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
where
formula_4 is a porosity of the porous medium, 
formula_5 is a water saturation,
formula_6 are saturations of liquid ("oil") 
and vapor ("gas") phases in the reservoir,
formula_7 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: 
formula_8 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),
formula_9 is a water formation volume factor 
(ratio of volume of water at reservoir conditions to volume of water at standard conditions),
formula_10 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),
formula_11 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), 
formula_12 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). 