Black-oil Equations

The black-oil equations are a set of partial differential equations that describe fluid flow in a petroleum reservoir, constituting the mathematical framework for a black-oil reservoir simulator. The term ''black-oil'' refers to the fluid model, in which water is modeled explicitly together with two hydrocarbon components, one (pseudo) oil phase and one (pseudo-)gas phase. This is in contrast with a compositional formulation, in which each hydrocarbon component (arbitrary number) is handled separately.

The equations of an extended black-oil model are:
:
\frac\left phi\left(\frac+\frac\right)\right+\nabla\cdot\left(
\frac\vec u_o+\frac\vec u_g\right)= 0

:
\frac\left phi\left(\frac\right)\right+\nabla\cdot\left(
\frac\vec u_w\right)= 0

:
\frac\left phi\left(\frac+\frac\right)\right+\nabla\cdot\left(
\frac\vec u_o+\frac\vec u_g\right)= 0

where
$\phi$ is a porosity of the porous medium,
$S_w$ is a water saturation,
$S_o,S_g$ are saturations of liquid ("oil")
and vapor ("gas") phases in the reservoir,
$\vec u_o,\vec u_w,\vec u_g$ are
Darcy velocities of the liquid phase, water phase and vapor phase in the reservoir.
The oil and gas at the surface ( standard conditions) could be produced from both liquid and vapor phases existing at high pressure and temperature of reservoir conditions. This is characterized by the following quantities:

$B_o$ is an oil
formation volume factor
(ratio of some volume of reservoir liquid
to the volume of oil at standard conditions
obtained from the same volume of reservoir liquid),

$B_w$ is a water formation volume factor
(ratio of volume of water at reservoir conditions to volume of water at standard conditions),

$B_g$ is a gas formation volume factor
(ratio of some volume of reservoir vapor
to the volume of gas at standard conditions obtained from the same volume of reservoir vapor),

$R_S$ is a solution of gas in oil phase
(ratio of volume of gas to the volume of oil at standard conditions
obtained from some amount of liquid phase at reservoir conditions),

$R_V$ is a vaporized oil in gas phase
(ratio of volume of oil to the volume of gas at standard conditions
obtained from some amount of vapor phase at reservoir conditions).

See also

*Porous medium
* Darcy's law
* Relative permeability
*Petroleum
*Hydrocarbon

References

Partial differential equations
Equations of fluid dynamics

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''Last updated by Jesse Gabriel, June 2021''
''Koma Kange''