The Modified Black Oil ( = Volatile Oil) flow model simulates 3-component fluid : water, liquid hydrocarbon (called "oil") and gaseous hydrocarbons ( called "gas") that flow in 3 possible phases (water, gasified oil and free gas) and defined by the following set of equations:

(1)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_W \bigg ] + \nabla \bigg ( \rho_{Ww} \ \mathbf{u}_w \bigg ) = q_{mW}(\mathbf{r})\end{array}$

(2)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_O \bigg ] + \nabla \bigg ( \rho_{Oo} \ \mathbf{u}_o + \rho_{Og} \ \mathbf{u}_g \bigg ) = q_{mO}(\mathbf{r})\end{array}$

(3)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_G \bigg ] + \nabla \bigg ( \rho_{Go} \ \mathbf{u}_o + \rho_{Gg} \ \mathbf{u}_g \bigg ) = q_{mG}(\mathbf{r})\end{array}$

(4)	$\begin{array}{l}\displaystyle \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} )\end{array}$

(5)	$\begin{array}{l}\displaystyle \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \mathbf{g} )\end{array}$

(6)	$\begin{array}{l}\displaystyle \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} )\end{array}$

(7)	$\begin{array}{l}\displaystyle P_o - P_w = P_{cow}(s_w)\end{array}$

(8)	$\begin{array}{l}\displaystyle P_o - P_g = P_{cog}(s_g)\end{array}$

(9)	$\begin{array}{l}\displaystyle s_w + s_o + s_g = 1\end{array}$

(10)

$\begin{array}{l}\displaystyle (\rho \,c_{pt})_m \frac{\partial T}{\partial t} - \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \nabla P + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T - \nabla (\lambda_t \nabla T) = \frac{\delta E_H}{ \delta V \delta t}\end{array}$

Equations (1) – (3) define the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component $\begin{array}{l}\{ m_W, \ m_O, \ m_G \}\end{array}$ during its transportation in space.

Equations (4) – (6) define the motion dynamics of each phase, represented as linear correlation between phase flow speed $\begin{array}{l}\bar u_\alpha\end{array}$ and partial pressure gradient of this phase $\begin{array}{l}\bar \nabla P_\alpha\end{array}$ (which is also called Darcy flow with account of the gravity and relative permeability).

Equations (7) – (8) define the hydrodynamic inter-facial balance between the phases with account of capillary pressure in porous formation $\begin{array}{l}P_{cow}, \ P_{cog}\end{array}$ . The key assumption is that capillary pressure at oil-water boundary is a function of water saturation alone $\begin{array}{l}P_{cow} = P_{cow}(s_w)\end{array}$ and capillary pressure at oil-gas boundary is a function of gas saturation alone $\begin{array}{l}P_{cog} = P_{cog}(s_g)\end{array}$

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 (9) implies that porous space is fully occupied by fluid at all times $\begin{array}{l}\{ s_w, s_o, s_g \}\end{array}$ .

Equation (10) 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 $\begin{array}{l}\frac{\delta E_H}{ \delta V \delta t}\end{array}$ defines the speed of change of heat energy $\begin{array}{l}E_H\end{array}$ volumetric density.

In impermeable rocks ( $\begin{array}{l}\phi =0, \; \bar u_\alpha = 0\end{array}$ ) heat flow is defined by heat conduction only:

(11)	$\begin{array}{l}\displaystyle \rho_r \, c_{pr} \frac{\partial T}{\partial t} - \nabla (\lambda_t \nabla T) = \frac{\delta E_H}{ \delta V \delta t}\end{array}$

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

(12)	$\begin{array}{l}\displaystyle (\rho \,c_{pt})_p = (1-\phi) \rho_r \, \ c_{pr} + \phi \ (s_w \rho_w \, c_{pw} + s_o \rho_o \, c_{po} + s_g \rho_g \, c_{pg} )\end{array}$

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

(13)	$\begin{array}{l}\displaystyle \lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g )\end{array}$

The term $\begin{array}{l}\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T\end{array}$ represents heat convection defined by the mass flow.

The term $\begin{array}{l}\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P\end{array}$ 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 $\begin{array}{l}\ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}\end{array}$ 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 (1) – (10) represent the system of 16 scalar equations on 16 unknowns:

$\begin{array}{l}\{ T, \ P_w, \ P_o, \ P_g, \ s_w, \ s_o, \ s_g, \ u_w^x, \ u_w^y, \ u_w^z, \ u_o^x, \ u_o^y, \ u_o^z, \ u_g^x, \ u_g^y, \ u_g^z \}\end{array}$ ,

which are all functions of time and space coordinates $\begin{array}{l}(t, \mathbf{r}) = (t,x,y,z)\end{array}$ .

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

(14)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_W \bigg ] + \nabla \bigg ( \rho_w \ \mathbf{u}_w \bigg ) = q_{mW}(\mathbf{r})\end{array}$

(15)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_O \bigg ] + \nabla \bigg ( {\tilde m}_{Oo} \ \rho_o \ \mathbf{u}_o + {\tilde m}_{Og} \ \rho_{g} \ \mathbf{u}_g \bigg ) = q_{mO}(\mathbf{r})\end{array}$

(16)	$\begin{array}{l}\displaystyle \partial_t \bigg [ \phi \ \rho_G \bigg ] + \nabla \bigg ( {\tilde m}_{Go} \ \rho_{o} \ \mathbf{u}_o + {\tilde m}_{Gg} \ \rho_g \ \mathbf{u}_g \bigg ) = q_{mG}(\mathbf{r})\end{array}$

(17)	$\begin{array}{l}\displaystyle \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} )\end{array}$

(18)	$\begin{array}{l}\displaystyle \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \mathbf{g} )\end{array}$

(19)	$\begin{array}{l}\displaystyle \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} )\end{array}$

(20)	$\begin{array}{l}\displaystyle P_o - P_w = P_{cow}(s_w)\end{array}$

(21)	$\begin{array}{l}\displaystyle P_o - P_g = P_{cog}(s_g)\end{array}$

(22)	$\begin{array}{l}\displaystyle s_w + s_o + s_g = 1\end{array}$

Substituting the values of mass densities and mass shares of fluid components (see Modified Black Oil fluid @model) and dividing each equation by density of corresponding component in standard conditions one gets the most popular form of Modified Black Oil Reservoir Flow @model

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