Transient two-phase flow behavior in wellbores and well control analysis for sour gas kick with high H₂S content

ABSTRACT

Complex two-phase flow phenomena in wellbores after sour gas kick may lead to severe well control incidents. By bridging the gas solubility, H₂S phase transition and flow equations, a two-phase flow model is established under conditions of transient and non-isothermal states in wellbores, which is well validated against the two-phase flow experimental data. The following conclusions are drawn in the article: (1) a higher dissolved quantity and larger density of invaded sour gas containing H₂S and CH₄ in drilling fluids makes the detection of gas influx with high H₂S content more difficult at an early stage; (2) when the mixed fluids move up to the wellhead, the release of the dissolved gas, the sharp decline of density of sour gas and the phase transition of H₂S reduce the annulus pressure and induce complicated well control risks; (3) during H₂S and CH₄ influx, there are five major parameters that impact the bottom-hole pressure, which are evaluated for well control analysis. The proposed model shows a strong capacity to exhibit proper control of the wellhead back pressure, which can help to prevent the phase transition of H₂S and reduce risks to well control.

KEYWORDS:

• Sour gas with high H₂S content
• wellbore transient flow
• well control
• H₂S phase transition
• two-phase flow

1. Introduction

Gas reservoirs with a high content of hydrogen sulfide (H₂S) are commonly referred to as sour gas or acid gas reservoirs, which are widely distributed around the world. More than 400 gas reservoirs containing H₂S have been identified, most of which are located in Canada, the United States, Russia, China and the Middle East (Al-Jaberi et al., 2017). About 60% of natural gas resources discovered in the world contain H₂S, and about 10% of the gas fields containing H₂S are high-sulfur gas fields, most of which are found in marine strata (Table 1; Zhou, 2017). In China, a quarter of natural gas fields are sour gas reservoirs, which are located in the Sichuan Basin, the Tarim Basin, the Bohai Bay Basin, the Ordos Basin, etc. (Zhu et al., 2004). For example, more than two-thirds of the gas fields in the Sichuan Basin contain H₂S, and the highest fraction concentration of H₂S exceeds 30% of the total volume. Carbonate reservoirs of the Tazhong area in the Tarim Basin contain H₂S, which are generally medium-high sulfur content natural gas, and the highest fraction concentration of H₂S exceeds 45% of the total volume. The maximum concentration of H₂S in the Zhaolanzhuang gas field located in the Bohai Bay Basin reaches 92% of the volume fraction (Yuan, 2006).

Table 1. Distribution of high-sulfur gas fields in the world.

	H2S (or maximum)		H2S (or maximum)
Sour gas reservoir	volume fraction (%)	Sour gas reservoir	volume fraction (%)
Zhaolanzhuang, China	62.0–92.0	Devonian, Canada	10.4
Wolonghe, China	12.8–17.1	Crossfield, Canada	34.4
Pugang, China	8.0–17.0	Leduc, Canada	53.5
Luojiazhai, China	7.1–13.7	Lacq, France	15.0
Zhongba, China	6.75–13.3	Sudoldenburg, Germany	20.5
Tazhong, China	45	Buchhoorst, Germany	4.8
Josephine, USA	78.0	Astrakhan, Russia	26.0
Smackover, USA	98.0	Arab, Saudi Arabia	37.0
Okotoks, Canada	15.4	Masjld Soliman, Iran	15.7

A kick happens when formation fluids invade the wellbore owing to the difference between pore pressure and annulus pressure (Vajargah & Van Oort, 2015). Sour gas reservoirs are typically located in marine carbonate formations and the formation conditions are complex with narrow safety pressure windows. Thus, kicks readily occur during the exploration of sour gas reservoirs, especially those with high H₂S content, which commonly causes complicated incidents. The phase transition of H₂S might change significantly because annulus temperature and pressure decline near the wellhead. Simultaneously, there is a great change in the sour gas solubility. Thus, complex two-phase flow phenomena in annuli after sour gas kick might lead to serious well control incidents. Multiple severe blowouts have occurred over the past few decades. A deadly blowout of pure H₂S was determined in the Zhao-48 well of the Bohai Bay Basin, China. In 1999, an underground blowout incident occurred in the Luojia2 well with a high H₂S concentration. In 2003, the Chongqing ‘12.23’ gas blowout was confirmed to be partially related to H₂S phase change, when the Luojia 16H well was being drilled (Dou et al., 2013; Huse & He, 2010). A sour gas blowout occurred in the Larq-3 well, which was followed by explosions and fires and was only controlled after 53 days. In Texas, USA, more than 200 blowout incidents were determined in the past decade, including 28 sour gas cases with a high H₂S content. Similar well control incidents of sour gas were reported in Canada, the Middle East and Russia (He, 2008). Therefore, two-phase transient flow and well control analysis during a sour gas kick have become critical for drilling operations in sour gas reservoirs with high H₂S content.

As is known, serious well control incidents could result from sour gas influx with high H₂S content. There is a lack of study of two-phase transient flows with high H₂S fractions in sour gas influx. In most existing two-phase flow models describing gas kick, only conventional natural gas and mud are considered in the analysis of wellbore flow and well control (Ambrus et al., 2015; Avelar et al., 2009; Choe et al., 2005; Ghalandari et al., 2019; Lage et al., 2003; Nickens, 1987; Perez-Tellez et al., 2003; Salih et al., 2019; Santos, 1991; Udegbunam et al., 2015; Xu et al., 2018), whereas the flow characteristics of sour gas with high H₂S content are significantly different from those of conventional gases. Therefore, these research results cannot benefit well control operations as sour gas invades. Limited research has been conducted on sour gas influx when it is mixed with H₂S. Few publications deal with kick or well control in the case of sour gas or pure H₂S (Dou et al., 2012, 2013; Shi et al., 2008; Sun et al., 2012, 2013; Yuan, 2006; Zhang et al., 2010). Previous research on this topic does not integrate the influx rate with reservoir seepage flow, which was considered to be a constant from sour gas reservoir to annulus. There is a lack of quantitative evaluation of the major factors that influence wellbore flow and well control safety. Furthermore, the effect of heat transfer, sour gas solubility in mud and the H₂S phase transition in the process of invaded mixed fluid traveling upwards have been neglected. Sun et al. (2012, 2013) and He et al. (2017) have included the solubility of H₂S in the study of fluid flow within wellbores, whereas other factors were not considered such as two-phase flow heat transfer in wellbores and reservoir–wellbore coupling. He et al. (2017) only simulated sour gas kick during Managed Pressure Drilling (MPD) operations, which is different from sour gas influx during conventional drilling. For supercritical CO₂ influx, Dou et al. (2019, 2021) studied the mechanisms of CO₂ phase transition along with transient flow. The physical properties, flow behavior and phase characteristics of H₂S and CO₂ are very different. Only pure CO₂ fluid was studied in their research.

After sour gas of high H₂S fraction invades the wellbore, H₂S dissolves in the water-based mud and subsequently changes from liquid to gas as the invaded fluid flows up through the wellbore. The annulus temperature and pressure of two-phase mixed fluids interact with each other. The properties and flow parameters of wellbore fluid vary significantly with the temperature and pressure changes. Therefore, based on three governing equations (continuity, momentum and energy conservation), this research article aims to establish a two-phase flow model under conditions of transient and non-isothermal states in wellbores to simulate sour gas with high H₂S content kick. Transient flow characteristics, sour gas properties and the phase change of H₂S could be investigated with the proposed model. Annulus pressure variation and well control risk could be analyzed in such research. Also, the effects of some of the main operating parameters of kick on the Bottom-Hole Pressure (BHP) and flow behavior during H₂S and CH₄ influx could be evaluated. This research could provide theoretical fundamentals and technical support during sour gas reservoir drilling.

2. Mathematical model development

2.1. Model hypothesis

Figure 1 demonstrates a schematic diagram of the transient flow process for two-phase fluid and heat transfer during high H₂S content sour gas influx. The governing equations are restricted to the following major assumptions.

1. The fluids are assumed to move in a one-dimensi-onal way in the wellbore, not considering radial flow.

2. The invaded fluid Q_i is assumed as the source term of the energy conservation equations.

3. We neglect heat transfer phenomena between the gas and the liquid.

4. It is assumed that drilling cuttings do not influence heat transfer and fluid flow within the annulus.

5. The properties of drilling mud are considered to remain constant within wellbores.

Figure 1. Schematic of two-phase flow during high H₂S content sour gas influx.

2.2. Flow and energy equations for transient flow in wellbores

Drilling fluids circulate normally within the wellbore before sour gas enters, as shown in Figure 1. This circulation process includes the following segments. First, the drilling fluid flows downwards within the drill string from Q(Z) to Q(Z + dZ) reaching the drill bit. After circulation out of the drill bit, the drilling fluid flows upwards within the annulus from Q(Z + dZ) to Q(Z). The drilling fluid circulates out of the annulus to the surface.

The flow through the wellbore during high H₂S content sour gas influx consists of the following three parts.

1. High H₂S content sour gas enters into the bottom hole.

2. The invaded sour gas mixes with the drilling fluid and then they move up together.

3. Only drilling fluid without gas invasion flows in the uppermost section of the annulus.

The continuity and momentum equations of part (1) are shown in Equations (1) and (3), respectively. There is a lack of a sour gas influx q_i in part (2) in comparison to part (1). part (3) is a simple drilling fluid single-phase flow.

2.2.1. Continuity conservations

Equation (1) is the continuity conservation equation of free gas, liquid and dissolved gas. Equation (3(4) $Q(Z+\text{d}Z)-Q(Z)=Q_{AD}-Q_F.$(4) ) is the momentum conservation equation of the liquid phase and the free gas phase. In the liquid phase, the dissolved gas is mixed up with the drilling fluid. (1) $\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\begin{array}{l}A{E}_g{\rho }_g\\ A{E}_s{\rho }_s\\ A{E}_d{\rho }_d\end{array}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\begin{array}{l}A{\rho }_g{v}_g{E}_g\\ A{\rho }_s{v}_s{E}_s\\ A{\rho }_d{v}_d{E}_d\end{array}\right)& =\left(\begin{array}{l}{q}_i-q_s\\ q_s\\ 0\end{array}\right)\end{array}$(1) (2) $\begin{array}{rl}{E}_g+E_s+E_d& =1\end{array}$(2) where A is annulus area (m²); E is the volume fraction of drilling fluids (dimensionless); $t$ represents time (s); $\rho$ is the fluid density (kg/m³); $z$ represents the axial position (m); $v$ is the superficial velocity (m/s); $q_i$ is the gas influx rate per unit length (kg/m·s); and $q_s$ is the dissolved amount of gas influx per unit length (kg/m·s). The subscript $g$ is free gas, $s$ is dissolved gas and $d$ is drilling fluid.

2.2.2. Momentum conservation equations

Equation (3) shows the momentum conservationequation: (3) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\begin{array}{l}{E}_g{\rho }_g{v}_g\\ E_s{\rho }_s{v}_s+E_d{\rho }_d{v}_d\end{array}\right)\\ & +\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\begin{array}{l}{E}_g{\rho }_g{v}_g^2+p+|dp{|}_{fr}\\ E_s{\rho }_s{v}_s^2+E_d{\rho }_d{v}_d^2+p+|dp{|}_{fr}\end{array}\right)\\ & +\left(\begin{array}{l}g\cos \theta (E_g{\rho }_g)\\ g\cos \theta (E_s{\rho }_s+E_d{\rho }_d)\end{array}\right)=0,\end{array}$(3) where $g$ represents the gravitational constant, 9.81 m/s² and $\theta$ represents the inclination angle with reference to the vertical direction (°). The subscript $fr$ represents friction pressure.

In this research, we apply the mechanistic model of Perez-Tellez et al. (2003) to estimate the pressure gradient of gas–liquid two-phase flow.

In addition, we apply the friction factor expression proposed by Chen (1979), which responds to the relative roughness and the Reynolds number.

2.2.3. Energy conservation equations

In Figure 1, the heat source of micro unit dZ in the annulus consists of two parts: the heat inflow Q(Z + dZ) from the lower annulus and the heat inflow Q_F from the formation. The heat loss of micro unit dZ also consists of two parts: the heat outflow Q(Z) from the annulus upside and heat Q_AD transfer from annulus to drill string. Note that the heat source Q_i brought by the sour gas invasion should be considered in the bottom hole. The following equations are derived based on the energy conservation equation.

Energy conservation equation for annulus fluid flow is given by Equation (4): (4) $Q(Z+\text{d}Z)-Q(Z)=Q_{AD}-Q_F.$(4) The invaded fluid Q_i from sour gas reservoirs into the annulus is considered as the additional heat source in the bottom hole, as shown in Equation (5): (5) $Q(Z+\text{d}Z)-Q(Z)=Q_{AD}+Q_i-Q_F.$(5) The steady-state energy equation of two-phase flow within the annulus is deduced as shown in Equation (6): (6) $\frac{\text{d}}{\text{d}z}\left[A\rho v\left(h+\frac{v^2}{2}-g\text{d}z\cos \theta \right)\right]=Q_{AD}-Q_F,$(6) where h represents the specific enthalpy (kJ/kg).

Similarly, the steady-state energy equation of two-phase flow in the bottom hole is given by Equation (7): (7) $\begin{array}{r}\frac{\text{d}}{\text{d}z}\left[A\rho v\left(h+\frac{v^2}{2}-g\text{d}z\cos \theta \right)\right]=Q_{AD}+Q_i-Q_F.\end{array}$(7) Therefore, the steady energy equation for single-phase flow within the drill string is given by Equation (8): (8) $\frac{d}{dz}\left[A_d\rho v\left(h+\frac{v^2}{2}+gdz\cos \theta \right)\right]=Q_{AD}$(8) where $A_d$ is the drill string area (m²).

We apply the unsteady-state heat transfer equation for Q_AD and the steady-state heat transfer equation for Q_F proposed by Kabir et al. (1996).

The research results summarized by Dou et al. (2013) and Abadi et al. (2020) on two-phase flow heat transfer coefficients for different flow patterns, such as bubble flow, dispersed bubble flow, slug flow, churn flow and annulus flow, are used in this article.

2.3. Auxiliary equation

2.3.1. Physical properties of H₂S and high H₂S content sour gas

H₂S is generally in the liquid state in gas reservoirs, which is caused by a high formation pressure and temperature. H₂S changes from the liquid state to the gaseous state owing to the decline in wellbore pressure and temperature when the mixed fluid flows up along the wellbore as shown in Figure 2. H₂S might become supercritical at deep or ultra-deep depths because the annulus temperature and pressure exceed the critical states of H₂S, as shown in Figure 2. The critical point of H₂S is about 100.22°C (373.37 K) temperature and 8.963 MPa pressure. In this case, H₂S initially has supercritical status, which is subsequently changed to liquid as it flows upwards and finally to gas in the wellhead area.

Figure 2. Phase state diagram of H₂S. The black line illustrates H₂S phase transition caused by change of annulus temperature and pressure as the invaded H₂S travels up along the wellbore.

2.3.2. Equation of state for H₂S

For pure or mostly pure H₂S, this article uses the Helmholtz free energy Equation Of State (EOS) for H₂S proposed by Sakoda and Uematsu (2004) (SU-EOS). SU-EOS is a function of reduced temperature and density with 23 terms on the basis of selected measurements. SU-EOS consists of the ideal-gas state contribution, ${\mathrm{\Phi }}^o(\tau ,\delta )$, and the residual real fluid contribution, ${\mathrm{\Phi }}^r(\tau ,\delta )$, and is expressed by Equation (9): (9) $\mathrm{\Phi }(\tau ,\delta )=\frac{a}{RT}={\mathrm{\Phi }}^o(\tau ,\delta )+{\mathrm{\Phi }}^r(\tau ,\delta ),$(9) where $\mathrm{\Phi }(\tau ,\delta )$ is the Helmholtz energy (dimensionless); $\tau =T_c/T$ is the inverse reduced temperature (dimensionless); $\delta =\rho /{\rho }_c$ is the reduced density (dimensionless); $T$ represents temperature (K); $\rho$ is the fluid density (kg/m³); $T_c$ is the critical temperature of H₂S (K); and ${\rho }_c$ is the critical density of H₂S (kg/m³). (10) $\begin{array}{rl}{\mathrm{\Phi }}^o(\tau ,\delta )& =\ln (\delta )+f_1+f_2\tau +f_3\ln (\tau )\\ & +\sum _{i=4}^5{f}_i\ln \{1-exp(-g_i\tau )\}\end{array}$(10) (11) $\begin{array}{rl}{\mathrm{\Phi }}^r(\tau ,\delta )& =\sum _{i=1}^{11}{n}_i{\tau }^{t_i}{\delta }^{d_i}+\sum _{i=12}^{16}{n}_i{\tau }^{t_i}{\delta }^{d_i}{e}^{(-\delta )}\\ & +\sum _{i=17}^{19}{n}_i{\tau }^{t_i}{\delta }^{d_i}{e}^{(-{\delta }^2)}+\sum _{i=20}^{21}{n}_i{\tau }^{t_i}{\delta }^{d_i}{e}^{(-{\delta }^3)}\\ & +\sum _{i=22}^{23}{n}_i{\tau }^{t_i}{\delta }^{d_i}{e}^{(-{\delta }^4)},\end{array}$(11) where $f_1$, $f_2$, $f_3$, $f_i$, $g_i$, $n_i$, $t_i$ and $d_i$ are non-analytic coefficients (dimensionless).

The SU-EOS is valid for temperatures from the triple point temperature (187.67 K) to 760 K at pressures up to 170 MPa based on selected experimental data for PρT, caloric and saturation properties. The predicted results with smooth behavior calculated by the model are verified by experimental thermodynamic property data of H₂S in the fluid phase. Taking H₂S density as an example, the maximum uncertainty of density is below 0.7%.

This research uses the prediction equation for viscosity and thermal conductivity of H₂S that was proposed by Chung et al. (1988) and summarized by Reid et al. (1987).

2.3.3. Properties equations of high H₂S content sour gas

Sour gas compressibility factors were calculated by combining the Dranchuk–Purvis–Robinson (DPR) model and the Dranchuk–Abu–Kassem (DAK) model along with the Wichert–Aziz (WA) method (Guo et al., 2008). Additionally, a fully predictive method due to Galliero et al. (2010) was applied for the calculation of sour gas viscosity.

2.3.4. Solubility of H₂S and CH₄

Currently, water-based drilling fluids are usually used for sour gas reservoirs. Water-based drilling fluids are mainly made up of fresh water or brine, bentonite and a small amount of other chemical agents. In this drilling fluid, fresh water or salt water are the most important components. Therefore, a thermodynamic model (Duan et al., 2007) calculating the solubility of H₂S with high prediction accuracy in pure water and in aqueous NaCl solutions (0–6 M, 273–500 K, 0–200 bar) is adopted to analyze the dissolution characteristics of H₂S and CH₄ in water-based drilling fluids. Compared with the experimental data, the prediction uncertainty of H₂S solubility from this model is below 7%.

As is known, CH₄ is the major part of conventional natural gas. For differentiating the dissolution characters between H₂S and CH₄, the previously published model of Duan and Mao (2006) is used to calculate CH₄ solubility in pure water and aqueous NaCl solutions. The model is valid for the temperature range 273–523 K, pressure up to 200 MPa and salt concentration range 0–6 mol·kg⁻¹. The approximate uncertainties of properties calculated with the model are verified by experimental data.

Figures 3 and 4 show the calculated solubility of H₂S and CH₄ in pure water using the Duan's model. The solubility amount of H₂S is much more than that of CH₄, which could put at risk well control operation.

Figure 3. Calculated H₂S solubility in pure water using the Duan model.

Figure 4. Calculated CH₄ solubility in pure water using the Duan model.

2.3.5. Formation percolation model

In this article, the sour gas influx rate was estimated from a model proposed by Zeng and Zhao (2008) that used the Forchheimer number to quantify non-Darcy flow accurately in gas reservoirs.

2.3.6. Slip equation

The velocities differ between the gas phase and the liquid phase, which is caused by gravitational effects, referred to as a ‘slip velocity’. The free gas rising velocity proposed by Zuber and Findlay (1965) is presented in Equation (12(13) $V_{dis}={10}^{-6}{S}_{\text{H2S}}{Q}_d{\rho }_{d}M,$(13) ): (12) $v_{tg}=c_o(E_s{v}_s+E_d{v}_d+E_g{v}_g)+v_{gr},$(12) where $v_{tg}$ is the gas flow velocity (m/s); $c_o$ is the profile parameter (dimensionless); and $v_{gr}$ is the gas slippage velocity (m/s).

There are many research articles on the profile parameter and slippage velocity. In this article, we use the relationship of $c_o$ and $v_{gr}$presented by Hasan et al. (2010).

3. Model solution and validation

3.1. Initial conditions

Sour gas of high H₂S fraction invades the annulus, which changes the single steady-state flow into transient two-phase flow. From the single steady-state flow model and the heat transfer state model for a single drilling fluid within the annulus, the annulus pressure and temperature are determined as the gas influx occurs. Initially, the gas velocity and the gas volume fraction of free gas and dissolved gas equal zero; the volume fraction of the drilling fluid equals one. The drilling fluid velocity is calculated from the flow rate over its annual cross-section area.

3.2. Boundary conditions

Some inlet boundary parameters are known: the wellhead back pressure or the outlet pressure $p_c$, the drilling fluid flow rate or the pump rate $q_{df}$ and the injection temperature $T_{inj}$. The pore pressure of the sour gas reservoirs remains constant when kick occurs.

3.3. Solution method

The two-phase flow and heat transfer of the gas–liquid phase are calculated along with mixed fluid properties. There are no closed form solutions for the Equations (1)–(8), which are nonlinear. Thus, the nonlinear equations are solved by a finite difference method with an implicit algorithm as shown in Figure 5. The difference format has second-order accuracy to meet the simulation requirements. Based on the slip equation, the sour gas front is tracked in order to divide the interface between the two-phase flow and the single-phase flow. Simulations are run on well depths from the bottom to the top of the wellbore.

Figure 5. Demonstration of the pace mesh and the time step for finite difference equations.

It is worth noting that the physical properties of drilling fluids, such as density, viscosity, etc. are constant during the sour gas invasion. The dissolved gas velocity is equal to the drilling fluid velocity. The sum of the two velocities is the liquid phase velocity.

The flow quantities within the wellbores are determined at time n. The solution procedures are given for the transient flow during sour gas influx from node i to node i + 1 in a time range n to n + 1.

(1) At time n + 1, assumptions of the BHP $p_0^{n+1}$ and bottom-hole temperature $T_0^{n+1}$ are given and the influx rate of the sour gas $q_i$ is calculated based on the formation percolation model.

(2) At node i + 1 and time n + 1, the pressure and temperature are assumed to be $p_{i+1}^{n+1(0)}$ and $T_{i+1}^{n+1(0)}$, respectively. The physical parameters of the sour gas are then calculated by the EOS.

(3) At node i + 1 and time n + 1, the volume fraction of the sour gas is assumed to be ${\alpha }_{i+1}^{n+1(0)}$. The velocities of liquid and free gas are calculated by the solubility equation and the continuity conservation equation.

(4) Re-calculate the volume fraction of the sour gas ${\alpha }_{i+1}^{n+1}$ by the drift-flux model. If $|{\alpha }_{i+1}^{n+1}-{\alpha }_{i+1}^{n+1(0)}|<\epsilon$ then go to the next step, or otherwise return to step (3).

(5) A new pressure $p_{i+1}^{n+1}$ is calculated based on the momentum conservation equation using existing parameters. A new temperature $T_{i+1}^{n+1}$ is calculated based on the energy conservation equations. If $|p_{i+1}^{n+1}-p_{i+1}^{n+1(0)}|<\epsilon$ and $|T_{i+1}^{n+1}-T_{i+1}^{n+1(0)}|<\epsilon$ and flow quantities at node i + 1 are present, continue to calculate the next node i + 2. If not, return to step (2) until the calculation accuracy is high enough.

(6) Continue the calculation using the previous solution procedures until the node reaches the wellhead. Using boundary conditions (the wellhead back pressure $p_c$ at the exit of the annulus and the drilling fluid injection temperature $T_{inj}$ are known) to judge its convergence. If $|p_c-p_c^{\ast }|<\epsilon$ and $|T_{inj}-T_{inj}^{\ast }|<\epsilon$ and the flow quantities at time n+1 as known amounts, continue to calculate the next time i + 2. Otherwise return to step (1).

3.4. Model validation

Sour gas of high sulfur fraction was added to the existing two-phase flow experiments, which had not been studied before owing to its high toxicity. In the two-phase flow experiments, air or N₂ and water were mostly combined, followed by a less-used air/kerosene combination. Previously, flow experiments in wellbores coupled with reservoirs were not observed to realize the dynamic two-phase transient flow test. We therefore compared our model with quasi-transient state flow influx experiment results at constant flow rate reported by Lage and Time (2002), who performed full-scale experiments in a vertical research wellbore. This vertical well consisted of a 177.8 mm Outside Diameter (OD) casing (Inner Diameter (ID) = 159.4 mm) and an 88.9 mm OD drilling string (ID = 70.2 mm), which was placed at the bottom of the well to a depth of 1275 m. Compressed N₂ and fresh water were pumped to generate the two-phase mixtures. The gas rate ranged from 0.142 to 0.472 m³/s, and the liquid rate ranged from 0.002,52 to 0.003,78 m³/s.

In comparison to experiments, the simulations set a constant invasion rate of two-phase mixtures that comprised CH₄ and fresh water. In contrast, transient flow in the annulus coupled with gas reservoirs seepage flow was also simulated. Figure 6 shows a comparison between the simulated results and the experimental data for the annulus pressure. Figure 7 shows a comparison between the measured data and the simulated results for the annulus pressure varying with rising time. From the two figures, the simulated results have a better agreement with the experimental data. However, the annulus pressure in the simulated results for transient flow is significantly lower than measured data, and the pressure drops faster in the simulated results for transient flow. This is because it had been assumed that the gas injection velocity was constant in the transient flow experiment (the triangular points in Figures 6 and 7) and simulation (the red curves in Figures 6 and 7). The gas invasion speed increases with the decrease of bottom-hole pressure in the simulation. Thus the annulus pressure is lower than the measured data, and the pressure declines faster. This phenomenon agrees with the real case of wellbore overflow. Hence, the investigation of transient flow coupled with gas reservoir seepage flow in wellbores is a crucial step for better well control and design, especially for high H₂S content sour gas kicks.

Figure 6. Demonstration of the annulus pressure from measurements and simulations.

Figure 7. Demonstration of the annulus pressure from measurements and simulations, which vary with rising time.

4. Calculations and results

Transient flow and well control were analyzed using the proposed model, which was based on the field drilling parameters of the G44–9 well in the Ordos Basin, China. Table 2 is a group of basic parameters applied to simulations for these cases.

Table 2. Basic data of well G44–9.

Basic parameters	Value	Basic parameters	Value
Well depth	2100 m	Initial pressure difference	2 MPa
Surface temperature	20°C	Formation permeability	10 mD
Back pressure	0.1 MPa	Geothermal gradient	0.036°C/m
PDC bit size	200 mm	Formation thermal conductivity	2.25 W/(m·°C)
Drill string ID	101.6 mm	Formation heat capacity	837 J/(kg·°C)
Drill string OD	127 mm	Formation density	2.64 g/cm^3
Drill collar length	120 m	Steel thermal conductivity	43.75 W/(m·°C)
Drill collar ID	76 mm	Steel heat capacity	400 J/(kg·°C)
Drill collar OD	165.1 mm	Steel density	7.8 g/cm^3
Drilling fluid density	1.2 g/cm^3	Tube roughness	0.045 mm
Drilling fluid viscosity	0.03 Pa.s	Roughness of borehole wall	2.5 mm
Drilling fluid heat capacity	3300 J/(kg·°C)	Circulation time	60 mins
Mud thermal conductivity	1.15 W/(m·°C)	Drainage radius of gas reservoir	300 m
Drilling fluid flow rate	30 L/s	Gas reservoir open thickness	5 m
Injection temperature	20°C	Skin factor	1.0

4.1. Transient flow behavior in wellbores

Figure 8 shows the annulus temperature distribution when the gas influx moves up to the wellhead. This is calculated by the proposed numerical solution for high H₂S content sour gas and CH₄ influx. The highest temperature is not at the bottom of the well. The temperature of drilling fluid is relatively low while it flows from the inside of drilling bits, and this temperature will significantly increase when it encounters the high-temperature formation, and the fluid temperature in the annulus will thus increase as well. As the drilling fluids and invaded gas flow upwards, the formation temperature will gradually decrease, which leads to the reduction of fluid temperature in the annulus. The heat transfer from annulus fluid to formation occurs while the fluid temperature is higher than the formation temperature in the upper part of the annulus. Thus, the temperature of the annulus mixed fluids is generally lower than the earth temperature, but it is higher than the earth temperature when the mixed fluids are near the wellhead. The annulus temperature slightly increases with the decrease of H₂S fraction of sour gas as the mixed fluid flows upwards to the wellhead. The differences of annulus temperature are probably caused by the increasing heat from the greater invaded fluid rate, and less exchange time due to a larger annulus flow rate when the H₂S content in the sour gas decreases.

Figure 8. Annulus temperature distribution for acid gas with different H₂S content when the invaded gas moves up to the wellhead.

Figure 9 illustrates the annulus pressure distribution generated by sour gas with different H₂S concentrations when the invaded gas front arrives at the wellhead. The wellbore pressure profile without gas intrusion is also given for comparison. The annulus pressure increases with increasing fraction of H₂S in the sour gas. As the H₂S concentration increases, the free gas mass rate decreases owing to the greater H₂S solubility and the increase in the density of free gas. These factors lead to an increase in the pressure gradient of the two-phase flow as the H₂S fraction increases. It is worth noting that a phase transition of H₂S occurs near the wellhead owing to the decline of temperature and pressure as shown in Figure 2. This phenomenon of phase transition is observed in Figure 11. When the pure H₂S moves up along the wellbore, a single-phase flow occurs before the occurrence of the phase change, followed by a two-phase flow after the phase transition near the wellhead. This transition results in a significant drop of annulus pressure as shown in Figure 9.

Figure 9. Annular pressure distribution for acid gas with different H₂S content when the invaded gas moves up to the wellhead.

Figure 10 illustrates the annulus pressure profiles of sour gas with different H₂S concentrations. At any depth above 700 m, the wellbore pressures are the same because only the drilling fluid flows. Below this depth, the wellbore pressures increase with increasing H₂S concentrations, which is similar to the case in Figure 9. From Figure 10, the annulus pressure during pure H₂S gas influx remains almost constant, compared with that without gas influx. This difference indicates that the identification of pure H₂S influx is difficult at the initial stage of kick.

Figure 10. Annular pressure distribution for acid gas with different H₂S content when the invaded gas front rises to a depth of 700 m from the wellhead.

Figure 11 shows the density distribution of the invaded fluids when the gas influx front arrives into the wellhead, which is generated from different H₂S concentrations of gas influx. Pure H₂S has a significantly larger density than pure CH_4. Natural gas contains different concentrations of H₂S before the H₂S phase transition occurs. The small variation of annulus pressure in Figure 8 is caused by the large density of H₂S shown in Figure 11, which leads to a significant increase in well control risk in the early stage of kick. As the invaded H₂S continues to flow upwards, with a decrease of temperature and pressure, the H₂S density decreases significantly from 760 kg/m³ to less than 30 kg/m³, but the H₂S volume increases dramatically by more than 27 times near the wellhead after the H₂S phase transition from liquid phase to gaseous phase in Figure 2. Figure 9 demonstrates a significant decrease of the pressure within the wellbore. Sour gas with an H₂S content of 10 or 30% has a similar density to that of the pure CH₄. This is mainly due to the dissolution of the majority of H₂S in the sour gas to the mud, which causes the major component of the free sour gas changing to be CH₄.

Figure 11. Annular fluid density profiles of acid gas with different H₂S content when the invaded gas front rises into the wellhead.

Figures 12 and 13 demonstrate the quantity distributions of free gas and dissolved gas under different H₂S concentrations. From Figure 12, the free H₂S quantity is zero in most well sections, except at the bottom of the wellbore and near the wellhead. The zero quantity of free H₂S in the wellbore sections is due to the intrusive fluid rate being less than or equal to its solubility. The free H₂S quantity in the uppermost section near the wellhead is greater than zero because the H₂S solubility decreases, as shown in Figure 13, and the dissolved gas transitions to free gas. The risk to well control is thus increased with the sudden release of H₂S. The free H₂S quantity in the lowermost section at the bottom hole is greater than zero because the H₂S intrusion flow rate increases beyond its solubility. Figure 12 also illustrates that the mass flow rates of the free CH₄ and the sour gas of different H₂S concentrations are large. The major component of sour gas is CH₄, the solubility of which is much smaller than that of H₂S, as shown in Figure 13. Thus, the quantity of free CH₄ and sour gas is significantly large.

Figure 12. Annular mass flow of free gas varying with different H₂S content in acid gas when the invaded gas moves up to the wellhead.

Figure 13. Annular dissolved quantity of invaded fluid varying with different H₂S content in acid gas when the invaded gas moves up to the wellhead.

The dissolved quantity of H₂S in the invaded fluid, V_dis, is given based on the H₂S solubility formula proposed by Duan et al. (2007). The formula for calculating the dissolved quantity of CH₄ is given in Equation (13): (13) $V_{dis}={10}^{-6}{S}_{\text{H2S}}{Q}_d{\rho }_{d}M,$(13) where V_dis is the H₂S dissolved quantity (kg/s); S_H2S is the H₂S solubility in the NaCl solutions (mol/kg); and M is the molar mass of H₂S (34 g/mol).

When the invaded quantity is less than the dissolved quantity, the quantity of free gas equals zero. If the quantity of gas influx is greater than the dissolved quantity, the free gas quantity equals the invaded gas quantity minus the dissolved quantity.

Figure 13 presents a much larger dissolved quantity of gas influx containing H₂S than that of pure CH₄ in the drilling fluids, which implies that the sour gas overflow is not easy to discover early. The dissolved quantity of sour gas with a 30% fraction of H₂S is more than that with pure H₂S, at the lower part of the wellbore. The reason is that the BHP for sour gas containing 30% H₂S is lower than the BHP for pure H₂S. Figure 13 illustrates that the dissolved quantity of H₂S rapidly reduces when the sour gas moves up to the wellhead, where the annulus pressure has a sharp decline caused by the partially dissolved H₂S being converted into free gas.

Figure 14 shows the distribution of the gas phase fraction over the whole well depth as the invasion front is at the wellhead. At any depth with a lack of H₂S phase change, the H₂S void fraction is zero as the H₂S is in the liquid state instead of gas. Above this point, H₂S is transformed from liquid to gas. The void fraction of the free H₂S gas remains at a low level owing to a smaller invasion rate and a relatively large solubility. As shown in Figure 14, the void fraction decreases with increasing concentration of H₂S. This is largely due to the fact that, the higher concentration of H₂S, the larger the dissolved quantity of invaded gas. In Figure 14, the sour gas void fraction in the intermediate part is less that that close to the wellhead. This is due to the considerable volumetric expansion and the decline of solubility, which are generated by the annulus pressure decrease. For the base section, the sour gas void fraction is larger due to the higher influx rate.

Figure 14. Void fraction of free gas as invaded gas moves up to the wellhead.

4.2. Well control analysis

The change of BHP is critical for, and might put at risk, well control operations, and special attention should be paid to this during drilling acid reservoirs. Therefore, BHP variation with increasing gas column and kick time are shown in Figures 15 and 16, respectively.

Figure 15. BHP varying with invaded sour gas rising height from the bottom hole.

Figure 16. BHP varying with kick time.

Figure 15 shows that BHP changes with increasing rise in height of gas from the wellbore bottom by surveying gas for varying concentrations of H₂S. In all cases, the BHP decreases with increasing height of the invaded fluid front. As the H₂S concentration increases, the decline amplitude of BHP significantly decreases. When the invaded pure H₂S flows upwards, the BHP remains almost constant at first and this is because the H₂S is dissolved in the liquid phase with a higher density. However, it drops significantly after the occurrence of the H₂S phase transition near the wellhead as shown in Figure 11. The sudden drop of the annulus pressure near the wellhead allows a very short operation time window for well control, increasing the risk to well control.

Figure 16 presents that BHP varies with kick time, i.e. the invaded gas ascending time during sour gas kicks with varying H₂S fractions. With increasing kick times, the BHP decreases substantially for all cases. The decrease amount of BHP becomes significantly more for CH₄ and sour gas containing H₂S (10 and 30% fractions) influx. The total time of kicking through all the wellbore significantly decreases for CH₄ and sour gas containing H₂S, which is less than 1000 s in contrast to pure H₂S influx. The BHP drops rapidly initially, which is easily detected when CH₄ and sour gas invade into the wellbores. An increase in the H₂S fraction of sour gas increases the kick time.

For pure H₂S influx, the total kick time is evidently longer than 1200 s, and for the initial 1100 s the BHP is almost constant. The BHP drops sharply by more than 5 MPa within 180 s near the wellhead. The shut-in operation seems not possible in such a short period of time, which results in a blowout.

As the H₂S phase transition occurs near the wellhead, significant volumetric expansion is determined in Figure 11 and a sharp BHP decrease is determined from Figures 15 and 16. These changes significantly increase the risk to well control. The operation response time for well control is so short that serious well control incidents may be induced. Therefore, much attention should be paid to this situation.

The main parameters affecting well control safety, especially BHP during the formation H₂S and CH₄ influx, are analyzed in order to optimize drilling designs and well control solutions. In this subsection, geological and engineering factors dependently affecting BHP are studied, including wellhead back pressure, drilling fluid flow rate, the pressure difference between the formation and the wellbore, reservoir open thickness and drilling fluid injection temperature. They are studied changing only one parameter at a time around the normal operating parameters, which are presented in Table 2. Their different influences on BHP could be inferred explicitly.

4.2.1. Wellhead back pressure

Figure 17 illustrates the BHP curves during the influx of H₂S and CH₄ with different wellhead back pressures (0.1, 1, 2, 3, 4 MPa), when the initial pressure difference remains constant. BHP increases significantly as the wellhead back pressure builds up, and BHP increases by 6.0 MPa for H₂S invasion and by 7.6 MPa for CH₄ invasion as the wellhead back pressure increases by 3.9 MPa. As shown from the figure, the wellhead back pressure could effectively control the annulus pressure.

Figure 17. BHP variation with different wellhead back pressures.

H₂S phase transition only occurs near the wellhead when the wellhead back pressure remains below 2 MPa, and the density of H₂S drops drastically, as shown in Figure 18. No phase transition occurs when the wellhead back pressure is no less than 2 MPa. Therefore, the wellhead back pressure dictates the H₂S phase transition. A wellhead back pressure of more than 2 MPa is recommended in the process of drilling sour gas reservoirs, which aims to restrain the sudden expansion of sour gas near the wellhead.

Figure 18. Annular density profiles for H₂S influx varying with wellhead back pressure.

4.2.2. Initial pressure difference

The initial pressure difference equals the reservoir pore pressure minus the BHP. Figure 19 shows the BHP curves during the influx of H₂S and CH₄ with different initial pressure differences (0, 1, 2, 3, 4, 5 MPa). The BHP during CH₄ influx first drops significantly, then a minimum is reached, and it slowly increases with increasing initial pressure difference. This difference has a more pronounced influence over the BHP during CH₄ influx than that during H₂S influx, though a similar tendency exists in both cases. The sharp decrease of the initial BHP is caused by the increasing invaded gas rate from the reservoirs, according to the formation percolation model. Later, there is a minor increase of BHP, which is generated by a great increase in the friction pressure within the two-phase flow. This results from a high invasion rate and gas expansion.

Figure 19. BHP variation with different initial pressure differences.

A more accurate prediction of the formation pressure gradient and precise Equivalent Circulation Density (ECD) control, which can reduce the initial pressure difference, have a determining impact on well control safety.

4.2.3. Drilling fluid flow rate and injection temperature

Figure 20 demonstrates the BHP variation curves with different drilling fluid flow rates (20, 25, 30, 35, 40 L/s) at a constant initial pressure difference. This indicates an increase in BHP with an increase in drilling fluid flow rate. The BHP variation is explained by the fact that a larger flow rate contributes to an increase in the annulus area occupied by the drilling fluid. The drilling fluid flow rate has a greater influence on BHP during CH₄ influx than during H₂S influx.

Figure 20. BHP variation with different drilling fluid flow rates.

The drilling fluid injection temperature has little effect on BHP during the gas kick, as shown in Figure 21.

Figure. 21. BHP variation with different drilling fluid injection temperatures.

4.2.4. Reservoir open thickness

Figure 22 presents the BHP curves during H₂S and CH₄ influxes with different reservoir open thicknesses (0, 1, 3, 5, 7, 9 m). During the CH₄ influx, the initial pressure difference influences the trend of BHP, which is similar to the influence caused by the reservoir open thickness. The BHP variation trend is affected by the reservoir open thickness during CH₄ influx, which is similar to the influence generated by the initial pressure difference.

Figure 22. BHP variation with different reservoir open thicknesses.

BHP decreases with increasing reservoir open thickness; the decrease of BHP from H₂S influx is much smaller than that from CH₄ influx.

A sensitivity analysis was conducted to evaluate the factors that influence BHP. This analysis could provide guidelines for drilling operations and well control designs in sour gas reservoirs.

5. Conclusions

This article successfully establishes a two-phase flow model for the transient state and studies sour gas kick having a large H₂S fraction. In simulations, the transient flow is coupled with the formation percolation model; two-phase flow and heat transfer are simulated along with mixed fluid properties. The solubility of H₂S and CH₄ in water-based drilling fluids and the phase transition of H₂S are considered in the research. The model is successfully verified by full-scale experimental data. Well control risk is evaluated considering the geological and engineering factors separately.

(1) As the mixed fluids flow upwards, their annulus pressure increases but the annulus temperature slightly decreases with increasing H₂S fraction in the sour gas.

(2) H₂S changes from liquid to gas as the invaded fluid front moves up along the wellbore. The density of pure H₂S is significantly larger than that of pure CH₄ gas and natural gas having different H₂S concentrations before H₂S phase transition occurs, whereas the H₂S density decreases significantly and its volume increases dramatically by more than 27 times after phase transition occurs. The sudden drop of annulus pressure results from the phase transition of H₂S when it moves close to the wellhead. This pressure drop gives a very short window of opportunity for well control operations, increasing the risk to well control.

(3) The quantity of free H₂S is zero in most wellbore sections, except at the bottom of the wellbore and near the wellhead. The dissolved quantity of invaded gas with H₂S is much more than that with CH₄, which implies that sour gas overflow will not be easily discovered at an early stage of kick.

(4) Below the well depth of the H₂S phase transition, the void fraction of H₂S is zero, which is in a liquid state rather than a gaseous state. Above this depth, H₂S changes from liquid to gas, which implies two-phase flow. An increase in the H₂S fraction of sour gas decreases the void fraction, which is caused by a large quantity of dissolved H₂S.

(5) In generally, BHP decreases for the following two reasons: the invaded fluid front moves up from the bottom of the well as the kick time elapses, whereas pure H₂S has an abrupt decline near the wellhead. As the H₂S content decreases, the declining amplitude of BHP significantly increases and the total kick time reduces.

(6) The proper control of wellhead back pressure can prevent the occurrence of H₂S phase transition and reduce the risk of sour gas kick. A wellhead back pressure higher than 2 MPa is recommended to restrain the sudden expansion of sour gas near the wellhead.

(7) When CH₄ influx occurs, BHP drops sharply, which then reaches a minimum point, but later increases slowly with an increase in the initial pressure difference. There is a greater influence of the BHP from this pressure difference during CH₄ influx than during H₂S influx. There is a similar tendency for both influxes. The trend of BHP variation affected by reservoir open thickness during CH₄ influx is similar to the effect caused by the initial pressure difference. There is a decrease of BHP during H₂S influx with an increase in reservoir open thickness.

(8) BHP increases with a rise in drilling fluid flow rate. The drilling fluid injection temperature has little influence on BHP during a gas kick.

This article does not consider the effect of cutting particles and the rotation of the drill string on the gas–liquid flow. These two factors might impact the recognition of the gas–liquid flow types and the accuracy of pressure drop in multi-phase calculations. There is a lack of previous publications on this topic, which is to be investigated in following research.

Abbreviations

H2S	=	Hydrogen Sulfide
CH4	=	Methane
CO2	=	Carbon Dioxide
MPD	=	Managed Pressure Drilling
BHP	=	Bottom-Hole Pressure
EOS	=	Equation of State
SU-EOS	=	Sakoda and Uematsu Equation Of State
DPR	=	Dranchuk–Purvis–Robinson
DAK	=	Dranchuk–Abu–Kassem
WA	=	Wichert–Aziz
N2	=	Nitrogen
OD	=	Outside Diameter
ID	=	Inner Diameter
ECD	=	Equivalent Circulation Density
PDC	=	Polycrystaline Diamond Compact

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was financially supported by the National Natural Science Foundation of China [Nos 52074221, 52020105001, 51974249].