The flow and heat transfer characteristics of compressed air in high-pressure air injection wells

Abstract

High-pressure air injection (HPAI) is a significant enhanced oil recovery (EOR) technology of light oils especially in deep, thin, low-permeability reservoirs. The flow and heat transfer behaviors of compressed air in wellbore is essential to maximize performance of air in EOR. Due to strong compressibility of air and high injection pressure, wellbore temperature and pressure are greatly affected by friction and gas compression. However, the available models of wellbore flow and heat transfer are only accurate for thermal fluid, such as saturated steam and superheated steam, injected at relatively low pressure and high temperature. In this paper, a novel model is proposed to characterize wellbore pressure and temperature distribution for HPAI wells with consideration of dynamic behaviors of injected air. Flow and heat transfer in depth direction are coupled with air properties by iterative technique, and heat transfer in radial direction is treated as steady state in wellbore and transient state in formation. The mathematical model is solved by employing finite difference method and it is validated by field data. Then, integrated analyses of flowing pressure, heat transfer mechanism, and interaction between pressure and temperature are conducted. Results indicate that (1) as well depth increases, temperature difference between formation and air tends to become constant, and the radial heat transfer tends to reach an equilibrium state. The higher the flow rate is, the deeper the equilibrium depth is. (2) Air temperature is dominated by heat transmission from formation at low flow rates and dominated by frictional heat and gas compression effect at high flow rates. Fictional heat begins to affect air temperature at an injection rate beyond the critical value, while gas compression effect can increase air temperature in the whole calculated injection rate range. (3) Interaction between wellbore temperature and pressure is mainly achieved by altering air density. The effect of injection pressure on air temperature can be negligible, while the influence of injection temperature shows strong dependency on injection rate.

Appendices

Appendix 1. Calculation of air density and specific enthalpy

Owing to strong compressibility of air, the interaction force among the air components can lead to significant error when calculating air properties under high-pressure condition. In this paper, the R-K-S state equation, which is widely applied in the hydrocarbon industry, is employed to estimate the properties of real gas. The R-K-S state equation can be given by:

$$ {Z_{\mathrm{air}}}^3-{Z_{\mathrm{air}}}^2+{Z}_{\mathrm{air}}\left(A-B-{B}^2\right)- AB=0 $$

(11)

$$ A=\frac{ap}{R^2{T}^2},B=\frac{bp}{RT},a={\left(\sum {y}_i{a}_i^{0.5}\right)}^2,b=\sum {y}_i{b}_i $$

(12)

where Z_air denotes the compression coefficient of air; R denotes the molar gas constant, J/(mol·K); 1, 2, a, and b denote constant of air; y_i denotes the mole fraction of the air components; a_i and b_i denote constant of the air components.

Based on the R-K-S state equation, air density can be calculated by the following expression:

$$ {\rho}_{\mathrm{air}}=\frac{M_{\mathrm{air}}p}{Z_{\mathrm{air}} RT} $$

(13)

where M_air denotes the molar mass of air (kg/mol).

The specific enthalpy of air can be calculated precisely by employing fugacity to replace pressure (Guo 1995; Du and Chen 2003). The expressions of can be given by:

$$ \ln {\varphi}_i=\frac{b_i}{b}\left({Z}_{\mathrm{air}}-1\right)-\ln \left({Z}_{\mathrm{air}}-B\right)\frac{A}{B}\left(2\frac{a_i^{0.5}{b}_i}{a^{0.5}b}\right)\ln \left(1+\frac{B}{Z_{\mathrm{air}}}\right) $$

(14)

$$ {f}_i={y}_i{\varphi}_ip $$

(15)

$$ {h}_{\mathrm{air}}=\sum {m}_i{h}_i\left({f}_i,T\right) $$

(16)

where f_i denotes the fugacity of air components (Pa); φ_i denotes the fugacity coefficient of air components; h_i denotes the specific enthalpy of air components (J/kg); m_i denotes the mass fraction air components.

Appendix 2. Calculation of friction coefficient

Friction coefficient is the key parameter for shear stress calculation of air flow in wellbore, which is a function of Reynolds N_Re number and relative roughness Δ. The expressions of friction coefficient corresponding to different flow patterns adopted by this paper are given by (Beggs and Brill 1973):

$$ {N}_{\mathrm{Re}}=\frac{2{r}_{ti}{v}_{\mathrm{air}}{\rho}_{\mathrm{air}}}{\mu_{\mathrm{air}}}=\frac{2{w}_{\mathrm{air}}}{\pi {r}_{ti}{\mu}_{\mathrm{air}}} $$

(17)

$$ \varDelta =\frac{\varepsilon }{2{r}_{ti}} $$

(18)

$$ \Big\{{\displaystyle \begin{array}{c}f=\frac{64}{N_{\mathrm{Re}}},{N}_{\mathrm{Re}}\le 2000\\ {}f={\left[1.14-2\lg \left(\varDelta +21.25{N}_{\mathrm{Re}}^{-0.9}\right)\right]}^{-2},{N}_{\mathrm{Re}}>2000\end{array}} $$

(19)

where μ_air denotes the viscosity of air (Pa·s); ε denotes the absolute roughness of tubing inner wall (m).

Appendix 3. Calculation of heat transfer from formation to wellbore

The steady-state heat transfer in wellbore can be expressed as following equations (Hasan and Kabir 2012):

$$ q=\frac{dQ}{dz}=2\pi {r}_{to}U\left({T}_{\mathrm{air}}-{T}_{\mathrm{cem}}\right) $$

(20)

$$ U={\left[\frac{r_{to}}{\lambda_{\mathrm{tub}}}\ln \frac{r_{to}}{r_{ti}}+{h}_c+{h}_r+\frac{r_{to}}{\lambda_{\mathrm{cas}}}\ln \frac{r_{co}}{r_{ci}}+\frac{r_{to}}{\lambda_{\mathrm{cem}}}\ln \frac{r_{\mathrm{cem}}}{r_{co}}\right]}^{-1} $$

(21)

where h_c and h_r denote the convective heat transfer coefficient and the radiative heat transfer coefficient of annular fluid respectively (W/(m²·K)) (Cheng et al. 2012b; Cheng et al. 2011b).

The unsteady radial heat conduction in formation can be expressed as following equations (Hasan and Kabir 1991; Cheng et al. 2012b):

$$ q=\frac{dQ}{dz}=2\pi {\lambda}_e\frac{T_{\mathrm{cem}}-{T}_{ei}}{f(t)} $$

(22)

$$ {T}_{ei}={T}_0+ az $$

(23)

where f(t) denotes the function of injection time (Guo 1995), which is given by,

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

(24)

where ω denotes the radio of the formation heat capacity to the wellbore heat capacity; τ_D denotes the dimensionless time, ${\tau}_D={\alpha}_rt/{r}_{\mathrm{cem}}^2$; u denotes the dummy variable for integration, which is given by,

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

(25)

where J₀ and J₁ denote the first-kind Bessel functions of zero and first orders, respectively; Y₀ and Y₁ denote the second-kind Bessel functions of zero and first orders, respectively.

If injection time is large enough, Eq. (26) can be simplified 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 denotes the Euler’s constant, C₁ = 0.5772.

Based on Eq. (20) and Eq. (22), the overall heat transfer from formation to wellbore can be described and the temperature of outer surface of cement can be calculated.

$$ {T}_{\mathrm{cem}}=\frac{\lambda_e{T}_{ei}+{T}_{\mathrm{air}}{r}_{to} Uf(t)}{\lambda_e+{r}_{to} Uf(t)} $$

(27)

Because both the convective heat transfer coefficient and the radiative heat transfer coefficient of annular fluid are strongly dependent on temperature, the heat transfer rate need to be calculated by iteration technique, presented as follows:

• 1. Make an initial guess of U.

• 2. Calculate T_cem by Eq. (27) and q by Eq. (22).

• 3. Calculate the temperature of tubing outer wall T_to and the temperature of casing inner wall T_ci by equations below.

$$ {T}_{to}={T}_{\mathrm{air}}-\frac{q}{2\pi {\lambda}_{\mathrm{tub}}}\ln \frac{r_{to}}{r_{ti}} $$

(28)

$$ {T}_{ci}={T}_{\mathrm{cem}}+\frac{q}{2\pi}\left(\frac{1}{\lambda_{\mathrm{cas}}}\ln \frac{r_{co}}{r_{ci}}+\frac{1}{\lambda_{\mathrm{cem}}}\ln \frac{r_{\mathrm{cem}}}{r_{co}}\right) $$

(29)

• 4. Calculate h_c and h_r by the equations below:

$$ {h}_c=\frac{0.049{\lambda}_a{\left({G}_r{P}_r\right)}^{0.33}{P}_r^{0.074}}{r_{bo}\ln \frac{r_{ci}}{r_{to}}} $$

(30)

$$ {h}_r=\sigma \omega \left({T}_{to}^2+{T}_{ci}^2\right)\left({T}_{to}+{T}_{ci}\right) $$

(31)

$$ \omega ={\left[\frac{1}{\omega_{to}}+\frac{r_{to}}{r_{ci}}\left(\frac{1}{\omega_{ci}}-1\right)\right]}^{-1},{P}_r=\frac{1000{C}_a{\mu}_a}{\lambda_a},{G}_r=\frac{{\left({r}_{ci}-{r}_{to}\right)}^3g{\rho}_a^2{\beta}_a\left({T}_{to}-{T}_{ci}\right)}{\mu_a^2} $$

(32)

where λ_a denotes the thermal conductivity of annular fluid; G_r denotes the Guerra Shchev number; P_r denotes the Prandtl number; σ denotes the Stefan-Boltzmann constant, σ = 5.67 × 10⁻⁸; ω_to and ω_ci denote the radiation rate of tubing outer wall and casing inner wall, respectively; C_a, μ_a, and λ_a denote the specific heat, viscosity, and thermal conductivity of annular fluid, respectively.

• 5. Calculate U^' by Eq. (21).

• 6. Calculate ∣U^' − U∣, if the deviation is small enough and meets the accuracy requirement, output q; otherwise, replace U with U^' and go back to step 2.