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The 3-phase water-oil-gas model is usually built as a superposition of gas-liquid model and then oil-water model:
Input & Output
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| Input | Output |
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| LaTeX Math Inline |
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| body | Ps, \, T_s, \ \{ q_w, q_o, \, q_g \} |
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as values at separator | | LaTeX Math Inline |
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| body | P(l), \, T(l), \, \{ s_w(l), \, s_o(l), \, s_g(l) \}, \, \{ q_w(l), \, q_o(l), \, q_g(l) \} |
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as logs along hole |
Application
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| Activity |
| Input | Output |
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| 1 | WPA – Well Performance Analysis | Optimizing the lift performance based on the IPR vs VLP models | | LaTeX Math Inline |
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| body | Ps, \, T_s, \ \{ q_w, q_o, \, q_g \} |
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as values at separator | | LaTeX Math Inline |
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| body | P_{wf}(l = l_{datum}) |
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as value at formation datum |
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| 2 | DM – Dynamic Modelling | Relating production rates at separator to bottom-hole pressure with VLP | | LaTeX Math Inline |
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| body | Ps, \, T_s, \ \{ q_w, q_o, \, q_g \} |
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as values at separator | | LaTeX Math Inline |
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| body | P_{wf}(l = l_{datum}) |
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as value at formation datum |
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| 3 | PRT – Pressure Testing | Adjust gauge pressure to formation datum | | LaTeX Math Inline |
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| body | P_{wf}(l = l_{gauge}) |
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as value at downhole gauge | | LaTeX Math Inline |
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| body | P_{wf}(l = l_{datum}) |
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as value at formation datum |
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| 4 | PLT – Production Logging | Interpretation of production logs | | LaTeX Math Inline |
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| body | \{ p(l), \, T(l), \, u_m(l), \, s_w(l), \, s_o(l), \, s_g(l) \} |
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as logs along hole | | LaTeX Math Inline |
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| body | \{ q_w(l), \, q_o(l), \, q_g(l) \} |
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as logs along hole |
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| 5 | RFP – Reservoir Flow Profiling | Interpretation of reservoir flow logs | | LaTeX Math Inline |
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| body | \{ p(l), \, T(l), \, u_m(l), \, s_w(l), \, s_o(l), \, s_g(l) \} |
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as logs along hole | | LaTeX Math Inline |
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| body | \{ q_w(l), \, q_o(l), \, q_g(l) \} |
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as logs along hole |
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Mathematical Model
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The multiphase wellbore flow in hydrodynamic and thermodynamic equilibrium is defined by the following set of 1D equations:
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| LaTeX Math Block |
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(\rho \,c_p)_m \frac{\partial T}{\partial t}
- \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \bigg) \frac{\partial T}{\partial l}
\ = \ \frac{1}{A} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial q_\alpha}{\partial l} |
where
| indicates a mixture of fluid phases |
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| water, oil, gas phase indicator |
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| measure length along wellbore trajectory |  Image Modified
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| in-situ velocity of -phase fluid flow |
| |
| cross-sectional average fluid density |
| wellbore trajectory inclination to horizon |
| cross-sectional average pipe flow diameter |
| in-situ cross-sectional area | LaTeX Math Inline |
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| body | A(l) = 0.25 \, \pi \, d^2(l) |
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| Darci flow friction coefficient |
| kinematic viscosity of -phase |
| temperature of -phase fluid flowing from reservoir into a wellbore |
Equations
| LaTeX Math Block Reference |
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–
| LaTeX Math Block Reference |
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define a closed set of 3 scalar equations on 3 unknowns: pressure
, temperature
and mixture-average fluid velocity
.
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| LaTeX Math Block |
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| anchor | rho_cp |
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| alignment | left |
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(\rho \,c_p)_m = \sum_\alpha \rho_\alpha c_\alpha s_\alpha |
The in-situ velocities
are usually expressed via the macroscopic flow velocity using the
| Expand |
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| LaTeX Math Block |
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| (\rho \,c_{pt})_p \frac{\partial T}{\partial t}
- \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
\ = \ \frac{\delta E_H}{ \delta V \delta t} |
Equation | LaTeX Math Block Reference |
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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 due to the inflow from formation into the wellbore.
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