Page 367 - Numerical Analysis and Modelling in Geomechanics

348 RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE
If compaction-induced deformation of the rock is sufficient, shear failure of
the rock mass may occur, thus generating new faults or fractures. The generation
of such failure surfaces, or the displacements on such new surfaces, are often
known as “localization” or “shear bands”. Although computational models exist
for the generation of shear bands, faults and fractures in rock, such simulations
are not yet sufficiently efficient for simulations involving a full scale
hydrocarbon reservoir. Therefore, inclusion of such geological features is based
on field evidence of their existence, or they are assumed to exist at specific
locations based on the field data, and the discontinuities explicitly included as
part of the model.
If compaction deformations are sufficiently large, layers of weak rock, such as
soft shale, can yield and undergo significant plastic flow, leading to relative
motion between harder, stiffer rock layers on either side of the weak rock layer.
Again, deformation is manifested as an offset in the centerline of the casing,
producing an observable “kink” in the casing string. The amount of offset in the
casing is dependent upon the properties of the surrounding rock and the
thickness of the weak rock zone, but has been observed typically to occur over
the order of several to tens of feet.
A model of the shear deformation of casing has been developed from analysis
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of compactive hydrocarbon reservoirs. The model was constructed from finite
elements available in the ABAQUS program for problems in which the
initial geometry is axisymmetric, but for which the deformations produce a non-
axisymmetric geometry. 55 The deformed geometry of the model must have a
plane of symmetry. The geometry of the shear-damaged casing satisfies the
requirements for the application of this element. In this case, the nodal
displacements can be expressed with a general interpolation function including a
Fourier expansion in the circumferential, or θ, direction as:

(11.45)

In the implementation of the ABAQUS program, the Fourier expansion terms
correspond to discrete locations around the circumference of an axisymmetric
solid, such as a pipe. Nodal displacements are calculated on these planes, rather
than on only one plane, as is usual for axisymmetric finite elements. As with
general Fourier approximations, increasing the number of terms included in the
expansion increases the accuracy of the approximation. Increasing the
summation index, Q, increases the number of terms of the expansion. In terms of
angular position around the model, Q=1 means that displacements are calculated
at θ= 0° and 180°. Q=2 corresponds to planes at θ=0°, 90° and 180°. It should be
noted, however, that while increasing the number of terms in the Fourier series
results in increased accuracy, there is a point of diminishing returns for the large