Environmental Engineering Reference
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in the corresponding FDM approach. However, it is not easy to implement
and is slower than FDM (King, 1992; Chen et al., 2006).
In finite volume method (FVM), physical quantities are calculated at
control volumes and the method is conservative (i.e., the flux entering a
given volume is identical to that leaving the adjacent volume). In a FVM the
unknown functions are represented in terms of average values over a set of
finite-volumes, over which the integrated PDE model is required to hold.
FVM can be viewed as a conservative finite-difference scheme that treats the
grid cells as control volumes. FVM is widely used in the petroleum industry
and can be formulated to allow for unstructured meshes. FVM is in between
FDM and FEM, faster and easier to implement than FEM; and more accu-
rate and versatile than FDM (King, 1992; Chen et al., 2006).
One of the simplest schemes of finite volume method is cell-centered
finite-volume method, also referred to as the two-point flux-approximation
(TPFA) scheme. The scheme uses two points, the cell-averages quantities,
to approximate the flux, and it is frequently used in reservoir simulation.
The pressure and saturation equations developed in here for two-phase
immiscible flow under pressure and electrical gradients are solved in a
sequential approach. In this approach, each equation is solved separately
using IMPES method. Different discretization methods were used to dis-
cretize the two fundamentally different equations. For the global pressure
and total velocity formulation, a sequential splitting method is used in the
following steps (Chen et al., 2006; Aarnes et al., 2007). First, the saturation
distribution from the previous time step (or initial data) is used to compute
the saturation-dependent coefficients in the pressure equation before the
equation is solved for global pressure and total velocity. Then, the total
velocity (
v
)
is kept constant as a parameter in saturation equation, while
the saturation is advanced in time. Next, the new saturation values are used
to update the saturation-dependent coefficients in the pressure equation,
and the pressure equation is solved again, and the computation is repeated.
To incorporate the flux resulting from the electrical gradient at a specific
time step; first the saturation distribution in the reservoir is found using
the TPFA scheme under the pressure gradient. Then, using the Archie's law,
the electrical conductivity of cells is approximated at that specific time step.
Archie's law is an empirical law in petro-physics that relates the in-situ resis-
tivity of a reservoir rock to its porosity, water saturation, and resistivity
of water through the following expression (Fleureau and Dupeyart, 1988):
05
.
=
⎛
⎞
⎟
R
R
(5.118)
S
w
Φ
⎜
w
t
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