Page 354 - Numerical Analysis and Modelling in Geomechanics

RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE 335
though more general forms can be developed (e.g., for element formulations
involving incompatible modes).
The pore pressure field is discretized such that a single scalar pore pressure is
computed at a defined node.

Linearization of variational forms

Linearization provides a means for properly and easily developing a linear theory
from the more general nonlinear equations. Linearization is also used to develop
exact expressions for the variations of the weak forms of the momentum and
mass conservation equations, G(u, p, δu) and H(u, p, δp), for use in a Newton-
type iterative solution of the governing equations. Newton procedures for the
solution of nonlinear, multivariate equations are described by Oden. 69 Briefly,
given a system of nonlinear equations, for example, F(x)=0, where F may be
scalar, vector, or tensor valued, the function is expanded as a Taylor series about
some suitably selected point:
(11.21)

where the point x . is a close approximation to the actual solution, x, after the i-th
i
iteration, and c is the difference between the actual solution and its estimation,
i+1
which should be small. The second term on the right-hand side, DF, is the
derivative of F at the point x . i Neglecting higher-order terms provides an
approximate solution of F after the i-th iteration using the iterate, c , to the
i+1
solution:

(11.22)

The matrix DF is referred to as the “stiffness matrix,” and is equivalent in this
case to the first variation of the function F. To solve the governing equations for
flow in a deforming porous medium requires the determination of the first
variations of the functions G(u, p, δu) and H(u, p, δp). As indicated by Equation
(11.21), these first variations can be obtained by linearizing the functions G and
H.
70
Following the procedures in Marsden and Hughes and Borja and Alarcón, 56
the first variation is obtained by the linearization of G(u, p, δu):

(11.23)