GLOSSARY

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Definition


Mathematical model of multiphase wellbore flow predicts the temperature, pressure and flow speed distribution along the hole 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

Stationary Temperature Model


In stationary conditions are defined as   \frac{\partial T}{\partial t} = 0 and \frac{\partial P}{\partial t} = 0 .


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



The temperature dynamic equation 

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 is going to be:

(1) \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:

(2) \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 then  q_\alpha^{k-1} \neq 0 and   T^{k-1} can be calculated iteratively from previous values of the wellbore temperature  T^k as:

(3) 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) }


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:

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

so that 

(5) \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}


References


Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617




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