A numerical model for wet steam circulating in horizontal wellbores during starting stage of the steam-assisted-gravity-drainage process

Abstract

Steam-assisted-gravity-drainage (SAGD) has been proved effective in heavy oil recovery. Preheating of the wellbore-surrounding reservoir is to circulate steam in the injector and producer so that heat can be conducted into surrounding oil layer. At this stage, the amount of steam injected into the reservoir is neglected. As a result, creating a large temperature difference between wellbore and annuli is key during the preheating process. A model is established for estimating steam properties in the wellbores so that the highest steam temperature in wellbores can be achieved. The model is comprised of mass, energy and momentum balance equations and the model is solved with numerical method. It is found that: (a) rich heat energy reflected in high steam quality does little effect on heat absorption rate of oil layer. The only effective method for temperature increase in oil layer is to increase the steam temperature in wellbores; (b) in order to increase the heat conduction rate to oil layer, a lower steam quality, a higher steam pressure and a lower mass flow rate is recommended.

Appendices

Appendix 1. Steam density calculation

The density of wet steam can be calculated by the weighted mean method. The weight coefficient is determined by the steam quality x, which can be expressed as [69, 70]:

$$ {\rho}_{wetsteam}={\left(\frac{1-x}{\rho_{water}}+\frac{x}{\rho_{steam}}\right)}^{-1} $$

(11)

$$ {\displaystyle \begin{array}{l}{\rho}_{water}=3786.31-37.2487\times T+0.196246\times {T}^2-5.04708\times 0.0001\times {T}^3\\ {}+6.29368\times 0.0000001\times {T}^4-{3.0848}^{\ast }0.0000000001\times {T}^5\end{array}} $$

(12)

$$ {\rho}_{steam}={e}^{-93.7072+0.833941\times T-0.00320809\times {T}^2+6.57652\ast 0.000001\times {T}^3-6.93747\times {10}^{-9}\times {T}^4+2.97203\times {10}^{-12}\times {T}^5} $$

(13)

Appendix 2. Shear force calculation

According to the theory of fluid mechanics, the shear force for wet steam flow in wellbores can be expressed as [70, 71]:

$$ {\tau}_f=\frac{1}{4}\pi {r}_{longtube,i}{f}_{wall}{\rho}_{longtube}{v}_{longtube}^2 dL $$

(14)

Where, f_wall is the function of Reynolds number (Re_i) and wellbore roughness (ε). The calculation method of friction coefficient is shown in Table 2.

Table 2 Basic parameters used for calculation [70, 71]

Appendix 3. Heat conduction from long tube to surrounding annuli

The heat conduction rate from long tube to surrounding annuli can be expressed as [70,71,72]:

$$ \frac{d{Q}_{longtube}}{dz}=2\pi {r}_{longtube,o}{U}_{longtube,o}\left({T}_{longtube}-{T}_{annuli}\right) $$

(15)

$$ {U}_{IT o}={\left[\frac{r_{IT o}}{\lambda_{IT}}\ln \frac{r_{IT o}}{r_{iITi}}+\frac{r_{IT o}}{h_{fITi}{r}_{IT i}}+\frac{1}{h_{fITo}}\right]}^{-1} $$

(16)

Appendix 4. Steam enthalpy calculation

The enthalpy of wet steam can be calculated by the weighted mean method. The weight coefficient is determined by the steam quality x, which can be expressed as [69, 70]:

$$ {h}_{wetsteam}=x{L}_v+{h}_w $$

(17)

$$ {L}_v=\sqrt{7184500+11048.6\times T-88.405\times {T}^2+0.162561\times {T}^3-1.21377\times {10}^{-4}\times {T}^4} $$

(18)

$$ {\displaystyle \begin{array}{l}{h}_w=23665.2-366.232\times T+2.26952\times {T}^2-0.00730365\times {T}^3\\ {}+1.30241\times {10}^{-5}\times {T}^4-1.22103\times {10}^{-8}\times {T}^5+4.70878\times {10}^{-12}\times {T}^6\end{array}} $$

(19)

Appendix 5. Wellbore heat loss rate

According to the assumption, the amount of steam injected into the oil layer is neglected. As a result, heat conduction from steam in annuli to oil layer is the only way that heat can be transferred to oil layer. The heat conduction rate can be expressed as [69]:

$$ \frac{d{Q}_{an nuli}}{dL}=2\pi {r}_{wo}{U}_{wo}\left({T}_{an}-{T}_h\right) $$

(20)

$$ {U}_{wo}={\left(\frac{r_{wo}}{f_{wi}{r}_{wi}}+\frac{r_{wo}}{\lambda_{cas}}\ln \frac{r_{wo}}{r_{wi}}+\frac{r_{wo}}{\lambda_{cem}}\ln \frac{r_{cem o}}{r_{wo}}\right)}^{-1} $$

(21)

$$ {U}_{wo}={\left(\frac{r_{wo}}{\lambda_{cem}}\ln \frac{r_{cem o}}{r_{wo}}\right)}^{-1} $$

(22)

Hasan et al. [37] found the function relationship between heat conduction rate and time, which was shown as:

$$ \frac{d{Q}_{an}}{dz}=2\pi {\lambda}_e\frac{T_h-{T}_{ei}}{f(t)} $$

(23)

For the f(t) in Eq. (23), many scholars presented different equations. In this paper, the expression of f(t) presented by Cheng et al. [31] is adopted.

$$ f(t)=\frac{16{\omega}^2}{\pi^2}{\int}_0^{\infty}\frac{1-\exp \left(-{\tau}_D{u}^2\right)}{u^3\varDelta \left(u,\omega \right)} du $$

(24)

$$ \varDelta \left(u,\omega \right)={\left[u{J}_0(u)-\omega {J}_1(u)\right]}^2+{\left[u{Y}_0(u)-\omega {Y}_1(u)\right]}^2 $$

(25)

When ${r}_{cem}^2/\left(4{\alpha}_rt\right)=1/4{\tau}_D<<1$ is satisfied, the f(t) can be expressed as:

$$ f(t)=\ln \left(2\sqrt{\tau_D}\right)-\frac{C_1}{2}+\frac{1}{4{\tau}_D}\left[1+\left(1-\frac{1}{\omega}\right)\ln \left(4{\tau}_D\right)+{C}_1\right] $$

(26)

Where C₁ (dimensionless) denotes the Euler’s constant (C₁=0.5772).

According to the principle of continuity, the heat conduction rate can be expressed as:

$$ \frac{d{Q}_{an}}{dL}=\frac{2\pi {r}_{wo}{U}_{wo}\left({T}_{an}-{T}_{ei}\right)}{1+\frac{r_{wo}{U}_{wo}}{\lambda_e}f(t)} $$

(27)

With the rapid development of modern computation technology [73,74,75], numerical methods are widely adopted in simulating the heat and mass transfer processes in nearly all aspects of engineering practices [76,77,78,79]. This paper is proposed based on previous works [17]. This model reveals the physical process of wet steam circulating in a SAGD horizontal well pair. The authors believe that the model will show its usefulness in the engineering applications.