Page 357 - Numerical Analysis and Modelling in Geomechanics

338 RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE

(11.29)

and

(11.30)

62
where, using the notation of Zimmerman, C is the compressibility of the bulk
bc
solid measured from a drained test, C is the compressibility of the rock or soil
r
grains (sometimes referred to as the matrix material), C is the compressibility of
f
the pore fluid, and C pp is the compressibility of the pores (i.e., the change in
volume of the pores due to a change in pore pressure).
Equations (11.26) and (11.27) make clear the coupling between pore pressure
and deformation of a porous, fluid-filled medium. The first two terms on the right-
hand side of Equation (11.26) are easily recognized as deriving from Hooke’s
law for a linear, elastic solid, while the third term arises from the effect of the
fluid pressure within the pores. When such a material is deformed, the pore
pressure will change according to the moduli E and H, and the stress will change
as well. Equation (11.27) expresses the effect of changes in mean stress, σ /3,
kk
and pore pressure upon the change in pore fluid volume in the sample.
For hydrocarbon reservoir compaction, the condition far from a well is that of
73
an undrained rock. A material parameter known as Skempton’s coefficient, B,
which characterizes the dependence between changes in pore pressure and mean
stress, may be recovered from the mass balance of the pore fluid:

(11.31)

where

(11.32)

It can be seen that for saturated soils and rocks, in which the pore fluid and rock
matrix are incompressible, B=1, but ranges between about 0.5 and 0.99 for actual
rocks. 71
A simple expression for the change of thickness of a reservoir due to
compaction resulting from a pore pressure reduction, or drawdown, ≥ p, can be
derived from Equation (11.27) by:

(1) assuming a reservoir of large lateral extent compared with its thickness,