Page 352 - Numerical Analysis and Modelling in Geomechanics

RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE 333
in Equation (11.10) becomes 1.0 when the rock material is assumed to be
incompressible, or when C is zero.
r

Mass balance
Considering the total mass of pore fluid within a control volume in the current,
or deformed, configuration, the mass continuity equation is:

(11.11)

where v is the average fluid velocity of the pore fluid.
f

Variational forms of governing equations
1
n
Let the space of configurations be C ={ф: B→ R |ф ≥ H , ф=ф on ∂B} and let
ф
d
i
1
n
the space of variations of the spatial variables be V =(η: B→R | η ≥ H , η=0 on
ф
i
1
∂B}, where H is the space of Sobelev functions of degree one. Now define the
1
n
space of pore fluid pressures as C ={Π: ф (B)→R |Π ≥ H , Π=Π on ∂ф } and
ф
t
t
d
1
the space of variations of V ={ψ: ф (B)→R n |ψ ≥ H ,ψ =0 on ∂ф }. The fluid
ф
t
t
potential Π is defined by:
(11.12)
where g:=−g ∂z/∂x is the gravitational acceleration and grad:=∂/∂x is the
gradient with respect to the current configuration. The space of configurations
may be identified with the deformation vector u and the variations of the
deformation field with η=δu.
A variational form, or weak form, of the linear momentum balance, written
with respect to the undeformed configuration, is:
(11.13)
where the identity δε:=grad δu=sym(grad δε) has been used, and t are the surface
tractions per unit area. The first term within the volume integral is the internal
virtual work, the second term exhibits the coupling between the deformation and
the pore pressure, and the third term accounts for the weight of the fluid.
The variational form for the mass balance of fluid, Equation (11.12), is:
(11.14)