Low level injection

2013-11-19T10:31:06Z

128.54.11.245:

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]].<ref>{{cite journal |last1=Trangenstein |first1=John A. |last2=Bell |first2=John B. |year=1989 |title=Mathematical Structure of the Black-Oil Model for Petroleum Reservoir Simulation |journal=SIAM Journal on Applied Mathematics |publisher=Society for Industrial and Applied Mathematics |volume=49 |issue=2 |pages=pp. 749–783 |jstor=2101984 |doi= }}</ref>

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
:<math>
\frac{\partial}{\partial t}\left[\phi\left(\frac{S_o}{B_o}+\frac{R_VS_g}{B_g}\right)\right]
+\nabla\cdot\left(
\frac{1}{B_o}\vec u_o+\frac{R_V}{B_g}\vec u_g\right)= 0
</math>

:<math>
\frac{\partial}{\partial t}\left[\phi\left(\frac{S_w}{B_w}\right)\right]
+\nabla\cdot\left(
\frac{1}{B_w}\vec u_w\right)= 0
</math>

:<math>
\frac{\partial}{\partial t}\left[\phi\left(\frac{R_S S_o}{B_o}+\frac{S_g}{B_g}\right)\right]
+\nabla\cdot\left(
\frac{R_S}{B_o}\vec u_o+\frac{1}{B_g}\vec u_g\right)= 0
</math>
where
<math>\phi</math> is a porosity of the porous medium,
<math>S_w</math> is a water saturation,
<math>S_o,S_g</math> are saturations of liquid ("oil")
and vapor ("gas") phases in the [[Petroleum reservoir|reservoir]],
<math>\vec u_o,\vec u_w,\vec u_g</math> are
[[Relative permeability|Darcy]] velocities of the liquid phase, water phase and vapor phase in the reservoir.
The oil and gas at the surface ([[Standard conditions for temperature and pressure|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:

<math>B_o</math> is an oil
[[Oil in place#Formation volume factor|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),

<math>B_w</math> is a water formation volume factor
(ratio of volume of water at reservoir conditions to volume of water at standard conditions),

<math>B_g</math> 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),

<math>R_S</math> 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),

<math>R_V</math> 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==
{{reflist}}

[[Category:Partial differential equations]]
[[Category:Equations of fluid dynamics]]
[[Category:Reservoir simulation]]

{{Petroleum-stub}}
{{fluiddynamics-stub}}

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Low level injection