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The Black Oil flow is specific type of the Volatile Oil flow with 

LaTeX Math Inline
bodyR_v=0
.


Mathematical Model

...


The Volatile Oil flow dynamics is defined by the following set of equations:


Section
Column
width25%
LaTeX Math Block
anchordivW1divW
alignmentleft
\partial_t \bigg [  \phi \ \rho_W bigg (  \frac{s_w}{B_w}  \bigg ) \bigg ]  +  \nabla  \bigg (   \rho_{Ww  \frac{1}{B_w} \ \mathbf{u}_w     \bigg )      = 
q_{mW}W (\mathbf{r})
LaTeX Math Block
anchordivO1divO
alignmentleft
\partial_t \bigg [  \phi \ \bigg (  \frac{s_o}{B_o} \rho_O+ \frac{R_v \ s_g}

{B_g}  \bigg ) \bigg ]  +  \nabla  \bigg (    \rho_{Oo \frac{1}{B_o} \ \mathbf{u}_o 

+    \rhofrac{R_v}{OgB_g} \   \mathbf{u}_g   \bigg )       = q_{mO}O(\mathbf{r})
LaTeX Math Block
anchordivG1divG
alignmentleft
\partial_t \bigg [  \phi \ \bigg (  \frac{s_g}{B_g} + \frac{R_s \rho_G  s_o}

{B_o}  \bigg ) \bigg ]  +  \nabla  \bigg (   \rho_{Go  \frac{1}{B_g} \   \mathbf{u}_og

+     \rhofrac{R_s}{GgB_o} \ \mathbf{u}_go  \bigg )     = q_{mG}G (\mathbf{r})
Column
width20%
LaTeX Math Block
anchorDarcyW1DarcyW
alignmentleft
\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla  P_w - \rho_w \  \mathbf{g} )
LaTeX Math Block
anchorDarcyO1DarcyO
alignmentleft
\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ (  \nabla P_o - \rho_o \ \mathbf{g} )
LaTeX Math Block
anchorDarcyG1DarcyG
alignmentleft
\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \ \mathbf{g} )
Column
width20%
LaTeX Math Block
anchorCapilarOW1CapilarOW
alignmentleft
P_o - P_w = P_{cow}(s_w)
LaTeX Math Block
anchorCapilarOG1CapilarOG
alignmentleft
P_o - P_g = P_{cog}(s_g)
LaTeX Math Block
anchorswsosg1swsosg
alignmentleft
s_w + s_o + s_g = 1
Column
width30%



Equations 

LaTeX Math Block-ref
anchordivW
 – 
LaTeX Math Block Reference
anchordivG
 suggest no sources of flow in the right side except the contacts between wells and reservroir which is specified by well models as boundary conditions (see below).


Expand
titleDerivation


The Volatile Oil flow dynamics is defined by the following set of equations:

Section
Column
width25%
LaTeX Math Block
anchordivW1
alignmentleft
\partial_t \bigg [  \phi \ \rho_W  \bigg ]  + \nabla  \bigg ( \rho_{Ww} \ \mathbf{u}_w     \bigg )      = q_{mW}(\mathbf{r})
LaTeX Math Block
anchordivO1
alignmentleft
\partial_t \bigg [  \phi \ \rho_O \bigg ]  + \nabla  \bigg (    \rho_{Oo}
divT
alignmentleft
(\rho \,c_{pt})_p \frac{\partial T}{\partial t}   - \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}   + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha
 \ \mathbf{u}_
\alpha \bigg) \nabla P  
o 

+
\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \
  \rho_{Og} \  \mathbf{u}_
\alpha
g \bigg )
\
\nabla
T
 
-
\nabla (\lambda
= q_{mO}(\mathbf{r})
LaTeX Math Block
anchordivG1
alignmentleft
\partial_t \
nabla
bigg
T)
[
=
 \phi \
frac{\delta E_H}{ \delta V \delta t}

The disambiguation fo the properties in the above equation is brought in The list of dynamic flow properties and model parameters.

Equations 

LaTeX Math Block Reference
anchordivW1
 – 
LaTeX Math Block Reference
anchordivG1
 define the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component
LaTeX Math Inline
body\{ m_W, \ m_O, \ m_G \}
 during its transportation in space. 

Equations 

LaTeX Math Block Reference
anchorDarcyW1
 – 
LaTeX Math Block Reference
anchorDarcyG1
 define the motion dynamics of each phase, represnted as linear correlation between phase flow speed  
LaTeX Math Inline
body\bar u_\alpha
 and partial pressure gradient of this phase 
LaTeX Math Inline
body\bar \nabla P_\alpha
 (which is also called Darci flow with account of the gravity and relative permeability).

Equations 

LaTeX Math Block Reference
anchorCapilarOW1
 – 
LaTeX Math Block Reference
anchorCapilarOG1
 define the hydrodynamic inter-facial balance between the phases with account of capillary pressure in porous formation 
LaTeX Math Inline
bodyP_{cow}, \ P_{cog}
. The key assumption is that capillary pressure at oil-water boundary is a function of  water saturation alone 
LaTeX Math Inline
bodyP_{cow} = P_{cow}(s_w)
 and capillary pressure at oil-gas boundary is a function of  gas saturation alone 
LaTeX Math Inline
bodyP_{cog} = P_{cog}(s_g)

In the absence of capillary pressure the inter-facial equilibrium simplifies and implies that all phases are at the same pressure at all times.  

Equations 

LaTeX Math Block Reference
anchorswsosg1
  implies that porous space is fully occupied by fluid at all times 
LaTeX Math Inline
body\{ s_w, s_o, s_g \}
.

Equation 

LaTeX Math Block Reference
anchordivT
  defines the heat flow continuity or equivalently represents heat conservation due to heat conduction and convection with account for adiabatic and Joule–Thomson throttling effect.

The term 

LaTeX Math Inline
body\frac{\delta E_H}{ \delta V \delta t}
 defines the speed of change of  heat energy 
LaTeX Math Inline
bodyE_H
 volumetric density.

In impermeable rocks (

LaTeX Math Inline
body\phi =0, \; \bar u_\alpha = 0
) heat flow is defined by heat conduction only:

 \rho_G  \bigg ]  +  \nabla \bigg (   \rho_{Go} \   \mathbf{u}_o

+     \rho_{Gg} \ \mathbf{u}_g  \bigg )     = q_{mG}(\mathbf{r})
Column
width20%
LaTeX Math Block
anchorDarcyW1
alignmentleft
\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} )
LaTeX Math Block
anchorDarcyO1
alignmentleft
\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ (  \nabla P_o - \rho_o \mathbf{g} )
LaTeX Math Block
anchorDarcyG1
alignmentleft
\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} )
Column
width20%
LaTeX Math Block
anchorCapilarOW1
alignmentleft
P_o - P_w = P_{cow}(s_w)
LaTeX Math Block
anchorCapilarOG1
alignmentleft
P_o - P_g = P_{cog}(s_g)
LaTeX Math Block
anchorswsosg1
alignmentleft
s_w + s_o + s_g = 1



Column
width30%


LaTeX Math Block
anchordivT
alignmentleft
(\rho \,c_{pt})_p
LaTeX Math Block
anchorJZ1IT
alignmentleft
\rho_r \, c_{pr}
 \frac{\partial T}{\partial t} 

 
- \
nabla
(
\
lambda_t
phi \
nabla T)
sum_{a =
\frac{\delta E_H}{ \delta V \delta t}

The effective specific heat capacity of formation with multiphase flow is a simple sum of its components:

LaTeX Math Block
anchorcpt
alignmentleft
(\rho \,c_{pt})_p  = (1-\phi) \rho_r \,
\{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}  
 
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{
pr
p \alpha}
+
\ \
phi
epsilon_\alpha \
(s_w \rho_w \, c_{pw} + s_o \rho_o \, c_{po} + s_g \rho_g \, c_{pg} )

The effective thermal conductivity of formation with multiphase flow is assumed to be a sum of its components:

LaTeX Math Block
anchor3MQCG
alignmentleft
\lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g )

The term 

LaTeX Math Inline
body\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T
 represents heat convection defined by the mass flow. 

The term 

LaTeX Math Inline
body\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P
 represents the heating/cooling effect of the multiphase flow through the porous media. This effect is the most significant with light oil and gases.

The term 

LaTeX Math Inline
body\ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
 represents the heating/cooling effect of the fast adiabatic pressure change. This usually takes effect in and around the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation). This effect is absent in stationary flow and negligible during the quasi-stationary flow and usually not modeled in conventional monthly-based flow simulations. 

The set 

LaTeX Math Block Reference
anchordivW1
 – 
LaTeX Math Block Reference
anchordivT
 represent the system of 16 scalar equations on 16 unknowns: 

LaTeX Math Inline
body\{ T, \ P_w, \ P_o, \ P_g, \ s_w, \ s_o, \ s_g, \ u_w^x, \ u_w^y, \ u_w^z, \ u_o^x, \ u_o^y, \ u_o^z, \ u_g^x, \ u_g^y, \ u_g^z \}
,

which are all functions of time and space coordinates 

LaTeX Math Inline
body(t, \mathbf{r}) = (t,x,y,z)
.

Expressing the molar densities with mass shares and phase density (see also "Volatile Oil Model") one gets:

...

Column
width25%
LaTeX Math Block
anchordivW1
alignmentleft
\partial_t \bigg [  \phi \ \rho_W  \bigg ]  + \nabla \bigg (     \rho_w \ \mathbf{u}_w     \bigg )      =  q_{mW}(\mathbf{r})
LaTeX Math Block
anchordivO1
alignmentleft
\partial_t \bigg [  \phi \ \rho_O \bigg ]  + \nabla \bigg (   {\tilde m}_{Oo} \ \rho_o \ \mathbf{u}_o 

+  {\tilde m}_{Og} \ \rho_{g} \  \mathbf{u}_g    \bigg )       =  q_{mO}(\mathbf{r})
LaTeX Math Block
anchordivG1
alignmentleft
\partial_t \bigg [  \phi \ \rho_G  \bigg ]  +  \nabla  \bigg (  {\tilde m}_{Go} \ \rho_{o} \ \mathbf{u}_o

+    {\tilde m}_{Gg} \ \rho_g \ \mathbf{u}_g  \bigg )     =  q_{mG}(\mathbf{r})
Column
width20%
LaTeX Math Block
anchorDarcyW1
alignmentleft
\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w  \mathbf{g} )
LaTeX Math Block
anchorDarcyO1
alignmentleft
\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o   \mathbf{g} )
LaTeX Math Block
anchorDarcyG1
alignmentleft
\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g  \mathbf{g} )
Column
width20%
LaTeX Math Block
anchorCapilarOW1
alignmentleft
P_o - P_w = P_{cow}(s_w)
LaTeX Math Block
anchorCapilarOG1
alignmentleft
P_o - P_g = P_{cog}(s_g)
LaTeX Math Block
anchorswsosg1
alignmentleft
s_w + s_o + s_g = 1
\mathbf{u}_\alpha \bigg)  \nabla P
 
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \  \nabla T 
 
 - \nabla (\lambda_t \nabla T) =  \frac{\delta E_H}{ \delta V \delta t}


The disambiguation fo the properties in the above equation is brought in The list of dynamic flow properties and model parameters.


Equations 

LaTeX Math Block Reference
anchordivW1
 – 
LaTeX Math Block Reference
anchordivG1
 define the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component 
LaTeX Math Inline
body\{ m_W, \ m_O, \ m_G \}
 during its transportation in space. 

Equations 

LaTeX Math Block Reference
anchorDarcyW1
 – 
LaTeX Math Block Reference
anchorDarcyG1
 define the motion dynamics of each phase, represnted as linear correlation between phase flow speed  
LaTeX Math Inline
body\bar u_\alpha
 and partial pressure gradient of this phase 
LaTeX Math Inline
body\bar \nabla P_\alpha
 (which is also called Darci flow with account of the gravity and relative permeability).


Equations 

LaTeX Math Block Reference
anchorCapilarOW1
 – 
LaTeX Math Block Reference
anchorCapilarOG1
 define the hydrodynamic inter-facial balance between the phases with account of capillary pressure in porous formation 
LaTeX Math Inline
bodyP_{cow}, \ P_{cog}
. The key assumption is that capillary pressure at oil-water boundary is a function of  water saturation alone 
LaTeX Math Inline
bodyP_{cow} = P_{cow}(s_w)
 and capillary pressure at oil-gas boundary is a function of  gas saturation alone 
LaTeX Math Inline
bodyP_{cog} = P_{cog}(s_g)

In the absence of capillary pressure the inter-facial equilibrium simplifies and implies that all phases are at the same pressure at all times.  


Equations 

LaTeX Math Block Reference
anchorswsosg1
  implies that porous space is fully occupied by fluid at all times 
LaTeX Math Inline
body\{ s_w, s_o, s_g \}
.


Equation 

LaTeX Math Block Reference
anchordivT
  defines the heat flow continuity or equivalently represents heat conservation due to heat conduction and convection with account for adiabatic and Joule–Thomson throttling effect.

The term 

LaTeX Math Inline
body\frac{\delta E_H}{ \delta V \delta t}
 defines the speed of change of  heat energy 
LaTeX Math Inline
bodyE_H
 volumetric density.

In impermeable rocks (

LaTeX Math Inline
body\phi =0, \; \bar u_\alpha = 0
) heat flow is defined by heat conduction only:

LaTeX Math Block
anchorJZ1IT
alignmentleft
 \rho_r \, c_{pr} \frac{\partial T}{\partial t}  - \nabla (\lambda_t \nabla T) =  \frac{\delta E_H}{ \delta V \delta t}

The effective specific heat capacity of formation with multiphase flow is a simple sum of its components:

LaTeX Math Block
anchorcpt
alignmentleft
(\rho \,c_{pt})_p  = (1-\phi) \rho_r \, \ c_{pr} + \phi \ (s_w \rho_w \, c_{pw} + s_o \rho_o \, c_{po} + s_g \rho_g \, c_{pg} )

The effective thermal conductivity of formation with multiphase flow is assumed to be a sum of its components:

LaTeX Math Block
anchor3MQCG
alignmentleft
\lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g )

The term 

LaTeX Math Inline
body\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T
 represents heat convection defined by the mass flow. 

The term 

LaTeX Math Inline
body\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P
 represents the heating/cooling effect of the multiphase flow through the porous media. This effect is the most significant with light oil and gases.


The term 

LaTeX Math Inline
body\ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
 represents the heating/cooling effect of the fast adiabatic pressure change. This usually takes effect in and around the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation). This effect is absent in stationary flow and negligible during the quasi-stationary flow and usually not modeled in conventional monthly-based flow simulations. 


The set 

LaTeX Math Block Reference
anchordivW1
 – 
LaTeX Math Block Reference
anchordivT
 represent the system of 16 scalar equations on 16 unknowns: 

LaTeX Math Inline
body\{ T, \ P_w, \ P_o, \ P_g, \ s_w, \ s_o, \ s_g, \ u_w^x, \ u_w^y, \ u_w^z, \ u_o^x, \ u_o^y, \ u_o^z, \ u_g^x, \ u_g^y, \ u_g^z \}
,

which are all functions of time and space coordinates 

LaTeX Math Inline
body(t, \mathbf{r}) = (t,x,y,z)
.


Expressing the molar densities with mass shares and phase density (see also "Volatile Oil Model") one gets

...

width30%

...

:


Section
divWdivOdivG
Column
width25%
LaTeX Math Block
anchor
divW1
alignmentleft
\partial_t \bigg [  \phi \ \
bigg ( \frac{s_w}{B_w}
rho_W  \bigg
) \bigg 
]  +
 \nabla \bigg (     \
frac{1}{B
rho_w
}
 \ \mathbf{u}_w     \bigg )      =  
q_
W 
{mW}(\mathbf{r})
LaTeX Math Block
anchor
divO1
alignmentleft
\partial_t \bigg [  \phi \ \
bigg ( \frac{s_o}{B_o} + \frac{R_v \ s_g} {B_g} \bigg )
rho_O \bigg ]  + 
\nabla \bigg (   {\tilde m}_{Oo} \ \
frac{1}{B
rho_o
}
 \ \mathbf{u}_o 

+  {\tilde m}_{Og} \
frac{R_v}{B_
 \rho_{g} \
  \mathbf{u}_g    \bigg )       =  q_
O
{mO}(\mathbf{r})
LaTeX Math Block
anchor
divG1
alignmentleft
\partial_t \bigg [  \phi \ \
bigg ( \frac{s_g}{B_g} + \frac{R_s \ s_o} {B_o} \bigg ) \
rho_G  \bigg ]  +  \nabla  \bigg (  {\tilde m}_{Go} \ \
frac{1}{B_g
rho_{o} \ \mathbf{u}_
g
o

+    {\
frac{R_s}{B_o}
tilde m}_{Gg} \ \rho_g \ \mathbf{u}_
o
g  \bigg )     =  q_
G 
{mG}(\mathbf{r})
DarcyWDarcyODarcyG
Column
width20%
LaTeX Math Block
anchor
DarcyW1
alignmentleft
\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla 
P_w - \rho_w
\
  \mathbf{g} )
LaTeX Math Block
anchor
DarcyO1
alignmentleft
\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o
\
  \mathbf{g} )
LaTeX Math Block
anchor
DarcyG1
alignmentleft
\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g
\
 \mathbf{g} )
CapilarOWCapilarOGswsosg
Column
width20%
LaTeX Math Block
anchor
CapilarOW1
alignmentleft
P_o - P_w = P_{cow}(s_w)
LaTeX Math Block
anchor
CapilarOG1
alignmentleft
P_o - P_g = P_{cog}(s_g)
LaTeX Math Block
anchor
swsosg1
alignmentleft
s_w + s_o + s_g = 1
Column
width30%

...

LaTeX Math Block Reference
anchordivW

...

LaTeX Math Block Reference
anchordivG

...



Substituting the values of mass densities and mass shares of fluid components (см. "Volatile Oil Model") and dividing each equation by density of corresponding component in standard conditions one gets the most popular form of Volatile Oil flow equations:




Initial Conditions

...


Начальное условие по температуре задается распределением температурного поля:

...