Synonym: = Volatile/Black Oil Reservoir Flow @model = Muskat - Leverett equation

Definition

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Mathematical Model

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The Volatile Oil flow dynamics is defined by the following set of 3D equations:

\partial_t \bigg [  \phi \ \bigg ( \rho_W  \frac{s_w}{B_w}  \bigg ) \bigg ]  +  \nabla  \bigg ( \rho_{Ww    \frac{1}{B_w} \ \mathbf{u}_w     \bigg )      = 
q_{mW}W (\mathbf{r})

\partial_t \bigg [  \phi \ \rho_O \bigg ] bigg (  \frac{s_o}{B_o} + \nablafrac{R_v  \bigg  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})

\partial_t \bigg [  \phi \ \rho_G  \bigg ]  +  \nabla \bigg (   \rhofrac{s_g}{GoB_g} \  + \mathbffrac{u}_o

+     \rho_{GgR_s \ s_o}

{B_o}  \bigg ) \bigg ]  +  \nabla  \bigg (     \frac{1}{B_g} \ \mathbf{u}_g

+    \frac{R_s}{B_o} \ \mathbf{u}_go  \bigg )     = q_{mG}G (\mathbf{r})

\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla  P_w - \rho_w \  \mathbf{g} )

\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ (  \nabla P_o - \rho_o \ \mathbf{g} )

\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \ \mathbf{g} )

P_o - P_w = P_{cow}(s_w)

P_o - P_g = P_{cog}(s_g)

s_w + s_o + s_g = 1

(\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
 
+ \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}rho_{\rm inj} \, c_{\rm inj} \, T_{\rm inj} \, q_{\rm inj}({\bf r})\, \delta({\bf r})

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

The right sides of equations Equations 

Equations 

Equations 

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 

Equation 

The term 

In impermeable rocks (

\partial_t \bigg [  \phi \ \rho_W  \bigg ]  + \nabla  \bigg ( \rho_{Ww} \ \mathbf{u}_w     \bigg )      = q_{mW}(\mathbf{r})

\partial_t \bigg [  \phi \ \rho_O \bigg ]  + \nabla  \bigg (    \rho_{Oo} \ \mathbf{u}_o 

+   \rho_{Og} \  \mathbf{u}_g \bigg )       = q_{mO}(\mathbf{r})

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

+     \rho_{Gg} \ \mathbf{u}_g  \bigg )     = q_{mG}(\mathbf{r})

\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} )

\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ (  \nabla P_o - \rho_o \mathbf{g} )

\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} )

P_o - P_w = P_{cow}(s_w)

P_o - P_g = P_{cog}(s_g)

s_w + s_o + s_g = 1

(\rho \,c_{pt})_m

 \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} 

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

 
 
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{

p \

alpha} \ \

epsilon_

\alpha \

\mathbf{

u}

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

\lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g )

The term 

The term 

The term 

The set 

which are all functions of time and space coordinates 

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

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\partial_t \bigg [  \phi \ \rho_W  \bigg ]  + \nabla \bigg (     \rho_w \ \mathbf{u}_w     \bigg )      =  q_{mW}(\mathbf{r}) 

\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})

\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}) 

\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w  \mathbf{g} )

\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o   \mathbf{g} )

\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g  \mathbf{g} )

P_o - P_w = P_{cow}(s_w)

P_o - P_g = P_{cog}(s_g)

s_w + s_o + s_g = 1

...

...

_\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}

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

(\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} )

\lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g )

\partial_t \bigg [  \phi \ \

rho_W  \bigg 

]  + 

\nabla \bigg (     \

rho_w

 \ \mathbf{u}_w     \bigg )      = 

 q_

{mW}(\mathbf{r}) 

\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})

\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}) 

\mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla 

P_w - \rho_w

  \mathbf{g} )

\mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o 

  \mathbf{g} )

\mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g 

 \mathbf{g} )

P_o - P_w = P_{cow}(s_w)

P_o - P_g = P_{cog}(s_g)

s_w + s_o + s_g = 1

Equations 

Initial Conditions

Initial Conditions

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Initial temperature distribution is set as inputНачальное условие по температуре задается распределением температурного поля:

T(0, \mathbf{r}) = T_0(\mathbf{r})

Начальное условие на давления, скоростей и насыщенности задается одним из двух вариантов.

Условие I  – Стационарный старт

In case the simulation is performed over the undisturbed reservoir then initial temperature distribution is geothermal.

The initial condition on phase pressure, phase velocities and phase saturations is set by one of the following options: Equilibrium Start and Non-equilibrium Start.

Condition I – Equilibrium Start

Equilibrium Start means that flow was not happening before the start: Стационарный старт означает, что до начального момента времени поле давлений

\nabla \cdot \bigg (     \frac{1}{B_w} \ \mathbf{u}_w     \bigg )_{t=0}      = 0

\nabla \cdot \bigg (     \frac{1}{B_o} \ \mathbf{u}_o 

+    \frac{R_v}{B_g} \   \mathbf{u}_g    \bigg )_{t=0}      = 0

\nabla \cdot \bigg (     \frac{1}{B_g} \ \mathbf{u}_g

+    \frac{R_s}{B_o} \   \mathbf{u}_o     \bigg )_{t=0}    = 0

\mathbf{u}_w(0, \mathbf{r}) = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ (\nabla  P_w(0, \mathbf{r}) - \rho_w \  \mathbf{g} ) = 0

\mathbf{u}_o(0, \mathbf{r}) = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ (  \nabla P_o(0, \mathbf{r}) - \rho_o \ \mathbf{g} ) = 0

\mathbf{u}_g(0, \mathbf{r}) = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g(0, \mathbf{r}) - \rho_g \ \mathbf{g} ) = 0

P_o(0, \mathbf{r}) - P_w(0, \mathbf{r}) = P_{cow}(s_w)

P_o(0, \mathbf{r}) - P_g(0, \mathbf{r}) = P_{cog}(s_g)

s_w + s_o + s_g = 1

Условие II – Нестационарный старт

Нестационарный старт означает, что к начальному моменту времени поле насыщенностей

s_w(0, \mathbf{r}) + s_o(0, \mathbf{r}) + s_g(0, \mathbf{r}) = 1

Condition II – Non-equilibrium Start

Non-equilibrium Start means that flow happening before the start: 

s

P_o(0, \mathbf{r}) - P_w(0, \mathbf{r}) = P_{cow}(s_w)

P+ s_o(0, \mathbf{r}) -+ Ps_g(0, \mathbf{r}) = P_{cog}(s_g).1

pressure distribution При этом начальное поле скоростей

P_o(0, \mathbf{u}r}) - P_w(0, \mathbf{r}) = - k_a \ \frac{k_{rwP_{cow}(s_w, s_g)}{\mu_w} \ ( \nabla  P_w)

P_o(0, \mathbf{r}) - P_g(0, \mathbf{r}) -= \rho_w \  \mathbf{g} )P_{cog}(s_g).

The phase velocities are initialized as:

\mathbf{u}_ow(0, \mathbf{r}) = - k_a \ \frac{k_{rorw}(s_w, s_g)}{\mu_ow} \ (  \nabla  P_ow(0, \mathbf{r}) - \rho_ow \  \mathbf{g} )

\mathbf{u}_go(0, \mathbf{r}) = - k_a \ \frac{k_{rgro}(s_w, s_g)}{\mu_go} \ (  \nabla P_go(0, \mathbf{r}) - \rho_g \ \mathbf{g} )

На практике, нестационарное начальное поле давлений, скоростей и насыщенностей является, как правило, результатом промежуточных расчетов этой же модели, либо более крупной модели.

Краевое условие на внешней границе

Краевое условие на температурное поле на внешней границе задается одним из двух вариантов

Термодинамическое Условие I  – Фиксированная температура 

T(t, \mathbf{r}) |_{\Gamma_e} = T_e( \mathbf{r}) 

...

{r}) - \rho_o \ \mathbf{g} )

\big( \mathbf{n}, \nabla T(tu}_g(0, \mathbf{r}) \big) \big |_{\Gamma_e} = \zeta \cdot \big( T(t= - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g(0, \mathbf{r}) - T_e(\rho_g \ \mathbf{rg}) \big) 

где 

Краевое условие на поле давления, скоростей  насыщенности на внешней границе задается одним из двух вариантов

Гидродинамическое Условие I  – Непроницаемая граница 

LaTeX Math Inline
body \Gamma_e

In practice, the non-equilibrium conditions before the start  is usually a result of previous flow simulations for the same reservoir, sometimes using a different grid-structure.

External Boundary Condition 

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The external boundary condition for the temperature is usually set by one if the two options:

External Temperature Boundary Condition I – Fixed Temperature

T(t, \mathbf{r}) 

\big( \mathbf{n}, \ (\nabla P_\alpha(t, \mathbf{r}) - \rho_\alpha  \mathbf{r}) \big) \big|_{\Gamma_e} = 0

где  

Гидродинамическое Условие II – Постоянное давление на границе 

LaTeX Math Inline
body \Gamma_e

T_e( \mathbf{r}) 

External Temperature Boundary Condition II – Fixed Heat Exchange

\big( \mathbf{n}, \nabla T

P_\alpha(t, \mathbf{r} \big) \big |_{\Gamma_e} = P_i = const

где  

Краевое условие на разломах

Предполагается выполнение одного из двух условий на разломах (индивидуально по каждому разлому).

 \zeta \cdot \big( T(t, \mathbf{r}) - T_e( \mathbf{r}) \big) 

where 

The external boundary condition for phase pressure, phase velocities and phase saturations is set by one of the two popular options:

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External Pressure Boundary Condition I – Non-permeable boundary 

LaTeX Math Inline
body \Gamma_

...

e

\big( \mathbf{n}, \ (

...

\nabla P_\alpha(t, \mathbf{r}) - \rho_\alpha  \mathbf{

...

r}) \big) \big|_{\Gamma_

...

e} = 0

...

where  

...

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normal vector to the boundary 

...

...

 and 

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External Pressure Boundary Condition

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II –

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constant-pressure boundary 

LaTeX Math Inline
body \Gamma_

...

e

...

...

P_\alpha(t, \mathbf{r}) \big|_{\Gamma_e} = P_i = const

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where  

Моделирование скважины

Well model

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Well flow model (don't get confused with Wellbore Flow Model) simulates the flow at the contact between well and reservoir thus relating the sandface flow rates and pressure distribution in reservoir around the wellМодель притока (или закачки) на каждой скважине связывает объемы добычи (закачки) каждой фазы и перепад давления в пласте и на забое скважины и задается следующей формулой:

P_w(t, \mathbf{r}) \big|_{\Gamma_{WRC}} = P_o(t, \mathbf{r}) \big|_{\Gamma_{WRC}} = P_g(t, \mathbf{r}) \big|_{\Gamma_{WRC}}= P_{wf}(t) \big|_{\Gamma_{WRC}}

where где 

Bottom-hole pressure Забойное давление

• Условие   I – Контроль по забойному давлению
• Условие  II – Контроль по жидкости
• Условие III – Контроль по нефти

...

 at the contact (at depth 

• Well Condition I – Pressure Control
  
  
• Well Condition II – Liquid Control
  
  
• Well Condition III – Oil Control
  
  

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P_{wf}(t) = P_{wf}(t, h_0) +  P_{\delta}(t, \delta h)

...

P_{wf}(t) = P_{wf}(t, h_{ref}) +  P_{\delta}(t, \delta h)

...

...

P_{wf}(t, h_{ref}) = 
\frac{ \int_{\Gamma_{WRC}}  \bigg(      
 
\frac{M_w (P_{ew}- \delta P_{wf})}{B^S_w} + \frac{M_o (P_{eo}- \delta P_{wf})}{B^S_o} + \frac{R_v M_g (P_{eg}- \delta P_{wf})}{B^S_g}
 
\bigg)  T_h  dh - q_L(t) }
 
{ \int_{\Gamma_{WRC}}  \bigg(  
 
\frac{M_w}{B^S_w} + \frac{M_o}{B^S_o} + \frac{R_v M_g}{B^S_g}
 
      \bigg)  T_h dh }

...

q_L(t) = q_W(t) + q_O(t) = 
 \int_{\Gamma_{WRC}}  \bigg(      

\frac{M_w (P_{ew} - P_{wf}(t, h_{ref}) -  P_{\delta})}{B^S_w} 
+ \frac{M_o (P_{eo} - P_{wf}(t, h_{ref}) - P_{\delta})}{B^S_o} 
+ \frac{R_v M_g (P_{eg} - P_{wf}(t, h_{ref}) - P_{\delta})}{B^S_g}

\bigg)  T_h  dh  

P_w(t, \mathbf{r}) \big|_{\Gamma_{WRC}} = P_o(t, \mathbf{r}) \big|_{\Gamma_{WRC}} = P_g(t, \mathbf{r}) \big|_{\Gamma_{WRC}} = P_{wf}^{min} = const

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...

...

...

...

...

P_{wf}(t) = P_{wf}(t, h_{ref}) + P_{\delta}(t, \delta h)

...

P_{wf}(t, h_{ref}) = 
\frac{ \int_{\Gamma_{WRC}}  \bigg(      

 \frac{M_o (P_{eo}- P_{\delta})}{B^S_o} + \frac{R_v M_g (P_{eg}- P_{\delta})}{B^S_g}

\bigg)  T_h  dh - q_O(t) }

{ \int_{\Gamma_{WRC}}  \bigg(  

 \frac{M_o}{B^S_o} + \frac{R_v M_g}{B^S_g}

      \bigg)  T_h dh }

q_O(t) = 
 \int_{\Gamma_{WRC}}  \bigg(      

\frac{M_o (P_{eo} - P_{wf}(t, h_{ref}) - P_{\delta})}{B^S_o} 
+ \frac{R_v M_g (P_{eg} - P_{wf}(t, h_{ref}) - P_{\delta})}{B^S_g}

\bigg)  T_h  dh 

P_o(t, \vec r) \big|_{\Gamma_{WRC}} = P_{wf}^{min} = const

The list of dynamic flow properties and model parameters

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