The 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: | Section |
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| \partial_t \bigg [ \phi \ \rho_W \bigg ] + \nabla \bigg ( \rho_{Ww} \ \mathbf{u}_w \bigg ) = q_{mW}(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \rho_O \bigg ] + \nabla \bigg ( \rho_{Oo} \ \mathbf{u}_o
+ \rho_{Og} \ \mathbf{u}_g \bigg ) = q_{mO}(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \rho_G \bigg ] + \nabla \bigg ( \rho_{Go} \ \mathbf{u}_o
+ \rho_{Gg} \ \mathbf{u}_g \bigg ) = q_{mG}(\mathbf{r}) |
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| anchor | DarcyW1 |
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| alignment | left |
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| \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} ) |
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| anchor | DarcyO1 |
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| alignment | left |
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| \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \mathbf{g} ) |
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| anchor | DarcyG1 |
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| alignment | left |
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| \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} ) |
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| anchor | CapilarOW1 |
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| alignment | left |
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| P_o - P_w = P_{cow}(s_w) |
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| anchor | CapilarOG1 |
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| alignment | left |
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| P_o - P_g = P_{cog}(s_g) |
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| anchor | swsosg1 |
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| alignment | left |
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| s_w + s_o + s_g = 1 |
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| (\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} |
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define the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component | LaTeX Math Inline |
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| body | \{ m_W, \ m_O, \ m_G \} |
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during its transportation in space. Equations | LaTeX Math Block Reference |
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define the motion dynamics of each phase, represnted represented as linear correlation between phase flow speed and partial pressure gradient of this phase (which is also called Darcy flow with account of the gravity and relative permeability).
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– | LaTeX Math Block Reference |
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define the hydrodynamic inter-facial balance between the phases with account of capillary pressure in porous formation . The key assumption is that capillary pressure at oil-water boundary is a function of water saturation alone | LaTeX Math Inline |
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| body | P_{cow} = P_{cow}(s_w) |
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and capillary pressure at oil-gas boundary is a function of gas saturation alone | LaTeX Math Inline |
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| body | P_{cog} = P_{cog}(s_g) |
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In the absence of capillary pressure the inter-facial equilibrium simplifies and implies that all phases are at the same pressure at all times.
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implies that porous space is fully occupied by fluid at all times .
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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 |
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| body | \frac{\delta E_H}{ \delta V \delta t} |
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defines the speed of change of heat energy volumetric density.In impermeable rocks ( | LaTeX Math Inline |
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| body | \phi =0, \; \bar u_\alpha = 0 |
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) heat flow is defined by heat conduction only:| LaTeX Math Block |
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| \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 |
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| (\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 |
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| \lambda_{t} = (1-\phi) \ \lambda_r + \phi \ (s_w \lambda_w + s_o \lambda_o + s_g \lambda_g ) |
The term | LaTeX Math Inline |
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| body | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T |
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represents heat convection defined by the mass flow. The term | LaTeX Math Inline |
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| body | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P |
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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 |
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| body | \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} |
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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 |
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represent the system of 16 scalar equations on 16 unknowns: | LaTeX Math Inline |
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| 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 \} |
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,which are all functions of time and space coordinates | LaTeX Math Inline |
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| body | (t, \mathbf{r}) = (t,x,y,z) |
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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}) |
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| \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}) |
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| \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}) |
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| anchor | DarcyW1 |
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| alignment | left |
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| \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} ) |
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| anchor | DarcyO1 |
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| alignment | left |
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| \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \mathbf{g} ) |
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| anchor | DarcyG1 |
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| alignment | left |
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| \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} ) |
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| anchor | CapilarOW1 |
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| alignment | left |
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| P_o - P_w = P_{cow}(s_w) |
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| anchor | CapilarOG1 |
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| alignment | left |
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| P_o - P_g = P_{cog}(s_g) |
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| anchor | swsosg1 |
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| alignment | left |
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| s_w + s_o + s_g = 1 |
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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:
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