Page 349 - Numerical Analysis and Modelling in Geomechanics

330 RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE
fundamental assumptions being stated, the governing equations for deformation
fluid-filled porous media are derived, with the unknown variables being
displacements and pore pressure. In the following section a mathematical
formulation for deformation of fluid-filled, porous media is presented. The
presentation follows that of Hibbitt, Karlsson and Sorensen 55 and Borja and
Alarcón. 56 A finite element approximation to these governing equations is
presented as implemented in the ABAQUS general-purpose, nonlinear finite
element software, which was used for simulations of the reservoir compaction
and casing damage discussed later in this chapter. The important effects of
coupling between overall stress on the solid-fluid volume and pore pressure are
discussed, as well as the effects of simplification by uncoupling the equations.
Linear and nonlinear constitutive models are presented; including a constitutive
model for fluid-filled porous media which incorporates a Drucker-Prager shear
failure surface and an elliptical Cap yield surface to capture the effects of
permanent deformation under volumetric strains.

Linear momentum balance
The material under consideration is assumed to be composed of a mixture of
solid and fluid. Grains, or particles, of soil or rock constitute the solid portion.
The particles may be cemented or bonded at contact points, or may slide,
translate and rotate relative to one another. The framework of solid material is
often referred to as the matrix. The interstitial space, or pore space, amongst the
particles is assumed in the following developments to be completely filled with a
single-phase fluid. That is, it is assumed that the media is saturated. It may be noted
that the assumption that the media is saturated is not completely necessary, as
only very general developments have been laid out for partially saturated media
and multiphase flows of pore fluid. 55–57 The processes that take place during
compaction are assumed to be quasi-static, inertia forces and contributions to
momentum from chemical reactions are assumed negligible.
n
Now consider a fluid-saturated body denoted by B ≥ R and let U be any open
n
1
subset of B with piecewise C boundary. Let ф : B → R be the motion, or set of
t
configurations of the body B. Let the motion of the fluid be denoted . Denote the
volume of the subset U in an assumed reference or undeformed configuration by
dV and the volume in the deformed configuration, after the motion ф (U), by dv.
t
The position of a material point in the body will be denoted by the X in the
reference configuration and by x=ф (X) in the deformed configuration. Let the
t
volume of solid matrix in the deformed configuration be denoted by dv . Then
g
the porosity may be defined as:
(11.1)