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The multiphase wellbore flow in hydrodynamic and thermodynamic equilibrium is defined by the following set of equations:

(1) \frac{\partial (\rho_m A)}{\partial t} + {\nabla } \bigg( A \, \sum_\alpha \rho_\alpha \, { \bf u}_\alpha \bigg) = 0
(2) \rho_\alpha \bigg[ \frac{\partial {\bf u}_\alpha}{\partial t} + ({\bf u}_\alpha \cdot \nabla) \ {\bf u}_\alpha - \nu_\alpha \Delta {\bf u}_\alpha\bigg] = - \nabla p + \rho_\alpha \, {\bf g} - \frac{ f_\alpha \, \rho_\alpha \, u_\alpha}{2 d} \ {\bf u}_\alpha
(3) (\rho \,c_{pt})_p \frac{\partial T}{\partial t}   - \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}   + \sum_\alpha \rho_\alpha \ c_{p \alpha} \ ({\bf u}_\alpha \cdot \nabla) \ T  \ =  \ \frac{\delta E_H}{ \delta V \delta t}

Equation (1)  defines the mass flow continuity or equivalently represents mass conservation during the mass transportation.

Equations (2)  are Navie-Stokes equations and define the momentum conservation during the mass transportation.

Equation (3)  defines the heat flow continuity or equivalently represents heat conservation due to heat exchange with surrounding rocks  and mass convection from reservoir inflow with account for wellbore adiabatic effects and Joule–Thomson reservoir throttling effect.

The term \frac{\delta E_H}{ \delta V \delta t}  defines the speed of change of  heat energy E_H volumetric density due to heat exchange with surrounding rocks and the inflow from formation into the wellbore.


Projecting the above equations to the well trajectory l(x,y,z) with down-pointing along-hole coordinate l:

(4) \frac{\partial (\rho_m A)}{\partial t} + \frac{\partial}{\partial l} \bigg( A \, \sum_\alpha \rho_\alpha \, u_\alpha \bigg) = 0
(5) \sum_\alpha \rho_\alpha \bigg[ \frac{\partial u_\alpha}{\partial t} + u_\alpha \frac{\partial u_\alpha}{\partial l} - \nu_\alpha \Delta u_\alpha\bigg] = - \frac{dp}{dl} + \rho_m \, g \, \sin \theta - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d}
(6) (\rho \,c_p)_m \frac{\partial T}{\partial t}   - \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha}\bigg) \ \frac{\partial p}{\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}


Equations (1) – (3) define a closed set of 3 scalar equations on 3 unknowns: pressure p(l), temperature T(l) and mixture-average fluid velocity u_m(l) .



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


Equation  (1) defines the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component \{ m_W, \ m_O, \ m_G \} during its transportation along wellbore. 

Equation (2) defines the motion dynamics of each phase (called Navier–Stokes equation), represented as linear correlation between phase flow speed  u_\alpha and pressure profile of mutliphase fluid p.



The term \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l} represents heat convection defined by the wellbore mass flow. 


The term \sum_\alpha \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 the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation). 


The term (\rho \,c_p)_m defines mass-specific heat capacity of the multiphase mixture and defined by exact formula:

(7) (\rho \,c_p)_m = \sum_\alpha \rho_\alpha c_\alpha s_\alpha

Stationary wellbore flow is defined as the flow with constant pressure and temperature:  \frac{\partial T}{\partial t} = 0 and \frac{\partial P_\alpha}{\partial t} = 0 .

This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks)  shut-in.


(8) (\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}

The wellbore fluid velocity u_\alpha can be expressed thorugh the volumetric flow profile q_\alpha and tubing/casing cross-section area \pi r_f^2 as:

(9) u_\alpha = \frac{q_\alpha}{\pi r_f^2}

so that 

(10) \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T = \frac{\delta E_H}{ \delta V \delta t}


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