A comprehensive mathematical model for estimating oil drainage rate in SAGD process considering wellbore/formation coupling effect

Abstract

The aim of this work is to present a comprehensive mathematical model for estimating oil drainage rate in Steam-assisted gravity drainage (SAGD) process, more importantly, wellbore/formation coupling effect is considered. Firstly, mass and heat transfer in vertical and horizontal wellbores are described briefly. Then, a function of steam chamber height is introduced and the expressions for oil drainage rate in rising and expanding steam chamber stages are derived in detail. Next, a calculation flowchart is provided and an example is given to introduce how to use the proposed method. Finally, after the mathematical model is validated, the effects of wellhead steam injection rate on simulated results are further analyzed. The results indicate that heat injection power per meter reduces gradually along the horizontal wellbore, which affects both steam chamber height and oil drainage rate in the SAGD process. In addition, when production time is the same, the calculated oil drainage rate from the new method is lower than that from Butler’s method. Moreover, the paper shows that when wellhead steam injection rate is low enough, the steam chamber is not formed at the horizontal well’s toe position and enhancing the wellhead steam injection rate can increase the oil drainage rate.

Appendix: Governing equations for thermophysical properties of steam in wellbores

The authors and their team [2, 6, 8–10] have done a series of researches on estimation of thermophysical properties of steam in both vertical wellbore and horizontal wellbore. In the following, the corresponding governing equations are listed briefly, and the detailed derivation can be found in Refs. [2, 6, 8–10].

1.1 In vertical wellbores

• 1. Mass balance equation

  $$\frac{{\partial w_{\text{t}} }}{\partial z} = \pi r_{\text{ti}}^{ 2} \frac{{\partial \left( {\rho_{\text{m}} \nu_{\text{m}} } \right)}}{\partial z} = 0$$

  (52)

• 2. Energy balance equation

  $$\frac{1}{{w_{\text{t}} }}\frac{{{\text{d}}Q_{\text{ver}} }}{{{\text{d}}z}} + \frac{{{\text{d}}h_{\text{m}} }}{{{\text{d}}z}} + \frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{v_{\text{m}}^{2} }}{2}} \right) - g\sin \theta_{\text{well}} = 0$$

  (53)

• 3. Steam pressure gradient in vertical wellbores

  $$\frac{{{\text{d}}p_{\text{ver}} }}{{{\text{d}}z}} = \frac{{\frac{1}{{w{}_{\text{t}}}}\frac{{{\text{d}}Q_{\text{ver}} }}{{{\text{d}}z}} - g\sin \theta_{\text{well}} }}{{C_{\text{Jm}} C_{\text{pm}} - C_{\text{pm}} \left. {\frac{{{\text{d}}f\left( p \right)}}{{{\text{d}}p}}} \right|_{{p = p_{\text{ver}} }} + \frac{{\nu_{\text{m}} \nu_{\text{sg}} }}{{p_{\text{ver}} }}}}$$

  (54)

• 4. The relationship between steam temperature and pressure

  $$\begin{aligned} T = f\left( p \right) = 280.034 + 14.0856{ \ln }\frac{p}{1000} + 1.38075\left( {{ \ln }\frac{p}{1000}} \right)^{2} - 0.101806\left( {{ \ln }\frac{p}{1000}} \right)^{3} + \hfill \\ 0.019017\left( {{ \ln }\frac{p}{1000}} \right)^{4} ,611{\text{Pa}} \le p \le 2.212 \times 10^{7} {\text{Pa}} \hfill \\ \end{aligned}$$

  (55)

• 5. Steam quality distribution in vertical wellbores

  $$xqua_{\text{ver}} \left( z \right) = e^{{ - \frac{{C_{2} }}{{C_{1} }}z}} \left( { - \frac{{C_{3} }}{{C_{2} }}e^{{\frac{{C_{2} }}{{C_{1} }}z}} + xqua_{\text{ver}} \left( 0 \right) + \frac{{C_{3} }}{{C_{2} }}} \right)$$

  (56)

where $w_{\text{t}}$, $\rho_{\text{m}}$,$\nu_{\text{m}}$ and $h_{\text{m}}$ are the wellhead mass flow rate, the density, the velocity and the specific enthalpy of steam/water mixture fluid, respectively; $p_{\text{ver}}$ is the steam pressure in the vertical wellbore; $C_{\text{pm}}$ and $C_{\text{Jm}}$ are the heat capacity at constant pressure and the Joule–Thomson coefficient of mixture fluid, respectively; $\nu_{\text{sg}}$ is the superficial gas velocity; $T$ and $p$ are the steam temperature and pressure, respectively; $xqua_{\text{ver}}$ is the steam quality in the vertical wellbore; $C_{1} = L_{\text{v}}$, $C_{2} = \left. {\frac{{{\text{d}}L_{\text{v}} }}{{{\text{d}}p}}} \right|_{{p = p_{\text{ver}} }} \frac{{{\text{d}}p_{\text{ver}} }}{\text{dz}}$, $C_{3} = \frac{1}{{w_{\text{t}} }}\frac{{{\text{d}}Q_{\text{ver}} }}{{{\text{d}}z}} + \left( {\left. {\frac{{{\text{d}}h_{\text{w}} }}{{{\text{d}}p}}} \right|_{{p = p_{\text{ver}} }} - \frac{{\nu_{\text{m}} \nu_{\text{sg}} }}{{p_{\text{ver}} }}} \right) \times \frac{{{\text{d}}p_{\text{ver}} }}{\text{dz}} - g\sin \theta_{\text{well}}$, $L_{\text{v}}$ is the latent heat of vaporization of steam, $h_{\text{w}}$ is the specific enthalpy of saturated water.

1.2 In horizontal wellbores

• 1. Energy balance equation

  $$Q_{{{\text{rad}},i}} + Q_{{{\text{c}},i}} = Q_{{{\text{in}},i}} - Q_{{{\text{out}},i}}$$

  (57)

• 2. Steam pressure gradient in horizontal wellbores

  $$\frac{{{\text{d}}p_{{{\text{t}},i}} }}{{{\text{d}}L}} = \frac{{\rho_{{{\text{m}},i}} g\sin \theta - \left( {f_{{{\text{wall}},i}} + f_{{{\text{perf}},i}} } \right)\frac{{\rho_{{{\text{ns}},i}} }}{{D_{\text{ci}} }}\frac{{\overline{{\nu_{i} }}^{2} }}{2}}}{{1 - \rho_{{{\text{m}},i}} \nu_{{{\text{m}},i}} \nu_{{{\text{sg}},i}} /p_{{{\text{t}},i}} }}$$

  (58)

• 3. Implicit equation for steam quality in horizontal wellbores

  $$I_{i} \overline{{\rho_{i} }} (\overline{{h_{{{\text{m}},i}} }} + \frac{{\nu_{{{\text{r}},i}}^{2} }}{2}) + \frac{{2\pi r_{\text{co}} U_{\text{co}} \lambda_{\text{e}} \Delta L}}{{r_{\text{co}} U_{\text{co}} f_{\bmod } (t) + \lambda_{\text{e}} }}\left( {\overline{{T_{i} }} - T_{\text{ei}} } \right) = w_{i - 1} \left( {h_{{{\text{m}},i - 1}} + \frac{{\nu_{i - 1}^{2} }}{2}} \right) - w_{i} \left( {h_{{{\text{m}},i}} + \frac{{\nu_{i}^{2} }}{2}} \right)$$

  (59)

where $Q_{{{\text{rad}},i}}$ is the energy transferred to the oil layer due to radial outflow; $Q_{{{\text{in}},i}}$ and $Q_{{{\text{out}},i}}$ are the energy carried by steam at the inlet and outlet, respectively; ${\text{d}}p_{{{\text{t}},i}} /{\text{d}}L$ is the total pressure drop; $f_{{{\text{wall,}}i}}$ and $f_{{{\text{perf,}}i}}$ are the friction factor for pipe flow and the friction factor of perforation roughness, respectively; $\rho_{{{\text{ns}},i}}$ is the no-slip density of mixture fluid; $D_{\text{ci}}$ is the inside diameter of casing; $\overline{{\nu_{i} }}$ is the average velocity of steam, $\overline{{\nu_{i} }} = \left( {\nu_{i - 1} + \nu_{i} } \right)/2$, $\nu_{i - 1}$ and $\nu_{i}$ are the velocities of steam at the inlet and outlet, respectively; $\nu_{{{\text{r}},i}}$ is the velocity of radial outflow from the horizontal wellbore to the oil layer, $\nu_{{{\text{r}},i}} = I_{i} /\left( {\pi r_{\text{ph}}^{2} } \right)$, $r_{\text{ph}}$ is the radius of perforation hole.