Environmental Engineering Reference
In-Depth Information
From these procedures, using the entropy inequality, it can be shown that the
average pressure of the three fluids acting on the solid phase is given by:
p = S
o
p
o
+ S
w
p
w
+ S
g
p
g
[14.5]
The effective stress, also called modified Bishop's stress [BIS 59], which is
responsible for the deformation of the reservoir rock, is then given by:
σ
=
σ′
-
m
p
[14.6]
where σ
is the total stress vector, σ′ the modified Bishop's stress vector [BIS 59],
and
m
a vector where the first three elements are equal to one and the remaining
three equal to zero. Traction values are positive for the stresses and compression for
pressures.
In the mass balance equations of the fluids, fluid velocity relative to the solid
phase is expressed by Darcy's law, which is actually a simplified form of the linear
momentum balance equation of the fluids. The momentum exchange term in this
law contains the relative permeability, which presents an important modeling aspect.
For example, three-phase relative permeability functions can be approximated from
a set of two-phase data for oil and water and for gas and oil. For the wetting phase
they may also be dependant on the degree of saturation of this phase alone. In this
case, the relative permeability of the non-wetting phase is then only the function of
the degree of saturation of the phase concerned. We remind ourselves here that there
are reservoirs where the wetting phase is water and others where the wetting phase
is oil. Hysteresis effects can be also observed for permeability functions.
Finally, we do not consider the case of fractured reservoirs, which can be
simulated with a double porosity model. Such a model can be obtained by extending
the model presented in this chapter (see [LEW 98a]).
14.3.2.
The mathematical model
Following the hypotheses made above, the final model is composed of the linear
momentum balance equation of the four-phase system (the solid matrix plus the
three fluids) and of four mass balance equations.
From equations [14.1] to [14.5] the following expressions are obtained:
dp
dp
w
dp
o
dp
g
[14.7]
'
'
o
'
g
=
S
+
S
+
S
w
dt
dt
dt
dt
