Page 353 - Numerical Analysis and Modelling in Geomechanics

334 RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE
where is the prescribed volumetric flow rate of fluid per unit undeformed area
across the boundary.
The constitutive model for pore fluid flow is assumed to be Darcy’s law:

(11.15)

where k is the permeability tensor of the fully saturated medium, and g is the
vector of gravitational acceleration. Substituting Equation (11.15) into Equation
(11.14) and using the divergence theorem, results in the variational form for the
mass balance as follows:

(11.16)

It can be seen that this equation is coupled due to the products of the pore
pressure and its variation and the Jacobian J.
The weak form of the boundary value problem for flow in a saturated porous
medium composed of an incompressible solid matrix is to find ф≥ C and p≥ C p
ф
such that
(11.17)

This weak form of the problem forms the basis for the finite element
approximation used in this chapter.

Finite element discretization
The finite element approximation of the Equation (11.13) and Equation (11.16)
is developed for this problem by using the following discretizations of the virtual
velocity and the virtual rate of deformation:

(11.18)

and
(11.19)

N
where N (S ) are the nodal interpolation functions with respect to the material
i
coordinates S Usually, the interpolation function is defined as:
i
(11.20)