Seepage–Diffusion Mechanism of Gas Kick Considering the Filtration Loss of Oil-Based Muds During Deepwater Drilling

Abstract

As oil and gas exploration gradually advances into deep waters, the combined effects of various types of gas kick and the accurate calculation of the gas-kick volume have gained increasing attention. This study focused on gas kicks from permeable gas-bearing formations, considering the mass transfer of gas in the filtration region of the drilling fluids and revealed the mechanisms of seepage-driven and diffusion-driven gas kicks. Based on seepage mechanics and diffusion theory, a comprehensive model for calculating gas-kick volume was established, considering the synergistic effect of gas-concentration-diffusion and negative-differential-pressure, as well as mass transfer in both the filtrate zone and the filter-cake zone. The new model showed high calculation accuracy. The sensitivity analysis showed that both the seepage-driven and diffusion-driven gas-kick volumes in the wellbore increased with increasing formation porosity and open-hole length, while the thickness of the filter cake had a strong inhibitory effect on both. Additionally, a “seepage–diffusion ratio” was introduced to reveal the gas-kick evolution pattern under a seepage–diffusion mechanism. Under specific case conditions, when the seepage–diffusion ratio was less than approximately 1%, diffusion-driven gas kick contributed more than seepage-driven gas kick; when the seepage–diffusion ratio exceeded 1%, seepage-driven gas kick contributed more than diffusion-driven gas kick. The research can provide crucial parameters for wellbore multiphase flow calculation and wellbore pressure prediction.

1. Introduction

Drilling operations carry risks to human safety, equipment, procedures, and the downhole environment, all of which can hinder performance. In severe cases, these risks may lead to significant resource loss and increased non-productive time, or even require plugging the well and starting a contingency side-track, adding environmental and economic challenges to the project [1]. One major downhole issue considered a risk event is the gas kick. Gas kick, which refers to formation gas entering the wellbore, is a prevalent problem during a large number of drilling operations [2]. Understanding the mechanism through which formation gas invades the wellbore is fundamental to solving problems such as the calculation of multiphase flow in the wellbore [3], hydrate-bearing sediments in the wellbore [4], and wellbore pressure control [5]. Moreover, kick tolerance is a crucial concept for ensuring drilling safety, especially in natural gas wells, where calculating it is one of the most important requirements in the deep-well design [6]. Kick tolerance can be determined by calculating the maximum allowable gas-kick volume at the bottomhole, making the prediction of gas-kick rate and gas-kick volume essential [7].

The main types of formation gas kicks into a wellbore include gravity-displacement gas kicks, negative-differential-pressure-driven gas kicks (seepage-driven gas kick), diffusion-driven gas kicks, and gas release from drill cuttings [8,9,10]. When drilling through a single vertical fracture or multiple connected fractures, due to the density difference between formation gas and drilling fluid, gas invades the wellbore from the upper part of the fracture, while the drilling fluid invades the formation from the lower part of the fracture. This phenomenon is known as gravity-displacement gas kick, which typically occurs in fractured formations. Shu et al. [11] and Li et al. [12,13] studied the co-existence of lost circulation and blowouts in fractured formations, deriving relationships for the drilling fluid loss under different conditions. However, they did not provide a correlation formula for gas-kick rate. Zhang et al. [14,15] assumed a constant gas-kick rate from gravity displacement and found that it is related to the porosity and permeability of the reservoir, as well as the length and radius of the open-hole section.

$$Q=f\left(K,\phi ,\alpha ,r_w\right)=\mathrm{constant}.$$

(1)

Negative-differential-pressure-driven gas kick is the phenomenon in which formation gas enters the wellbore when the bottomhole pressure is lower than the formation pressure, which is commonly referred to as seepage-driven gas kick. Therefore, its models are all based on seepage theory [16,17,18,19,20,21]. For example, Rommetveit [8] was one of the earliest to conduct field tests on seepage-driven gas kicks, determining the impact of various factors on gas kicks. Zhou [18] developed a gas-kick model for underbalanced drilling in which the gas-kick volume is a function of the differential pressure at the bottom of the wellbore, the formation characteristics, and the thickness of the opened part of the reservoir.

$$Q={\displaystyle {\int }_0^t\frac{-X+\sqrt{X^2+4Y\left[P_e^2-P_w^2\left(t\right)\right]}}{2Y}}Hdt,$$

(2)

where $X={\displaystyle \frac{1.27\times {10}^{-6}T\mu Z}{K}}\ln {\displaystyle \frac{0.472r_e}{r_w}}$ and $Y={\displaystyle \frac{2.282\times {10}^{-21}\beta {\rho }_{g}gTZ}{R}}\left[{\displaystyle \frac{1}{r_w}}-{\displaystyle \frac{1}{r_e}}\right]$. Jia et al. [21] applied a steady-state gas seepage formula to calculate the gas-kick volume when studying co-existing lost circulation and blowouts in multi-pressure formations.

$$Q={\displaystyle \frac{774.6KH}{T\mu Z}}\cdot {\displaystyle \frac{\left(P_e^2-P_w^2\right)}{\ln \left(r_e/r_w\right)}}.$$

(3)

When drilling into gas-bearing formations, the gas concentration in the wellbore is much lower than that in the formation. As a result, the formation gas gradually invades the wellbore, a phenomenon known as diffusion-driven gas kick. Research on this is relatively limited. Wu et al. [22] and Wang et al. [23] revealed the gas-kick patterns under different operating conditions based on Fick’s law of diffusion.

$$Q=AH{C}_e^{*}\left(1-e^{-{\displaystyle \frac{2}{r_w}}{K}_{D}t}\right).$$

(4)

During the normal drilling process, the drill bit breaks the rock, releasing gas from the rock pores, which then directly invades the wellbore. However, the amount of gas release from drill cuttings is limited and can generally be ignored [16]. Yuan et al. [20] developed a simplified model for calculating the gas-kick volume from drill cuttings based on the principle of gas release from the pores of drill cuttings.

Most existing gas-kick models focus on a single type of gas kick. However, with the development of deep and deepwater wells [24,25,26] and the use of oil-based drilling fluids [27,28,29], the combined effects of multiple types of gas kick, such as the synergies of both seepage-driven and diffusion-driven gas kick, and the accurate calculation of gas-kick volumes have gradually become more important. Due to the micro-differential pressure and density difference between the formation and the wellbore, this coupling mechanism is particularly relevant in deep and deepwater drilling scenarios with permeable gas-bearing formations and oil-based drilling fluids, where both types of gas kick can occur simultaneously. Additionally, the effect of drilling fluid filtration loss on the seepage-driven and diffusion-driven gas kick should be considered. This is crucial in oil-based mud systems, where the filter cake can significantly impact the formation permeability and gas-kick characteristics.

On the basis of previous research, this study aimed to provide a comprehensive and accurate method for calculating gas-kick volumes under the seepage–diffusion mechanism in permeable gas-bearing formations during deepwater drilling. Firstly, a more realistic and accurate representation of gas-kick dynamics was established by considering the effect of filtration loss and the synergistic effect of negative-differential-pressure and gas-concentration-diffusion. Then, the validation of the model and a sensitivity analysis were conducted, revealing variations of gas-kick volume with time, formation porosity, open-hole length, filter-cake thickness, and other parameters. Finally, a seepage–diffusion ratio was introduced to distinguish between diffusion-driven and seepage-driven gas kicks. This study offers a nuanced understanding of gas-kick behavior and a foundation for multiphase flow and wellbore pressure prediction.

2. Seepage–Diffusion Mechanism and Physical Model of Gas Kick

During drilling, free water from the drilling fluid penetrates the fractures or pores of the rock around the wellbore wall under the differential pressure between the wellbore and the formation, causing filtration loss and forming a filter cake on the wellbore wall [30,31]. When the filter cake is already stable and the gas-kick volume is relatively small in that case, the effect of the filter cake on the gas kick cannot be ignored [23]. Figure 1 presents a physical model of seepage–diffusion-driven gas kicks, considering the effects of filtration loss. The open-hole section is divided into five parts from the wellbore to the formation: a drilling fluid zone, an outer filter cake, an inner filter cake, a filtrate zone, and an unpolluted gas-bearing zone.

At position C (x = h), located h meters from the bottom of the wellbore,

$$P_w\left(h\right)=P_e.$$

(5)

If the position x at the BC segment,

$$P_w\left(x\right)=P_e+{\rho }_{m}g\left(h-x\right)\ge P_e, 0\le x\le h.$$

(6)

If the position x at the AC segment,

$$P_w\left(x\right)=P_e+{\rho }_{m}g\left(h-x\right)<P_e, h<x\le H.$$

(7)

Under the influence of a micro-differential pressure, a seepage-driven gas kick occurs in the open-hole section, and a diffusion-driven gas kick also occurs due to the gas concentration difference, forming a seepage–diffusion-driven gas-kick process. Based on the characteristics of both the seepage- and diffusion-driven gas kicks, and considering Equations (6) and (7), it is evident that, for an open-hole section of length $H$, only a diffusion-driven gas kick occurs in the BC segment, whereas a seepage–diffusion-driven gas kick occurs in the AC segment.

3. Mathematical Model of Seepage–Diffusion-Driven Gas Kick

3.1. Model Assumptions

Based on the principles of seepage–diffusion-driven gas kicks, the following assumptions were made when constructing a mathematical model [16,23]: (1) the gas-bearing formation in the open-hole section is homogeneous; (2) the drilling fluid that has invaded the formation reaches a quasi-static state, and the interface between the drilling fluid filtrate and the formation gas remains stable; (3) the pressure in the open-hole section of the wellbore follows a linear distribution in the axial direction; (4) the seepage and diffusion processes are isothermal; (5) an average value for both the viscosity and compressibility factor is applied during the radial transfer of a gas kick.

3.2. Model Construction

3.2.1. Model of Seepage-Driven Gas Kick Considering Filtration Loss

In homogeneous formations, negative-differential-pressure-driven gas kicks occur in the form of radial steady-state seepage. The gas volumetric flow rate changes with pressure during the seepage process, but the mass flow rate remains constant. For a control unit, dx, along the wellbore axis, a physical model of seepage-driven gas kick is established, as shown in Figure 1.

According to Darcy’s law, the mass flow rate of gas seepage in the control unit is represented by

$$dG=A{\rho }_{g}v=-A{\rho }_g{\displaystyle \frac{K}{\mu }}{\displaystyle \frac{dP}{dr}}.$$

(8)

A pressure function is introduced as

$$\tilde{P}=2{\displaystyle {\int }_{P_0}^P\frac{P}{\mu Z}dP}.$$

(9)

By substituting Equation (9) into Equation (8) and then rearranging, we obtain

$$dG=-{\displaystyle \frac{\pi K{T}_a{Z}_a{\rho }_a}{P_{a}T}}dx\cdot r{\displaystyle \frac{d\tilde{P}}{dr}}.$$

(10)

Since the seepage is assumed to be isothermal, $\mu Z$ is a function of pressure. When calculating the difference in the pressure function between two pressures that are close to each other, the average $\mu Z$ value of the corresponding pressures is used. Thus,

$${\tilde{P}}_1-{\tilde{P}}_2={\displaystyle \frac{1}{\overline{\mu }\overline{Z}}}\left(P_1^2-P_2^2\right).$$

(11)

By substituting Equation (11) in Equation (10), separating the two variables in $r{\displaystyle \frac{d\tilde{P}}{dr}}$, and performing a definite integration of the inverse of r/dr, we obtain

$$dG={\displaystyle \frac{\pi K}{\overline{\mu }\overline{Z}}}{\displaystyle \frac{T_a{Z}_a{\rho }_a}{P_{a}T}}\cdot {\displaystyle \frac{P_2^2-P_1^2}{\ln \frac{r_2}{r_1}}}dx.$$

(12)

Thus, the gas volumetric flow rate between any two pressure points in the control unit is

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{\left(P_2^2-P_1^2\right)}{\frac{\overline{\mu }\overline{Z}}{K}\ln \frac{r_2}{r_1}}}dx.$$

(13)

According to Equation (13), the following is valid in the unpolluted gas-bearing zone of the control unit:

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{\left(P_e^2-P_3^2\right)}{\frac{\overline{{\mu }_e}\overline{Z_e}}{K_e}\ln \frac{r_e}{r_3}}}dx.$$

(14)

In the filtrate zone of the control unit, the following is valid:

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{\left(P_3^2-P_2^2\right)}{\frac{\overline{{\mu }_3}\overline{Z_3}}{K_3}\ln \frac{r_3}{r_2}}}dx.$$

(15)

In the inner filter cake of the control unit, the following is valid:

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{\left(P_2^2-P_1^2\right)}{\frac{\overline{{\mu }_2}\overline{Z_2}}{K_2}\ln \frac{r_2}{r_1}}}dx.$$

(16)

In the outer filter cake of the control unit, the following is valid:

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{\left(P_1^2-P_w^2\left(x\right)\right)}{\frac{\overline{{\mu }_1}\overline{Z_1}}{K_1}\ln \frac{r_1}{r_w}}}dx.$$

(17)

By combining Equations (14) through (17), the volumetric flow rate of gas percolating from the formation into the wellbore is obtained:

$$dQ=\pi {\displaystyle \frac{T_a{Z}_a}{P_{a}T}}\cdot {\displaystyle \frac{P_e^2-P_w^2\left(x\right)}{C}}dx.$$

(18)

where $C={\displaystyle \frac{\overline{{\mu }_e}\overline{Z_e}}{K_e}}\ln {\displaystyle \frac{r_e}{r_3}}+{\displaystyle \frac{\overline{{\mu }_3}\overline{Z_3}}{K_3}}\ln {\displaystyle \frac{r_3}{r_2}}+{\displaystyle \frac{\overline{{\mu }_2}\overline{Z_2}}{K_2}}\ln {\displaystyle \frac{r_2}{r_1}}+{\displaystyle \frac{\overline{{\mu }_1}\overline{Z_1}}{K_1}}\ln {\displaystyle \frac{r_1}{r_w}}$. By ignoring the differences between $\overline{{\mu }_e}\overline{Z_e}$, $\overline{{\mu }_3}\overline{Z_3}$, $\overline{{\mu }_2}\overline{Z_2}$, and $\overline{{\mu }_1}\overline{Z_1}$, and replacing them with the average value $\overline{{\mu }_{ew}}\overline{Z_{ew}}$ corresponding to $P_e$ and $P_w$, an overall seepage coefficient is introduced as follows:

$${\displaystyle \frac{1}{K_P}}={\displaystyle \frac{1}{K_e}}\ln {\displaystyle \frac{r_e}{r_3}}+{\displaystyle \frac{1}{K_3}}\ln {\displaystyle \frac{r_3}{r_2}}+{\displaystyle \frac{1}{K_2}}\ln {\displaystyle \frac{r_2}{r_1}}+{\displaystyle \frac{1}{K_1}}\ln {\displaystyle \frac{r_1}{r_w}}.$$

(19)

By substituting Equation (19) into Equation (18) and rearranging, we obtain

$$dQ={\displaystyle \frac{\pi T_a{Z}_a}{P_{a}T\overline{{\mu }_{ew}}\overline{Z_{ew}}}}{K}_P\cdot \left(P_e^2-P_w^2\left(x\right)\right)dx.$$

(20)

3.2.2. Model of Diffusion-Driven Gas Kick Considering Filtration Loss

Three factors affect mass diffusion: the concentration difference, temperature difference, and pressure difference. Typically, thermal diffusion and pressure diffusion have a significant impact on mass transfer only when the temperature difference or pressure difference is large. This paper only considers concentration diffusion under isothermal, uniform-pressure conditions. By selecting a control unit, dx, along the wellbore axis, a physical model of the diffusion-driven gas kick is established, as shown in Figure 1.

The formation gas is assumed to be fully dissolved into the drilling fluid upon diffusion into the wellbore, and the diffusion-driven gas kick is assumed to follow Fick’s first law. At radius r, for a microelement with thickness dr, the diffusion flux per unit time is

$$dG=-D\cdot 2\pi rdx{\displaystyle \frac{dC}{dr}}.$$

(21)

Through variable separation and integration in Equation (21), we can obtain the diffusion flux between any two positions with different concentrations:

$$dG=2\pi dx\cdot D{\displaystyle \frac{C_1-C_2}{\ln \frac{r_2}{r_1}}}.$$

(22)

For the filtrate zone in the control unit, we have

$$dG=2\pi dx\cdot D_{e3}{\displaystyle \frac{C_e-C_3}{\ln \frac{r_3}{r_2}}}.$$

(23)

For the inner filter cake in the control unit, we have

$$dG=2\pi dx\cdot D_{e2}{\displaystyle \frac{C_3-C_2}{\ln \frac{r_2}{r_1}}}.$$

(24)

For the outer filter cake in the control unit, we have

$$dG=2\pi dx\cdot D_{e1}{\displaystyle \frac{C_2-C_1}{\ln \frac{r_1}{r_w}}}.$$

(25)

According to dual-porosity theory [32], the diffusion flux at the boundary between the filtrate zone and the unpolluted gas-bearing zone is

$$dG=2\pi dx\cdot r_3\phi k_e\left(C_e^{*}-C_e\right).$$

(26)

The diffusion flux at the boundary between the outer filter cake and the drilling fluid zone is

$$dG=2\pi dx\cdot r_w{k}_w\left(C_1-C_w\right).$$

(27)

By combining Equations (23) through (27), the mass flux of gas diffusing from the formation into the wellbore is

$$dG=2\pi dx\cdot {\displaystyle \frac{C_e^{*}-C_w}{\frac{1}{r_3\phi k_e}+\frac{1}{D_{e3}}\ln \frac{r_3}{r_2}+\frac{1}{D_{e2}}\ln \frac{r_2}{r_1}+\frac{1}{D_{e1}}\ln \frac{r_1}{r_w}+\frac{1}{r_w{k}_w}}}.$$

(28)

An overall diffusion coefficient is introduced:

$${\displaystyle \frac{1}{K_D}}={\displaystyle \frac{1}{r_3\phi k_e}}+{\displaystyle \frac{1}{D_{e3}}}\ln {\displaystyle \frac{r_3}{r_2}}+{\displaystyle \frac{1}{D_{e2}}}\ln {\displaystyle \frac{r_2}{r_1}}+{\displaystyle \frac{1}{D_{e1}}}\ln {\displaystyle \frac{r_1}{r_w}}+{\displaystyle \frac{1}{r_w{k}_w}}.$$

(29)

By substituting Equation (29) into Equation (28) and converting the mass flux per unit time into volumetric flux under standard conditions, we obtain

$$dQ={\displaystyle \frac{2\pi }{{\rho }_a}}{K}_D\cdot \left(C_e^{*}-C_w\right)dx.$$

(30)

3.2.3. Model of Seepage–Diffusion-Driven Gas Kick

Gas seepage from the formation increases the concentration of gas in the drilling fluid, which in turn affects the diffusion gas kick. Therefore, the model for the gas-kick volume under a seepage–diffusion mechanism in the control unit can be expressed as

$$dQ=d{Q}_P+d{Q}_D.$$

(31)

The gas concentration in the drilling fluid at a certain time is taken as the initial condition for the next iteration of the gas seepage–diffusion calculations. By substituting Equations (20) and (30) into Equation (31) and integrating over the open-hole section of length H at time t + 1, we obtain

$$Q^{t+1}={\displaystyle \frac{\pi T_a{Z}_a}{P_{a}T\overline{{\mu }_{ew}}\overline{Z_{ew}}}}{K}_P\cdot {\displaystyle {\int }_h^H\left(P_e^2-P_w^2\left(x\right)\right)}dx+{\displaystyle \frac{2\pi H}{{\rho }_a}}{K}_D\cdot \left(C_e-C_w^t\right).$$

(32)

By assuming that the initial gas concentration in the drilling fluid is zero and ignoring axial diffusion above the open-hole section, the gas concentration in the open-hole section increases over time until it reaches the maximum gas concentration. Thus, the initial and boundary conditions are

$$\left\{\begin{array}{l}{C}_w^t{|}_{t=0}=0\\ C_w^t{|}_{t=T}=C_w\left(\max \right)\\ C_w^t={\displaystyle \frac{Q^t}{H\pi r_w^2}}\cdot {\rho }_a\end{array}\right..$$

(33)

3.3. Auxiliary Equations

When solving the equations of the mathematical model described above, the following auxiliary equations are needed. The DPR empirical formula is used for calculating the gas compressibility factor [33]. The semi-empirical Lee–Gonzalez–Eakin method is used for calculating the gas viscosity [34]. The maximum gas concentration can be determined based on the solubility of methane in drilling fluids, with the solubility of methane in water-based drilling fluids and oil-based drilling fluids calculated using the Fan method [35] and the Obryan method [36], respectively. The effective diffusion coefficient of gas in the formation is $D=D_0\phi /\tau$ [23]. Herein, the molecular diffusion coefficient (D₀) of methane in oil at 25 °C and 0.1 MPa ranges between 2 × 10⁻⁹ and 9 × 10⁻⁹ m²/s [37]. When the pore size of the formation is comparable and uniformly distributed, $\tau$ = 3; when the pore size of the formation varies significantly and is unevenly distributed, $\tau$ > 3; when the pore size varies significantly and is distributed in an elongated pattern, $\tau$ < 3 [38,39]. The permeability of the formation can be estimated approximately based on the porosity using the relationship between the porosity and the permeability [40]. It should be noted that since the main component of natural gas is methane, which constitutes about 90%, the above parameters for methane can be approximately used for those of natural gas, as demonstrated in reference [23]. For details on the main auxiliary equations, please refer to Appendix A or the cited references above.

4. Evolution Pattern of Seepage–Diffusion-Driven Gas Kick

4.1. Model Validation and Case Study

The basic parameters of the case study include geometric and boundary parameters (

Table 1. Geometric and boundary parameters.

Geometric Parameter	Parameter Value
open-hole radius of wellbore (m)	0.108
outer radius of outer filter cake (m)	0.109
outer radius of inner filter cake (m)	0.116
outer radius of filtrate zone (m)	0.128
outer radius of gas-bearing formation (m)	1000
natural gas concentration in formation (kg/m3)	180
initial gas concentration in drilling fluid (kg/m3)	0
length of open-hole section (m)	100
bottomhole pressure (MPa)	36.0
formation pressure (MPa)	36.5
formation temperature (°C)	100.2
critical pressure of natural gas (MPa)	4.59
critical temperature of natural gas (°C)	−82.6
drilling fluid density (kg/m3)	1400

) and formation properties (

Table 2. Formation parameters.

Parameter	Outer Filter Cake	Inner Filter Cake	Filtrate Zone	Gas-Bearing Formation
porosity	0.01	0.02	0.20	0.20
tortuosity	3	3	3	3
molecular diffusion coefficient (m2/s)	11 × 10−9	11 × 10−9	11 × 10−9	11 × 10−9

), which can be found in references [22,23].

To ensure the validity of the models proposed in this work, a comparison of computational results between our models and other models was obtained, as shown in

Table 3. Comparison of computational results between our models and other models. Condition 1 considers diffusion-driven gas kick, with the length of open-hole section of 100 m. Condition 2 considers seepage-driven gas kick, with the length of open-hole section of 5 m.

Condition	Model	Gas-Kick Volume (m3) (10 min, a Short Time)	Gas-Kick Volume (m3) (10 d, a Long Time)	Type of Gas Kick	Source
condition 1	Equation (4)	0.00936	13.83	diffusion-driven	reference [23]
	Equation (30)	0.00899	12.86	diffusion-driven	this work
condition 2	Equation (2)	0.00882	12.43	seepage-driven	reference [16]
	Equation (3)	0.00865	11.96	seepage-driven	reference [21]
	Equation (20)	0.00874	12.59	seepage-driven	this work
	Equation (32)	0.00899	12.78	seepage–diffusion-driven	this work

. Equations (4) and (30), which consider diffusion-driven gas kick, were used in the calculations under condition 1. Equations (2), (3), and (20), which consider seepage-driven gas kick, and Equation (32), which considers seepage–diffusion-driven gas kick, were used in the calculations under condition 2. Based on the calculated values from Equation (4), the errors for the 10-min and 10-day calculations using Equation (30) are 3.95% and 7.01%, respectively. Based on the calculated values from Equation (2), the errors for the 10-min and 10-day calculations using Equation (20) are 0.91% and 1.29%, respectively, and for Equation (32) they are 1.93% and 2.82%, respectively. Similarly, based on the calculated values from Equation (3), the errors for the 10-min and 10-day calculations using Equation (20) are 1.03% and 5.01%, respectively, while for Equation (32) they are 3.93% and 6.86%, respectively. It should be noted that the calculation result of Equation (32) is larger than that of Equation (20), as Equation (32) incorporates the diffusion effect in its calculation of seepage-driven gas kick, which is a reasonable consideration. In short, as shown in the table and the above analysis, the new models show high calculation accuracy and can be employed to reveal the evolution pattern and to conduct sensitivity analysis of seepage–diffusion-driven gas kick.

4.2. Evolution Pattern and Sensitivity Analysis of Seepage-Driven Gas Kick

4.2.1. Distribution of Gas-Kick Volume in Wellbore Under Seepage Effect

The evolution pattern of the seepage-driven gas kick was analyzed from both spatial and temporal perspectives (Figure 2). In terms of the spatial distribution, within approximately 36.5 m along the axial direction of the wellbore (from the bottomhole to the wellhead), the wellbore pressure exceeded the formation pressure, so no gas kick occurred in this open-hole section. In the region between 36.5 and 100 m, where the wellbore pressure was less than the formation pressure, the cumulative gas-kick volume increased gradually along the axial direction as the differential pressure increased. In terms of the temporal distribution, the cumulative gas-kick volume in the wellbore increased linearly over time.

4.2.2. Effect of Formation Porosity on Seepage-Driven Gas Kick

Figure 3 shows the variation in the cumulative gas-kick volume as a function of the formation porosity. The cumulative gas-kick volume in the wellbore increased in an S-shaped curve with increasing porosity, meaning it initially grew slowly, then quickly, and then slowed down again. When the formation porosity was less than 0.02, it was comparable to the porosity of the filter cake, and the permeability of the entire formation was very low, resulting in almost no gas kick. As the formation porosity increased, its influence on the gas kick began to dominate, causing the gas-kick volume to increase rapidly. When the formation porosity increased further to a value significantly larger than the filter-cake porosity, the effect of the filter-cake porosity began to dominate, with the rate of increase in the gas-kick volume slowing down and eventually stabilizing.

4.2.3. Effect of Open-Hole Length on Seepage-Driven Gas Kick

Figure 4 shows the variation in the cumulative gas-kick volume with the open-hole length. In the section within 36.5 m, where the wellbore pressure was greater than the formation pressure, no gas kick occurred, so the open-hole length had no effect. In the open-hole section beyond 36.5 m, the cumulative gas-kick volume increased non-linearly with increasing open-hole length. The greater the open-hole length was, the larger the pressure difference between the wellbore and the gas-bearing formation became. Since the gas-kick volume was not proportional to the pressure difference but rather proportional to its square, the gas-kick volume increased non-linearly with the open-hole length.

4.2.4. Effect of Drilling Fluid Filtration Loss on Seepage-Driven Gas Kick

To analyze the effect of drilling fluid filtration loss on the seepage-driven gas kick, the last four combinations of filter-cake thicknesses from

Table 4. Combinations of different drilling fluid filter cakes.

Parameter	Combination 1	Combination 2	Combination 3	Combination 4	Combination 5
thickness of cake outer filter (mm)	0	0.5	1.0	1.5	2.0
thickness of cake inner filter (mm)	0	3.5	7.0	10.5	14.0
thickness of the filtrate zone (mm)	0	6	12	18	24

were selected. Combination 1 did not consider drilling fluid filtration loss, while combinations 2, 3, 4, and 5 represent increasing thicknesses for the outer filter cake, the inner filter cake, and the filtrate zone. Figure 5 shows the variation in the cumulative gas-kick volume with filter-cake thickness. The cumulative gas-kick volume decreased with increasing filter-cake thickness. When the outer filter-cake thickness increased from 0.5 to 2.0 mm, the gas-kick volume decreased from 40 m³ to approximately 10 m³. This meant that when the outer filter-cake thickness increased by a factor of 4, the gas-kick volume decreased by a factor of 4, demonstrating that the drilling fluid filtration loss significantly inhibited the seepage-driven gas kick, similar to its the effect on the diffusion-driven gas kick.

4.3. Evolution Pattern and Sensitivity Analysis of Diffusion-Driven Gas Kick

4.3.1. Distribution of Gas-Kick Volume in Wellbore Under Diffusion Effect

Figure 6 shows the variations in the cumulative gas-kick volume and gas-kick rate over time. It can be seen that the cumulative gas-kick volume increased approximately linearly with time, while the gas-kick rate decreased approximately linearly. For water-based drilling fluids, the maximum allowable gas-kick volume in the wellbore and the maximum gas-kick time were approximately 15.16 m³ and 10 days, respectively (Point A). Therefore, diffusion gas kick ceased after 10 days. For oil-based drilling fluids, the maximum allowable gas-kick volume and the maximum gas-kick time were approximately 121.70 m³ and 100 days, respectively (Point B). Thus, the diffusion gas kick ceased after 100 days. This demonstrated that the diffusion gas kick under oil-based drilling fluid conditions was much greater than that under water-based drilling fluid conditions. Therefore, diffusion-driven gas kick should be given more attention when drilling with oil-based drilling fluids.

4.3.2. Effect of Formation Porosity on Diffusion-Driven Gas Kick

Figure 7a shows the variation in the cumulative gas-kick volume over time for different formation porosities, while Figure 7b shows the variation in the 10-day cumulative gas-kick volume as a function of the porosity. The higher the formation porosity was, the faster the cumulative gas-kick volume increased. Since the gas concentration in the wellbore did not reach saturation after 10 days, the cumulative gas-kick volume increased with the increase in the porosity, but the rate of increase in the cumulative gas-kick volume gradually decreased. When the formation porosity was low, the effect of the porosity on the diffusion-driven gas kick was dominant. When the formation porosity was high, the porosity of the filter cake played a major role in influencing the diffusion-driven gas kick, because the filter-cake porosity was much smaller than the formation porosity. The effect of the filter-cake porosity resembled a “throttling effect”, which inhibited the influence of the formation porosity on the gas-kick volume in the wellbore.

4.3.3. Effect of Open-Hole Length on Diffusion-Driven Gas Kick

Figure 8a shows the variation in the cumulative gas-kick volume over time for different open-hole lengths, and Figure 8b shows the variation in the 10-day cumulative gas-kick volume as a function of the open-hole length. The figures show that the cumulative gas-kick volume increased linearly over time for a given open-hole length. For a fixed gas-kick time, the cumulative gas-kick volume also increased linearly with increasing open-hole length. The longer the open-hole length was, the faster the cumulative gas-kick volume increased.

4.3.4. Effect of Drilling Fluid Filtration Loss on Diffusion-Driven Gas Kick

Five different combinations of filter-cake thicknesses (Table 4) were selected to analyze the effect of drilling fluid filtration loss on the diffusion-driven gas kick. Figure 9a shows the variation in the cumulative gas-kick volume over time for the different filter-cake combinations, and Figure 9b shows the variation in the 10-day cumulative gas-kick volume as a function of the filter-cake thickness. Figure 9a shows that when drilling fluid filtration loss was absent, the gas-kick volume in the wellbore quickly reached its maximum value, meaning the gas concentration in the drilling fluid was saturated, and no further diffusion gas kick occurred. When drilling fluid filtration loss was present, the gas-kick volume decreased as the filter-cake thickness increased. Figure 9b shows that even a few millimeters of increase in the filter-cake thickness could result in an order-of-magnitude change in the gas-kick volume, demonstrating that the filter-cake thickness had a significant inhibitory effect on the diffusion gas kick.

4.4. Evolution Pattern Under Seepage and Diffusion Effects

Since a seepage-driven gas kick only occurs when the wellbore pressure is less than the formation pressure, while a diffusion-driven gas kick occurs throughout the entire open-hole section, the evolution pattern of the gas kick under the seepage–diffusion mechanism was analyzed by introducing the “seepage–diffusion ratio (λ)”, defined as the ratio of the length of the open-hole section where the seepage-driven gas kick occurs to the total length of the open-hole section. Figure 10a shows the variation in the cumulative gas-kick volume with the seepage–diffusion ratio, where the range was 0% ≤ λ ≤ 5%. When the seepage–diffusion ratio was large, most of the gas-kick volume was attributed to the seepage-driven gas kick. Only when the seepage–diffusion ratio was small did the diffusion-driven gas kick become a significant contributor to the gas-kick volume. When the seepage–diffusion ratio was less than approximately 1%, the diffusion-driven gas kick contributed more than the seepage-driven gas kick. When the seepage–diffusion ratio was greater than 1%, the seepage-driven gas kick contributed more. Figure 10b shows the variation in the cumulative gas-kick volume with time when the seepage–diffusion ratio was 1%. The cumulative gas-kick volume in the wellbore was determined by both seepage-driven and diffusion-driven gas kicks. When the seepage–diffusion ratio was less than 1% or when the bottomhole pressure was greater than the formation pressure, the diffusion-driven gas kick dominated, and it could not be ignored when calculating the gas-kick volume. When the seepage–diffusion ratio exceeded 10%, the seepage-driven gas kick dominated, and the diffusion-driven gas kick could be neglected in the calculation.

4.5. Summary and Model Limitation

Table 5. Sensitivity analysis summary of the seepage–diffusion-driven gas-kick model.

Parameters	Impact Type	Influence
seepage-driven gas kick
formation porosity	positive correlation	Increased porosity significantly raises gas-kick volume, especially when formation porosity exceeds filter-cake porosity.
open-hole length	positive correlation	Longer open-hole lengths increase gas-kick volume due to greater pressure differentials.
filter-cake thickness	negative correlation	Thicker filter cake reduces gas-kick volume by impeding flow.
diffusion-driven gas kick
formation porosity	positive correlation	Increased porosity significantly boosts gas-kick volume, especially at lower levels.
open-hole length	positive correlation	Longer open-hole sections raise gas-kick volume by exposing more formation.
filter-cake thickness	negative correlation	Thicker filter cake reduces gas diffusion, lowering gas-kick volume.
overall impact
seepage–diffusion ratio (λ)		When λ < 1%, diffusion-driven gas kick dominates. when λ > 1%, seepage-driven gas kick dominates.

summarized the influence of various parameters on the gas-kick volume during drilling, as analyzed using the seepage–diffusion model. For seepage-driven gas kick, higher formation porosity and longer open-hole length facilitate greater gas flow, leading to increased gas-kick volume. However, thicker filter cake hinders gas flow, resulting in decreased gas-kick volume. Meanwhile, for diffusion-driven gas kick, higher formation porosity and longer open-hole length increase the exposed formation area, leading to increased gas-kick volume. Thicker filter cake significantly reduces gas diffusion, limiting the gas-kick volume. The seepage–diffusion ratio (λ) identifies the predominant mechanism of gas kicks: diffusion is the main driver when λ is less than 1%, whereas seepage becomes dominant when λ exceeds 1%. This ratio offers valuable insights into the comparative significance of each mechanism under specific wellbore conditions.

These analyses indicate that, under the conditions of the case study, the model remains accurate and reliable, but the model also has some limitations: (1) The model depends on certain input parameters, such as filter-cake thickness, which may not be readily available or may require logging operations or other methods to obtain. As a result, it cannot be used in real-time. (2) The assumptions of homogeneous formation, static filter cake, linear pressure distribution, and isothermal process in this model also limit its applicability. For example, the model assumptions are a reasonable approximation for shorter open-hole lengths. For open-hole lengths up to 100 m, we mainly discuss diffusion-driven gas kick, which is a very slow process, and the assumptions remain valid. For seepage-driven gas kick, it does not occur throughout the entire open-hole section but depends on the positive or negative pressure difference. However, longer lengths of the open-hole may require more sophisticated assumptions and extra considerations. Additionally, the seepage–diffusion-driven gas kick involves a relatively small pressure difference between the wellbore pressure and the formation pressure (less than 0.5 MPa in the case study), resulting in a relatively slow mass transfer process. The model shows good accuracy in this condition. However, the large pressure difference effect is a relatively fast dynamic process, which may reduce its accuracy.

5. Conclusions

• For permeable formations, a comprehensive model for calculating gas-kick volumes under the seepage–diffusion mechanism was established, which considered the synergistic effect of gas-concentration-diffusion and negative-differential-pressure, and mass transfer in both the filtrate zone and the filter-cake zone. It improves the prediction accuracy of the model
• For the diffusion-driven gas kick, the cumulative gas-kick volume in the wellbore increased with increasing formation porosity and open-hole length, while the thickness of the filter cake had a strong inhibitory effect.
• For the seepage-driven gas kick, the cumulative gas-kick volume in the wellbore also increased with increasing formation porosity and open-hole length. The filter-cake thickness inhibited the seepage-driven gas kick.
• The introduction of a “seepage–diffusion ratio” factor was beneficial for exploring the evolution pattern of gas kicks under the seepage–diffusion mechanism. Under specific case conditions, when the seepage–diffusion ratio was less than 1%, the diffusion-driven gas kick contributed more than the seepage-driven gas kick; when the seepage–diffusion ratio exceeded 1%, the negative-differential-pressure-driven gas kick contributed more. This ratio provides a valuable tool for understanding the relative contributions of seepage and diffusion mechanisms to overall gas kick.