Definition
Mathematical model of multiphase wellbore flow predicts the temperature, pressure and flow speed distribution along the wellbore trajectory with account for:
- 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
(1) | (\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
– define 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 in space.Equations
– define the motion dynamics of each phase, represented as linear correlation between phase flow speed \bar u_\alpha and partial pressure gradient of this phase \bar \nabla P_\alpha .Equation (1) 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 \frac{\delta E_H}{ \delta V \delta t} defines the speed of change of heat energy E_H volumetric density due to the inflow from formation into the wellbore.
The term
\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T represents heat convection defined by the wellbore mass flow.
The term \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P 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 \ \phi \sum_{a = \{w,o,g \}} \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 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
Stationary wellbore flow is defined as the flow with constant pressure and temperature: \frac{\partial T}{\partial t} = 0 and \frac{\partial P}{\partial t} = 0 .
This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks) shut-in.
The temperature dynamic equation (1) is going to be:
(2) | \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} |
and its discrete computational scheme will be:
(3) | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k = \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k ) |
where \delta p^k = p_e^k - p_{wf}^k is drawdown, p_e^k – formation pressure in k-th grid layer, p_{wf}^k – bottom-hole pressure across k-th grid layer, T_r^k – remote reservoir temperature of k-th grid layer.
The l-axis is pointing downward along hole with (k-1)-th grid layer sitting above the k-th grid layer.
If the flowrate is not vanishing during the stationary lift ( \sum_{a = \{w,o,g \}} |q_\alpha^{k-1}| > 0) then T^{k-1} can be calculated iteratively from previous values of the wellbore temperature T^k as:
(4) | T^{k-1} = \frac{\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k + \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k )}{\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) } |
References
Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617