Environmental Engineering Reference
In-Depth Information
5.8.1
Incompressible Single-Phase Flow Under Applied
Pressure Gradient
The basic equation describing the flow of a single fluid through a porous
medium is the continuity equation which states that mass is conserved and
expressed as:
∂
∂
(
)
+∇
[]
(5.56)
q
=
n
r
r
v
t
where,
n
is the porosity,
r
is fluid density,
v
is the fluid velocity, and
q
is the source/sink term (i.e. outflow and inflow per volume). For low-flow
velocities, flow through porous media is modeled through the empirical
Darcy's law in which the fluid velocity
is related to pressure and gravity
forces as:
−
K
m
(
)
(5.57)
v
=
∇− ∇
P
r
g
z
where, K
is the intrinsic permeability,
μ
is the viscosity,
g
is the gravita-
tional constant, and
z
is the spatial coordinate in vertical direction. In most
real field geological reservoir models,
K
is an anisotropic diagonal tensor
(Aziz and Settari, 1979; Allen et al., 1988; Green, 1998; Chen et al., 2006;
Aarnes et al., 2007). There are two driving forces in porous media flow
including gravity, and the pressure gradient as evident from equation 5.57.
Since gravity forces are approximately constant inside a reservoir domain,
pressure gradient is considered the main driving force and can be consid-
ered as the main unknown parameter for single-phase flow problems. To
solve for the pressure, Darcy's equation (eq. 5.57) and the continuity equa-
tion (eq. 5.56) are combined. Assuming the constant porosity in time and
incompressible fluid, the temporal derivative term in eq. 5.56 vanishes and
the following equation for the water (
w
) pressure is obtained:
⎡
⎢
K
m
⎤
⎥
=
q
(
)
∇=−
.
v
∇− ∇
P
r
g
z
w
(5.58)
w
w
w
r
w
w
To complete the single phase flow model, initial and boundary condi-
tions are specified. In reservoir engineering, the common practice is to
use no-flow boundary conditions (Aziz and Settari, 1979; Allen et al.,
1988, Green, 1998; King, 1992; Chen et al., 2006; Aarnes et al., 2007). For
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