
manner in which a temperature change,
T, induces stress in a poroelastic body:

S ij = λδ ij ε 00 + 2Gε ij − α T δ ij P 0 − Kα T δ ij
T (3.21)
1δL
where α T = is the coefficient of linear thermal expansion and defines the change
LδT
in length, L,ofa sample in response to a change in temperature δT.
Figure 3.14 shows the magnitude of α T for different rocks as a function of quartz con-
tent (Griffith 1936). Because quartz has a much larger coefficient of thermal expansion
than other common rock forming minerals, the coefficient of expansion of a given rock
is proportional to the amount of quartz. To put this in a quantitative perspective, changes
in temperature of several tens of C can occur around wellbores during drilling in many
◦
reservoirs (much more in geothermal reservoirs or steam floods, of course), which has
non-negligible stress changes around wellbores and implications for wellbore failure
as discussed quantitatively in Chapters 6 and 7.