Drilling ultra-deep wells is faced with complex geological conditions, high temperature, high pressure, and large well depth etc., so formation fluid invasion into the wellbore is inevitable. Higher formation pressure often results in higher casing pressure after overflow well shut-in, so the drill string in the well would bear large uplift force under the action of high fluid pressure. When the gravity of the drill string and the friction of the shut-in seal are insufficient to suppress the uplift force caused by overflow, the drilling tool may be broken or washed out of the wellbore, resulting in blowout out of control. In the drilling process of an ultra-deep well in Xinjiang, when drilling to 693 m deep, an overflow of 0.5 m³ was found. When the well was shut down according to the “five-seven” code, the well had a standpipe pressure of zero, casing pressure soaring to 21.5 MPa, and the drill string was pushed up 2.7 m. When a well in Daqing Oilfield was drilled to 211 m, the drill tools uplifting happened, resulting in bending of drill pipe and damages to water hose and traveling car^[1]. There is some understanding on the drill tools uplift caused by annulus pressure build-up^[2,3], but there is still a lack of theoretical analysis on the mechanism of the drill string uplift, the mechanism of high pressure fluid acting on the drill string to produce the uplift force, and the relationship between the uplift force and buoyancy. Therefore, this work has attempted to find out the mechanics mechanism of drill string uplift from the basic forces exerting on drill string in hydraulic environment. Meanwhile, the non-circulating multiphase flow theory was used to analyze the change of wellbore pressure, and the evolution and density distribution of gas-liquid two-phase flow during shut-in to obtain the variation of axial load on drill string during overflow well shut-in and the formation conditions of the uplift of drill string, which would provide theoretical support for wellbore safety during ultra-deep well drilling.

Suppose there is one set of bottomhole assembly (BHA) composed of drill pipes of different sizes of n sections, as shown in Fig. 1. The density of drilling fluid inside and outside the drill string are different. The cross-section method for analyzing internal force in material mechanics and the pressure area method in hydrostatics were used to establish the model to solve the axial force at any cross-section a below the wellhead.

The BHA below section a is considered as a whole (including drill string and drilling fluid in the drill string), and the force model is shown in Fig. 2. The mass of each drill string section is composed of its own mass and the fluid mass in it. Therefore, the mass of the first and second drill string sections are as follows:

For each drill string section below section a, the resultant force of liquid pressure acting on the sidewall of the drill tool is zero, so only the pressure acting on the section and shoulder side of the drill string are considered, and the resultant force is positive in the opposite direction of gravity. The pressure of the fluid is:

According to the static equilibrium relationship, the pulling force at the cross section a can be expressed as:

The equations (1)-(5) are substituted into (6):

where ${{K}_{i}}=\text{1}-\frac{R_{i}^{2}{{\rho }_{\text{o}}}-r_{i}^{\text{2}}{{\rho }_{\text{i}}}}{\left( R_{i}^{2}-r_{i}^{\text{2}} \right){{\rho }_{\text{s}}}}$

In equation (7), the ΔH_iq_iK_i is the gravitational force of each drill string section in drilling fluid, that is, the buoyant weight. Since q_i is the average weight per unit length, the gravity of joints has been taken into account, and the influence of shoulders and joints need not be considered in the calculation of buoyant weight^[4,5].

The first item on the right side of equation (7) is the buoyant weight of the drill string, and the second item is deformed to obtain:

F_x in equation (8) is the virtual force generated by hydraulic pressure. The concept and calculation method of virtual force are derived from the mechanical analysis of column instability under hydraulic environment^[6,7,8]. The virtual force is the axial force really exists, not virtual. The American Petroleum Society named it effective bucking force. The two kinds of axial force can be connected by virtual force, that is, the real axial force is equal to the difference between the effective axial force and the virtual force^[9]. In equation (7), T_a is the true axial force of cross-section a.

In equation (9), T_e is the effective axial force, which is the accumulation of the buoyant weight of the drill string from each section, and the direction is consistent with the direction of gravity. Therefore, the uplift force on the drill string can only be generated by the virtual force. When the virtual force is opposite to the gravity direction, it is the uplift force on the drill string. Therefore, the uplift force on the drill string is derived from hydraulic pressure rather than buoyancy.

In ultra deep well drilling, the blowout preventer is installed in the drill string, when the standpipe pressure is zero, the equation (8) is simplified into:

At this time, F_x is the uplift force, but the existence of uplift force on the drill string does not necessarily mean the appearance of drill string uplift. Only when the uplift force on the drill string is larger than the drill string buoyant weight, the uplift of the drill string will appear. Therefore, the condition

for preventing the uplift of the drill string is:

Assuming that overflow occurred in a well and the well was shut in. When the well was shut in, the drilling collar (with outer diameter of 120.7 mm) remained in the well is 500 m, the drilling fluid density is 1.5 g/cm³, and the drilling collar weight per unit length is 683 N/m. The relationship between wellhead axial load and wellhead casing pressure as shown in Fig. 3 can be calculated by equations (1)-(10).

When the casing pressure is greater than 24.16 MPa, the axial load of the wellhead is negative, and the uplift force is greater than the buoyant weight of the drill string, so the drill string uplift will occur. In order to prevent the drill string uplift, the casing pressure at the wellhead should be controlled at less than or equal to 24.16 MPa, that is to say, the maximum casing pressure at the top of the drill string is 24.16 MPa. As for when the maximum casing pressure would be reached, it is necessary to conduct dynamic analysis of fluid migration and wellbore pressure evolution during shut-in.

During the overflow well shut-in stage, the change of casing pressure and wellbore overflow state will lead to constant change of wellhead axial load. On the one hand, the increase of casing pressure leads to the increase of uplift force, on the other hand, the change of density of mixture in the wellbore causes the change of the buoyant weight of drill string. In order to accurately calculate the change of axial load of the drill string during overflow shut-in, it is necessary to analyze the change process of wellbore pressure and gas-liquid mixture state during shut-in. The change of wellbore pressure and wellbore overflow state after shut-in is the result of both wellbore after-flow and gas slippage effects^[10,11,12]. In the early stage of shut-in, the bottom hole pressure is less than formation pressure. Under the action of pressure difference, the formation gas continues to invade into the wellbore, the wellbore fluid is compressed, the bottom hole pressure and wellhead pressure increase. When the bottom hole pressure equals to the formation pressure, the formation gas stops invading into the wellbore, which is the wellbore after-flow effect. Since the gas density is lower than the density of drilling fluid, the gas will slip off and the gas volume will expand and compress the wellbore fluid during the ascending process, resulting in an increase of the bottomhole and wellhead pressure, which is the gas slippage rise effect.

Under the effect of wellbore after-flow, the formation fluid continuously invades into the wellbore under the pressure difference. The influx amount is related to the thickness and permeability of the reservoir, and the gas properties. The infiltration velocity can be calculated by the theory of seepage mechanics:

Since the volume of the wellbore annulus is constant, the formation fluid influx volume per unit time is equal to the volume of fluid reduced in the wellbore per unit pressure, which is:

The bottom hole pressure at time t during after-flow can be calculated:

where $Y=\frac{{{p}_{\text{e}}}+{{p}_{\text{w0}}}}{{{p}_{\text{e}}}-{{p}_{\text{w0}}}}\exp \left[ \frac{1.076{{p}_{\text{e}}}{{K}_{\text{r}}}{{h}_{\text{r}}}{{B}_{\text{g}}}t}{\left( {{V}_{\text{g}}}{{C}_{\text{g}}}+{{V}_{\text{l}}}{{C}_{\text{l}}} \right)T{{Z}_{\text{g}}}\mu \ln \frac{{{r}_{\text{e}}}}{{{r}_{\text{w}}}}} \right]$

Since the density of formation gas is lower than the density of drilling fluid, the gas slips and rises, and the density and pressure at each position of the wellbore change with time and space, which should be calculated with the time and space grid method. Moreover, the position and length of the gas- liquid two-phase segment change during the rise of gas. Therefore, in this study, two grid systems were adopted. For the wellbore annulus, the static mesh analysis method was used to divide the wellbore annulus into X grids, and the grid size and number were constant. For the gas-liquid two-phase segment, the dynamic mesh analysis method was used to divide the two-phase segment into Z meshes. The number of meshes didn’t change with time, but the mesh size h_g(t,z) changes continuously with the rise of gas. The volume increment of gas due to expansion equals to the sum of the volume of the drilling fluid reduced by compression and the volume of the drilling fluid lost to the formation, which is:

The calculation of gas holdup in the gas-liquid two-phase unit is:

The formula for calculating the unit pressure of gas-liquid two-phase is:

The calculation formula of gas-liquid two-phase unit length is:

The formula of gas slippage velocity is as follows:

When the wellbore pressure is greater than the formation pressure, the drilling fluid will be filtered into formation through the mud cake^[13], and the drilling fluid loss is:

The wellbore volume is constant throughout the shut-in process. Within a unit time, the volume of fluid flowing into the wellbore minus the volume of fluid flowing out plus the volume of slipping and swelling equals to the volume of fluid in the wellbore reduced by the increase of wellbore pressure, which is:

The simultaneous equations (12)-(21) were solved by Gauss-Seidel iteration method to get the relationship between fluid density and wellbore pressure in the wellbore with time, and then the relationship was substituted into equation (9) to obtain the dynamic change of the uplift force of the drill string during shut-in.

Overflow well shut-in happened in a well in Xinjiang during the trip and the overflow rate was 3 m³ when the well was shut-in. The well structure and BHA are shown in Tables 1 and 2. The formation parameters were: the thickness of drilled gas reservoir of 2.5 m, the permeability of the gas reservoir of 10×10^-3 μm², the formation pressure of 120 MPa, and the geothermal gradient of 0.02 °C/m. The parameters of drilling fluid and mud cake were: the density of drilling fluid of 1.8 g/cm³, the viscosity of drilling fluid of 40 mPa·s, the solid content of drilling fluid of 5%, the solid content of mud cake of 15%, the volume coefficient of drilling fluid of 0.000 4 MPa^-1, and the permeability of mud cake of 0.000 2×10^-3 μm².

As shown in Fig. 4, the bottom hole pressure and casing pressure increase with time. In the first 20 min, the wellbore after-flow effect was dominant, followed by the gas slippage effect. After 747 min, the gas slipped to the wellhead and the casing pressure was 25.5 MPa. When the gas reached the wellhead, because the gas didn’t rise anymore, the gas volume would not change, casing pressure would remain unchanged, and at this time the casing pressure was the maximum casing pressure during shut-in. The calculated pressure changes in this study are the same as those obtained by Lage et al.^[14]

Since there is a maximum value of the shut-in casing pressure, there will be a critical drill string length under a certain working condition, that is, when the length of the drill string in the well exceeds the critical drill string length, there is no risk of uplift of the drill string. Under the maximum casing pressure, the relationship between the axial load of wellhead and the length of drill string in the well is shown in Fig. 5.

Fig. 5 shows that the axial load at the wellhead is zero when the drill string length is reduced to 690 m. Afterward, the axial load direction of the wellhead turns from downward to upward, and the drill string pushes up. When the uplift force exceeds the blowout preventer sealing friction, the drill string would be pushed out of the wellhead. Therefore, under the condition of this example, the critical length of the drill string is 690 m, if the length of the drill string assembly is more than 690 m, there is no risk of drill string uplift; otherwise, it is necessary to analyze the critical shut-in time under the condition of this example.

Fig. 6 shows the relationship between wellbore axial load and shut in time in the case of drill string length of 500 m. When the well is closed for 568 min, the axial load of the wellhead is zero, and then the direction of the axial load of the wellhead changes from downward to upward, and the drill string uplift occurs. In view of the overflow condition and the length of drill string of 500 m in this example, in order to prevent the drill string from uplift, the shut-in time should be controlled at less than 568 minutes, that is, the critical shut-in time is 568 min.

By calculating the critical shut-in time at different drill string lengths, the risk distribution map of drill string uplift can be obtained (Fig. 7). According to the critical shut-in time and critical drill string length, the risk of drill string uplift can be divided into five regions. The axial loads of inner wellhead are negative in regions ① and ② and there is a risk of drill string uplift; the axial loads of inner wellhead are positive in regions ③, ④ and ⑤, and there is no risk of drill string uplift. The lengths of drill string in regions ④ and ⑤ are larger than the critical drill string length. Gas has risen to the wellhead in regions ① and ②, casing pressure remains constant at the maximum, while gas still migrates in the wellbore in regions ③, ④ and ⑤, the casing pressure continues to rise. In order to reduce the risk of the drill string uplift, the length of drill string and the shut-in time in the well should be controlled within regions ③, ④ and ⑤.

The risk of drill string uplift could occur during overflow well shut-in in ultra-deep wells. The shorter the drill string length is, the greater the risk. The longer the shut-in time, the greater the risk. Especially when the overflow occurs during the trip process, it is necessary to strictly prevent the uplift of drill string. Fig. 8 shows a set of management procedure for preventing drill string from uplift in ultra-deep wells, which can control drill string uplift risk more rigorously and scientifically.

The source of drill string uplift force is the virtual force produced by hydrostatic force in the well. When the virtual force direction is opposite to the gravity direction, it is the drill string uplift force. When the uplift force is greater than the buoyant weight of drill string, the uplift of drill string would appear.

There are two key parameters: the critical drill string length and critical shut-in time. When the drill string length in the wellbore is less than the critical drill string length and the shut-in time exceeds the critical shut-in time, there is the risk of drill string uplift. On this basis, the risk distribution chart of drill string uplift can be plotted, and the length of drill string in the well and the shut-in time can be controlled in the risk-free zone to prevent drill string uplift.

A set of drill string uplift risk management program after well shut-in caused by overflow in ultra-deep wells is recommended to make the management of drill string uplift risk more rigorous and scientific.

A_a(t,z)—annular cross-sectional area, m²;

A_ai, A_ao—inner and outer cross-sectional area of drill string, m²;

A_c(k)—filter area of mud cake in open hole section, m²;

B_g—gas volume factor, dimensionless;

C_g, C_l—isothermal compressibility of gas and drilling fluid, MPa^-1;

d_c, d_t—outer and inner diameter of annulus, m;

E_g(t,z)—gas holdup of gas-liquid two-phase segment, %;

f₁, f₂—upward and downward hydraulic pressure on the first section of drill string, N;

f₃—the downward hydraulic pressure on the second section of drill string, N;

f_sc, f_sm—solid content of mud cake and drilling fluid, %;

F_x—virtual force, N;

g—acceleration of gravity, 9.8 m/s²;

h_g(t,z)—length of gas-liquid two-phase segment, m;

h_low(t)—length of drilling fluid column at the lower end of gas-liquid two-phase segment, m;

h_r—thickness of gas layer, m;

H_a—well depth at section a, m;

H_i—well depth at the bottom of i-th section of drill string, m;

ΔH_i—length of the i-th section drill string, m;

k—grid number of open hole segment;

K—mesh quantity of open hole segment;

K_c—mud cake permeability, μm²;

K_i—buoyancy coefficient of i-th drill string, dimensionless;

K_r—permeability of gas reservoir, 10^-3 μm²;

m_g(z)—gas mass in gas-liquid two-phase segment, kg;

n—number of drill strings;

p(t,z)—pressure of gas-liquid two-phase segment, MPa;

p_ai, p_ao—internal and external pressure of drill string, Pa;

p_c—casing pressureat wellhead, Pa;

p_c(t,k), p_e(k)—wellbore and formation pressure in open hole section, MPa;

p_e—supply pressure, MPa;

p_w—bottom hole pressure, MPa;

p_w0—bottom hole pressure at initial moment, MPa;

p_x(t)—pressure at x position in wellbore, MPa;

q_i—unit weight of i-th section drill string, N/m;

Q_g—gas flow rate, m³/min;

r_e—supply boundary, m;

r_i, R_i—inner and outer radius of i-th section drill string, m;

r_w—borehole radius, m;

t—time, min;

Δt—time step, min;

T—formation temperature, K;

T_a—true axial force at section a, N;

T_e—effective axial force, N;

v—gas slippage velocity, m/s;

v₁(t,z), v₂(t,z)—gas slippage velocity at upper and lower ends of gas-liquid two-phase segment, m/s;

V_f(t)—filtrate loss of drilling fluid in unit time step, m³;

V_l—drilling fluid volume in well, m³;

V_lx(t)—drilling fluid unit volume in wellbore, m³;

V_g—overflow volume, m³;

ΔV_g—the increase volume of gas due to expansion in unit time during slippage rise, m³;

W_i—mass of i-th section drill string, kg;

x—grid number of wellbore annulus;

X—grid quantity in wellbore annulus;

z—grid number of gas-liquid two-phase segment;

Z—grid quantity of gas-liquid two-phase segment;

Z_g—natural gas compressibility factor, dimensionless;

μ—gas viscosity, mPa·s;

μ_m—drilling fluid viscosity, mPa·s;

ρ_g(t,z)—gas density in gas-liquid two-phase segment unit, kg/m³;

ρ_i, ρ_o—drilling fluid density inside and outside drillstring, kg/m³;

ρ_s—drill string density, kg/m³;

σ(t,z)—surface tension of gas-liquid, 10^-3 N/m.