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- tubing head pressure which is set by gathering system or injection pump
- wellbore design
- pump characterisits
- fluid friction with tubing /casing walls
- interfacial phase slippage
- heat exchange between wellbore fluid and surrounding rocks
Flow Model
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(\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 disambiguation fo the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
Equations
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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
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– | LaTeX Math Block Reference |
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define the motion dynamics of each phase, represented as linear correlation between phase flow speed and partial pressure gradient of this phase .
Equation
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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
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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.
The term
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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 wellbore mass flow. The term
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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 oils and gases.
The term
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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
Stationary Flow Model
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Stationary wellbore Stationary wellbore flow is defined as the flow with constant pressure and temperature:
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| body | \frac{\partial T}{\partial t} = 0 |
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and
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| body | \frac{\partial P}{\partial t} = 0 |
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This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks) shut-in.
Stationary Flow Model
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The temperature dynamic equation
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| anchor | divT | page | Volatile Oil and Black Oil dynamic flow models |
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is going to be:
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\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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