Evolution Law of Wellbore Instability Risk under Fluctuating Pressure

Abstract

The bottom hole fluctuating pressure has a large influence on the wellbore instability. To address this problem, firstly, according to the principle of seepage mechanics, we established a calculation model of the change in pore pressure around the wellbore radius under fluctuating pressure; then, through laboratory rock mechanics test and rock damage mechanics theory analysis, the change law of formation strength under the action of static hydration and dynamic damage are determined; finally, based on the theory of rock mechanics in porous media, a quantitative evaluation method for the risk of wellbore instability under fluctuating pressure is established and the changing pattern of wellbore instability risk is analyzed. The results show that the pore pressure around the well shows a trend of fluctuation increase under fluctuating pressure, and there is a certain lag in the fluctuation of pore pressure inside the formation; the longer the muddy shale is immersed in drilling fluid, the greater the reduction in strength; the reduction is greater in the early stage of immersion, and the reduction in strength in water-based drilling fluid is greater than that in oil-based drilling fluid. At the beginning of the pressure cycle, the formation damage variable and compressive strength gradually increase and decrease with the increase of the pressure cycle number; after several cycles, the magnitude of change gradually decreases with the increase of the cycle number. When the bottom hole pressure fluctuates at a certain period, the greater the fluctuation, the shorter the period of wellbore stability; when the bottom hole pressure fluctuates at a certain range, the smaller the fluctuation period, the faster the borehole enters the high-risk period, while the shortest period of wellbore stability occurs when the fluctuation period is smallest; and when the wave cycle is in the middle, the wellbore stability period is the longest.

1. Introduction

The problem of wellbore instability is common in the drilling process. Wellbore instability usually causes serious economic losses. For plastic formation, the borehole shrinkage is usually caused by the low density of drilling fluid, so the casing is crushed or the borehole is abandoned. In brittle formations, improper use of drilling fluid density often causes rock collapse around the hole, leading to complex problems such as stuck drills. Wellbore instability not only affects drilling safety and increases drilling costs, but also affects logging and cementing quality, as well as the successful production of oil and gas resources. In severe cases, it can lead to hole abandonment. It is estimated that the annual loss due to wellbore instability is about 10% [1] of the total cost of drilling in the world; more than 40% [2,3,4] of all non-drilling time is lost.

In drilling engineering, because the pipe string (drill string, casing, tubing) can easily produce fluctuating pressure when moving in a fluid-filled hole [5,6], this can lead to the pressure in the well dropping or rising and the pressure system in the well losing balance, causing blowouts, losses, well collapses and other complex downhole conditions and accidents, bringing difficulties to oil reservoir protection [7].

The above mechanism analysis of wellbore instability can be attributed to the static damage of drilling fluid to rock strength and the influence of shale deformation and stress change around the well. In addition to the above reasons, this study also identified the disturbance of formation pressure caused by pressure fluctuation in the well and dynamic damage of formation strength in order to quantitatively describe the variation law of wellbore instability risk under the influence of the above factors. In this paper, a quantitative evaluation method of wellbore instability risk under fluctuating pressure is established on the basis of considering the disturbance of formation pressure caused by wellbore pressure fluctuation, dynamic damage of formation strength and analyzing the variation law of wellbore instability risk.

2. Research Status of the Subject

In order to solve this problem of wellbore instability, many scholars have carried out a lot of useful research. In 1988, Detournay [8] applied the elastic theory of Biot porous media to the analysis of borehole stability of vertical wells in isotropic formations, obtained the analytical solutions of borehole perimeter stress, displacement and pore pressure and analyzed the fracture and collapse conditions of borehole walls. In 1995, Cui [9] extended the results of the Detournay theory from vertical wells in isotropic formations to directional wells and obtained the distribution of per-well stress and pore pressure under different well angles, using the Drucker–Prager strength criterion to analyze the shear failure around wells. In 1995, Abousleiman [10] further applied the elastic theory of porous media to transverse isotropic strata. After a period of practice, it was found that in addition to the influence of the original stresses, such as ground stress, pore pressure and mud column pressure, the seepage action, temperature change and hydration in the process of drilling will also affect the stress state around the well, thus affecting the stability of the wellbore. In 1996, Van Oort [11] comprehensively summarized the driving forces of fluids under the interaction between drilling fluid and muddy shale formation, including liquid column pressure, chemical potential and electric potential difference between drilling fluid and muddy shale formation and temperature, and considered the semi-permeable film efficiency of low permeability mud shale to be about 1~10%. In 2001, Abousleiman [12] established the fluid–solid thermal coupling model of a porous medium in muddy shale, derived the relevant governing equations and finally obtained the analytical solution. In 2005, Abousleiman and Ekbote [13] studied the effect of anisotropic material parameter combination on pore pressure, effective stress and shear failure potential and obtained the analytical solution of pore thermoelasticity without considering the effect of thermal permeability and thermal filtration. In 2014, Yuan Cao [14] studied and analyzed the mechanism and prevention measures of wellbore instability caused by sandstone creep. In 2017, Cao et al. [15] studied Offshore wellbore stability using porothermoelastic theory, considering the special temperature and environmental conditions of the ocean.

Due to the influence of pressure fluctuation in the borehole, the instability of the wellbore has an obvious time effect. Previously, scholars have analyzed the time effect of the wellbore instability of muddy shale. Abousleiman et al. [16] (2000) studied the time-dependent wellbore stability prediction method of muddy shale and successfully applied it to the drilling practice of extended-reach wells. A wellbore stability analysis model coupled with poroelastic, temperature gradient, chemical potential and viscous rheological factor was established to guide the selection of drilling fluid density and fluid activity, as well as the time effect analysis of wellbore stability. Tare et al. [17] (2001) studied the time effect of water-based mud on muddy shale properties. The effects of 20%CaCl₂ + 2%KCl solution and 20%NaCl solution on the pore pressure and mechanical properties of mud shale were tested using a Pore Pressure Transfer test (PPT). Cheng Y F et al. [18] (2007) applied mechanical–chemical coupling theory to analyze the mechanism of delayed wellbore collapse of muddy shale. Based on the seepage mechanics of porous media, the influence of hydraulic pressure difference and the chemical potential difference were mainly considered, combined with the balance equation and the activity diffusion equation; the calculation model of pore pressure around the well under the mechanical–chemical coupling effect was obtained; and the variation law of pore pressure with time and space was analyzed. Mohiuddin M A [19] (2007) analyzed wellbore instability in vertical, directional and horizontal wells using field data. Yury V. Ilyushin and Vadim Festisov [20] (2022), considering high-paraffin reservoirs, proposed an automated process for extracting high-paraffin oil from marginal deposits. Shale reservoirs, high paraffin reservoirs, fractured reservoirs and other reservoirs with complex rock or fluid composition or complex structure, with more complex drilling fluid–reservoir/rock–reservoir fluid mechanics and chemical reaction mechanisms, is the future research direction to tackle.

3. Fluctuating Pressure

Intra-well fluctuation pressure is the additional pressure generated by the movement of the tubular string (drill string, casing, tubing) in a fluid-filled borehole. The reduction in pressure in the well due to suction during start-up is known as suction pressure, and the additional pressure increase in the well due to the displacement effect of the tubular string during down-drilling is known as fluctuating pressure. Fluctuating pressures can unbalance the pressure system in a well and cause downhole complications and accidents such as blow-outs, well leaks and well collapses, and can also cause difficulties in reservoir protection. Furthermore, as well depths increase and well diameters decrease, the operational problems associated with starting a trip or lowering casing become increasingly complex and severe. The three main causes of fluctuating pressure are the static shear force of the drilling fluid, the viscous force of the drilling fluid, and the inertia of the drilling tool.

3.1. Fluctuating Pressure Due to Static Shear Force of Drilling Fluid

The flow of drilling fluid must first overcome the static shear force and therefore generates fluctuating pressure. The fluctuating pressure that occurs at the moment of resumption of circulation to open pumping and the start of drilling down or casing processes is mainly caused by the hydrostatic shear force of the drilling fluid. The fluctuating pressure due to the static shear force of the drilling fluid can be calculated by the following equation:

$$\Delta P_t=\frac{4\times {10}^{-4}{\tau }_{s}L}{\left(D_h-D_{po}\right)}$$

(1)

where $\Delta P_t$ is the fluctuating pressure generated by the static shear force of the drilling fluid, MPa; ${\tau }_s$ is the static shear force of the drilling fluid in the wellhole, Pa; L is the length of the moving tubular column, m; $D_h$ is the diameter of the borehole, cm; and $D_{po}$ is the outside diameter of the moving tubular column, cm.

As can be seen above, the fluctuating pressure due to static shear force is proportional to the static shear of the drilling fluid for a given borehole geometry structure. With a high hydrostatic shear force of drilling fluid, pumping on and drilling down will often result in large pressure fluctuations.

3.2. Fluctuating Pressure Due to Inertial Forces in Moving Pipe Columns

If the drilling tool is accelerated in the borehole, the drilling fluid in the annulus is subjected to an inertial force; that is, the fluctuating pressure caused by the inertial force of the drilling tool, calculated by the following equation:

Total blockage of the lower casing and bit water way (plugged pipe):

$$\Delta P_i=\frac{1\times {10}^{-3}\rho La{D}_{po}^2}{D_h^2-D_{po}^2}.$$

(2)

For no blocked tube:

$$\Delta P_i=\frac{1\times {10}^{-3}\rho La\left(D_{po}^2-D_{pi}^2\right)}{D_h^2-D_{po}^2+D_{pi}^2}$$

(3)

where $\Delta P_i$ is the fluctuating pressure, MPa; $a$ is the axial acceleration of the annular fluid along the well body, m/s²; $\rho$ is the drilling fluid density, g/cm³; and $D_{pi}$ is the internal diameter of the motion string bore, cm.

When using the above formula, it is specified that acceleration is positive downwards and negative upwards. $\Delta P$ is the fluctuating pressure when it is positive and is the suction pressure when it is negative.

3.3. Fluctuating Pressure Due to Viscous Drag of Drilling Fluid

The frictional losses caused by the flow of drilling fluid upwards in the annulus due to displacement and adhesive attraction on the surface of the tubular column during the drilling down or casing process is due to the fluctuating pressure caused by the viscous force of the drilling fluid. The calculation process involves first identifying the flow pattern, then using different formulae for different flow patterns to calculate the coefficient of friction and finally calculating the fluctuating pressure.

(1)

Calculating the Reynolds number and determining the flow pattern

Bingham Fluids:

$$R_e=\frac{\rho \left(D_h-D_{po}\right)V_a}{{\mu }_p\left(1+\frac{{\tau }_o\left(D_h-D_{po}\right)}{8{\mu }_p{V}_a}\right)}.$$

(4)

Power-law fluids:

$$R_e=\frac{{\left(D_h-D_{po}\right)}^n{V}_a^{2-n}\rho }{{12}^{n-1}K{\left(\frac{2n+1}{3n}\right)}^n}$$

(5)

where $R_e$—Reynolds number, dimensionless number; $\rho$—drilling fluid density, kg/m³; ${\mu }_p$—plastic viscosity of Bingham fluid, Pa·s; ${\tau }_0$—the dynamic shear force of Bingham fluid, Pa; n—flow index of power-law fluid, dimensionless; K—consistency coefficient of power-law fluid, Pa·sⁿ; $D_h$—the diameter of the borehole, m; $D_{po}$—outside diameter of moving tubular column, m. $V_a$—total average flow velocity in the annulus when tripping string, m/s.

For Bingham fluids, $R_e$ < 2000 is laminar flow, and $R_e$ ≥ 4000 is turbulent flow. For power-law fluids, $R_e$ ≥ 3470–1370 n is turbulent.

(2)

Calculating the coefficient of friction

Laminar flow:

$$f=\frac{24}{R_e}.$$

(6)

Turbulent:

Bingham Fluids:

$$f=\frac{0.3164}{{R_e}^{0.25}}.$$

(7)

Power-law fluids:

$$f=\frac{a}{{R_e}^b}$$

(8)

where f—Friction coefficient, dimensionless number.

$$a=\left(\lg n+3.93\right)/50$$

(9)

$$b=\left(1.75-\lg n\right)/7$$

(10)

(3)

Calculation of fluctuating pressures

$$\Delta P_a=\frac{0.196f\rho L{V}_a^2}{D_h-D_{po}}$$

(11)

where $\Delta P_a$—the fluctuating pressure, MPa; $\rho$—the drilling fluid density, g/cm³; L—the length of the moving tubular column, m; $V_a$—total average flow velocity in the annulus when the tripping string, m/s; $D_h$—the diameter of the borehole, cm; $D_{po}$—outside diameter of moving tubular column, cm.

4. Pore Pressure Variation around the Well under Fluctuating Pressure

During the drilling process, the pressure at the bottom of the well dynamically changes due to factors such as tripping; factors such as pressure and velocity change with time in the process of drilling fluid seepage; and the boundary conditions carry time variables. Since the flow lines of the drilling fluid into the formation diverge from the bottom of the well to the surrounding area, the percolation of drilling fluid in the borehole into the formation is simplified to a planar radial percolation, and the basic differential equation is expressed as:

$$\frac{k}{u{C}_t}\left(\frac{{\partial }^{2}P}{\partial r^2}+\frac{1}{r}\frac{\partial P}{\partial r}\right)=\frac{\partial P}{\partial t}$$

(12)

where $C_t$ is the comprehensive compression factor; $u$ is the fluid viscosity; $k$ is the permeability; $r$ is the radial distance from the wellbore in time.

The initial and boundary conditions in a well under fluctuating pressure are expressed as:

$$\left\{\begin{array}{ll}t=0,& P(r,0)=P_0\\ r=r_w,& P(r_w,t)=P_w(t)\\ r=\infty ,& P(\infty ,t)=P_0\end{array}\right..$$

(13)

In order to describe the effect of drilling fluids on formation pore pressure under fluctuating pressure conditions, the variation of pore pressure with time at different locations around the borehole under different pressure fluctuations was analyzed and calculated. The simulated fluctuating pressure scenarios include the following and the results of the calculations are shown in Figure 1:

(1)

Sine wave, amplitude 0.5 MPa, period 20 h;

(2)

Sine wave, amplitude 2.0 MPa, period 20 h;

(3)

Sine wave, amplitude 2.0 MPa, period 10 h;

(4)

Sine wave, amplitude 2.0 MPa, period 30 h.

Based on the calculation results, it can be seen that when overbalance drilling, the pore pressure of the formation around the well fluctuates, increasing propulsion with time due to the seepage of drilling fluid into the formation under the conditions of fluctuating bottomhole pressure, and there is certain lag in the fluctuation of pore pressure inside the formation. The pore pressure in the formation within a short distance of the borehole fluctuates to a certain extent with the fluctuation of bottomhole pressure, and the median value of the fluctuation increases gradually as time goes on. As the distance from the borehole gradually increases, the amplitude of formation pore pressure fluctuations gradually decreases. The larger the fluctuation range of the bottom hole pressure is, the larger the area around the well formation pressure fluctuates; the shorter the fluctuation period is, the smaller the area around the well formation pressure fluctuates, but the faster the frequency of fluctuations in the formation near the wellbore.

5. Evolution Law of Formation Strength Damage under Fluctuating Pressure

5.1. Static Hydration

On the one hand, as the muddy shale formation matrix or fracture filling contains active clay minerals, the drilling fluid and its filtrate intruding into the fracture will chemically interact with the formation and fracture filler, weakening the strength of the formation; on the other hand, drilling fluid which intrudes into weak surface structures, such as cracks in shale, plays a lubricating role, changes the frictional properties of the wall and weakens the strength of the formation.

In order to study the variation law of muddy shale strength with soaking time, water-based and oil-based drilling fluids were configured, respectively, and drilling fluid soaking tests were conducted on mud shale rock blocks obtained from the field.

The water-based drilling fluid formulation used in the test was 3% bentonite + 4% sodium carbonate soil + 0.8% ammonium + 0.5% PLH + 0.2% FA367 + 0.3% XY-27 + 4% SMC + 4% SMP-1 + 5% sulfonated bitumen + 3% fine mesh calcium + 3% lubricant + 2% NW-1 + 3% KCL + 3% polymeric alcohol (40 to 80 °C) + 2% non-permeable treatment agent.

The oil-based drilling fluid formulation used in the test was 70% diesel + 4% SP-80 + 2% oleic acid + 5% organoclay + 2.5% sulfonated bitumen + 3% lime + 30% calcium chloride solution (51% concentration) + 4% fine-mesh calcium.

The strength parameters of the mud shale blocks in Well A and Well B after 24 h, 48 h, 72 h and 96 h of drilling fluid immersion are shown in

Table 1. Strength test results of mud shale in Well A after soaking in mud.

Number of Tests	Strength of Different Types of Drilling Fluids at Different Immersion Times (MPa)
	Water-Based 24 h	Oil-Based 24 h	Water-Based 48 h	Oil-Based 48 h	Water-Based 96 h	Oil-Based 96 h
1	36.6	32.1	52.1	33.1	21.3	69.2
2	72.1	136.5	21.0	112.2	46.3	55.6
3	25.8	28.8	22.6	57.9	10.5	97.5
4	45.5	86.9	39.7	92.3	16.2	23.5
5	61.5	62.2	56.4	45.6	33.8	73.6
6	31.9	118.5	--	--	--	--
Average value	45.6	77.5	38.4	68.2	25.6	63.9

and

Table 2. Strength test results of mud shale in Well B after soaking in mud.

Number of Tests	Strength of Different Types of Drilling Fluids at Different Immersion Times (MPa)
	Water-Based 24 h	Oil-Based 24 h	Water-Based 72 h	Oil-Based 72 h
1	58.7	72.1	60.2	20.9
2	23.7	53.5	13.1	37.3
3	38.1	113.1	19.6	126.5
4	127.3	38.2	56.5	59.2
5	26.2	35.1	12.9	103.3
6	17.5	91.3	20.7	33.6
Average value	48.6	67.2	30.5	63.5

, respectively. According to the test results, after soaking drilling fluid, the uniaxial strength decreases, and the longer the soaking time, the greater the strength reduction range.

Based on the comparative results, it can be seen that the oil-based drilling fluid is better than the water-based drilling fluid in maintaining formation strength significantly. The average uniaxial strength of the muddy shale block in Well A decreased by about 70% after 4 days of immersion in the water-based drilling fluid; the uniaxial strength decreased by only about 22% after 4 days of immersion in the oil-based drilling fluid. The average uniaxial strength of the muddy shale block in well B decreased by about 60% after soaking the water-based drilling fluid for 3 days; the uniaxial strength only decreased by about 17% after 3 days of immersion in the oil-based drilling fluid.

The analysis of the variation pattern of muddy shale strength with water-based and oil-based drilling fluid immersion time for Wells A and B is shown in Figure 2. According to the experimental results, it was found that when the mud shale was immersed in the drilling fluid, the strength decreased more at the initial stage and less at the later stage. The analysis suggests that this is mainly due to the fact that when the mud shale is immersed in drilling fluid, the driving effect of physicochemical factors such as capillary force and chemical potential causes the drilling fluid to penetrate the rock along its fractures, changing its frictional properties and causing hydration reactions, making the formation strength gradually weaken over time. When the filtrate of drilling fluid is saturated in the rock, the muddy shale water absorption stops and the drilling fluid filtrate no longer undergoes subsequent physicochemical reactions with the rock matrix, thus reducing the formation strength to its minimum value. Based on the results of the analysis, it was determined that the pattern of variation in the strength of the mud shale formation over time when exposed to drilling fluid was:

$$UCS\left(t\right)=\left(UC{S}_{\mathrm{I}}-UC{S}_{\mathrm{f}}\right)\cdot ercf\left(a\times t\right)+UC{S}_{\mathrm{f}}$$

(14)

where:

• $UCS\left(t\right)$ is the strength of the formation at the time t, MPa;
• $UC{S}_{\mathrm{I}}$ is the strength of the formation before contact with the drilling fluid, MPa;
• $UC{S}_{\mathrm{f}}$ is the final strength of the formation after contact with the drilling fluid, MPa;
• $a$ is a constant related to the properties of the formation and the physicochemical properties of the mud;
• $t$ is the contact time between the drilling fluid and the formation around the well, hrs.

5.2. Dynamic Damage

The rock around the well is subjected to cyclic loading and unloading as the pressure at the bottom of the well changes dynamically. A triaxial rock mechanics experimental test system was used to perform cyclic loading and unloading experiments on the cores in the field. In order to avoid damage to the cores caused by the upper-stress limit of the cyclic load, the strength of muddy shale cores with similar properties at the same depth was first tested before the experiment, and the uniaxial compressive strength of the cores used for this experiment was estimated to be about 70 MPa. During the test, the upper-stress limits of the four cores were 69%, 74%, 70% and 79% of the uniaxial compressive strength, respectively, and the experimental parameters and experimental results are shown in

Table 3. Experimental parameters and results of mud shale compression under cyclic load.

Sample Number	Upper and Lower Limits of Cyclic Stress (MPa)	Relative to Uniaxial Compressive Strength	Cycle Period (h)	Cycle Index	Damage Condition
1#	2~48	2.9~68.6%	0.07	6	unbroken
2#	2~52	2.9~74.3%	0.07	6	unbroken
3#	2~49	2.9~70.0%	0.32	4	unbroken
4#	2~55	2.9~78.6%	0.07	4	unbroken

. Based on the experimental results, it can be seen that the fatigue damage phenomenon did not occur after several cycles of the mud shale at stress levels of around 80% of the uniaxial compressive strength.

The cyclic loading and unloading did not cause fatigue damage to the muddy shale due to the stress level and the number of cycles; however, analysis of the experimental results revealed that fatigue damage occurred to the muddy shale under cyclic loading, the uniaxial compressive strength weakened over time and the risk of causing wellbore instability increased under cyclic loading. This is mainly due to the development of internal fractures in the muddy shale, which is caused by the continuous cyclic loading and unloading, resulting in the continuous closure and opening of the original fractures and the gradual creation, expansion and penetration of new fractures.

According to damage mechanics theory, during the cyclic loading and unloading process, the rock sample has already produced local damage before macroscopic damage occurs, resulting in a reduction in rock strength. Based on the strain equivalence hypothesis of the damage mechanics theory, and considering the damage process of the rock, the damage constitutive equation can be expressed by the following equation [21]:

$$\sigma =E(1-D)\epsilon$$

(15)

where $D$ is the damage variable; the larger the value of $D$, the more severe the rock damage level and the higher the strength attenuation value.

Using the modulus of elasticity method, the damage variable can be expressed by the following equation:

$$D=1-\frac{\tilde{E}}{E}$$

(16)

where $\tilde{E}$ is the modulus of elasticity of the rock after damage.

The modulus of elasticity after rock damage in the above method is usually taken as the unloading stiffness, which is only suitable for elastic damage and not applicable to elastoplastic damage. Based on the modulus of elasticity method, the constitutive equation for elastoplastic damage can be expressed by the following equation [14]:

$$D=1-\frac{\epsilon -{\epsilon }_P}{\epsilon }\frac{E^{\prime }}{E}$$

(17)

where ${\epsilon }_P$ is the plastic strain and $E^{\prime }$ is the modulus of elasticity at unloading.

After rock sample damage, some of its internal infinitesimals lose their bearing capacity and the effective bearing area of the rock macroscope is reduced; therefore, the strength is attenuated and the uniaxial compressive strength of the rock after damage can be expressed by the following formula [22]:

$${\sigma }_{c1}=(1-D){\sigma }_c$$

(18)

where ${\sigma }_{c1}$ is the uniaxial compressive strength after damage and ${\sigma }_c$ is the uniaxial compressive strength before damage.

The damage variables and compressive strength development patterns of the experimental mud shale cores, compared with the number of cycles of loading and unloading, were analyzed using the above method. The results of the analysis are shown in Figure 3. According to the analysis results, it can be seen that, after several cycles, a certain degree of fatigue damage occurred in the rock, and the uniaxial compressive strength decreased. With an increase in the number of cycles, the damage variables and the development of strength show two stages: at the beginning of the cycle, the damage variables and compressive strength gradually increase and decrease with the increase in the number of cycles; after several cycles, the rangeability of the damage variables and compressive strength gradually decrease with the increase in the number of cycles after several cycles.

6. Methodology for Evaluating the Risk of Wellbore Instability under Fluctuating Pressure

According to the elastic mechanic theory of porous media, concentrated stress is induced surrounding the borehole due to stress redistribution after wells were drilled. Various authors have addressed methods for calculating the concentrated stress state under the effect of in situ ground stress and bottom hole pressure [23,24,25]. Many wells being drilled for oil and gas production are either horizontal, highly deviated from vertical or have complex trajectories, and for an arbitrarily deviated well, the principal stresses acting in the vicinity of the wellbore wall are generally not aligned with the wellbore axis, so the stress state surrounding an arbitrarily deviated well should be calculated through coordinate system transformation. At first, the in situ ground stress should be transformed from the geodetic coordinate systems (1, 2, 3) to the borehole coordinate system (x, y, z), and the coordinate conversion is presented in Figure 4.

The stress tensor in the borehole coordinate system can be given by transforming in situ ground stresses:

$$\left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{yx}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{zx}& {\sigma }_{zy}& {\sigma }_{zz}\end{array}\right]\hspace{1em}=\hspace{1em}\left[L\right]\hspace{1em}\left[\begin{array}{ccc}{\sigma }_H& 0& 0\\ 0& {\sigma }_h& 0\\ 0& 0& {\sigma }_v\end{array}\right]\hspace{1em}{\left[L\right]}^{\mathrm{T}}$$

(19)

where ${\sigma }_H$ and ${\sigma }_h$ correspond to the maximum and minimum horizontal principal stress, respectively, ${\sigma }_v$ is vertical principal stress and L is the coordinate system transformation matrix:

$$[L]=\hspace{1em}\left[\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \beta \\ -\sin \beta & \cos \beta & 0\\ \sin \alpha \cos \beta & \sin \alpha \cos \beta & \cos \beta \end{array}\right]$$

(20)

where $\alpha$ is the well deviation angle and $\beta$ is the well azimuth angle relative to the maximum horizontal principal stress orientation.

Assuming plane strain is normal to the borehole axis, concentrated stress components solutions for the permeable formation surrounding an arbitrarily deviated well in terms of cylindrical polar coordinates can be written as:

$$\left\{\begin{array}{l}{\sigma }_r=\frac{R_w{}^2}{r^2}{P}_i+\frac{\left({\sigma }_{xx}+{\sigma }_{yy}\right)}{2}\left(1-\frac{R_w{}^2}{r^2}\right)+\frac{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}{2}\left(1+\frac{3R_w{}^4}{r^4}-\frac{4R_w{}^2}{r^2}\right)\cos 2\theta \\ \begin{array}{cc}& \end{array}+{\sigma }_{xy}\left(1+\frac{3R_w{}^4}{r^4}-\frac{4R_w{}^2}{r^2}\right)\sin 2\theta +\delta \left[\frac{\alpha \left(1-2\nu \right)}{2\left(1-\nu \right)}\frac{\left(r^2-R_w{}^2\right)}{r^2}-f\right]\left(P_i-P_P\right)\\ {\sigma }_{\theta }=-\frac{R_w{}^2}{r^2}{P}_i+\frac{\left({\sigma }_{xx}+{\sigma }_{yy}\right)}{2}\left(1+\frac{R_w{}^2}{r^2}\right)-\frac{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}{2}\left(1+\frac{3R_w{}^4}{r^4}\right)\cos 2\theta \\ \begin{array}{cc}& \end{array}-{\sigma }_{xy}\left(1+\frac{3R_w{}^4}{r^4}\right)\sin 2\theta +\delta \left[\frac{\alpha \left(1-2\nu \right)}{2\left(1-\nu \right)}\frac{\left(r^2+R_w{}^2\right)}{r^2}-f\right]\left(P_i-P_P\right)\\ {\sigma }_z={\sigma }_{zz}-\nu \left[2\left({\sigma }_{xx}-{\sigma }_{yy}\right){\left(\frac{R_w}{r}\right)}^2\cos 2\theta +4{\sigma }_{xy}{\left(\frac{R_w}{r}\right)}^2\sin 2\theta \right]+\delta \left[\frac{\alpha \left(1-2\nu \right)}{2\left(1-\nu \right)}-f\right]\left(P_i-P_P\right)\\ {\sigma }_{r\theta }={\sigma }_{xy}\left(1-\frac{3R_w{}^4}{r^4}+\frac{2R_w{}^2}{r^2}\right)\cos 2\theta -\frac{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}{2}\left(1-\frac{3R_w{}^4}{r^4}+\frac{2R_w{}^2}{r^2}\right)\sin 2\theta \\ {\sigma }_{\theta z}={\sigma }_{yz}\left(1+\frac{R_w{}^2}{r^2}\right)\cos \theta -{\sigma }_{xz}\left(1+\frac{R_w{}^2}{r^2}\right)\sin \theta \\ {\sigma }_{zr}={\sigma }_{xz}\left(1-\frac{R_w{}^2}{r^2}\right)\cos \theta +{\sigma }_{yz}\left(1-\frac{R_w{}^2}{r^2}\right)\sin \theta \end{array}\right.$$

(21)

where $R_w$ represents wellbore radius, r represents the distance from the borehole axis, θ is the azimuth angle relative to the x-axis and z is the position along the borehole axis.

The principal stresses should be calculated by solving the three roots of the characteristic equation:

$${\sigma }_n^3-I_1{\sigma }_n^2+I_2{\sigma }_n^{}-I_3=0$$

(22)

where

$$\begin{array}{l}{I}_1={\sigma }_r+{\sigma }_{\theta }+{\sigma }_z\\ I_2={\sigma }_r{\sigma }_{\theta }+{\sigma }_{\theta }{\sigma }_z+{\sigma }_z{\sigma }_r-{\tau }_{r\theta }^2-{\tau }_{\theta z}^2-{\tau }_{zr}^2\\ I_3=\left|\begin{array}{ccc}{\sigma }_r& {\tau }_{r\theta }^{}& {\tau }_{rz}^{}\\ {\tau }_{\theta r}^{}& {\sigma }_{\theta }& {\tau }_{\theta z}^{}\\ {\tau }_{zr}^{}& {\tau }_{z\theta }^{}& {\sigma }_z\end{array}\right|\end{array}.$$

The above cubic characteristic equation has three solutions, representing the three principal stresses, respectively, and denoted as ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ from large to small values.

Shear failure will be generated when the Mohr’s circle constituted by the maximum and minimum effective principal stress on the wellbore wall exceeds the shear strength; at this time, borehole collapse occurs. According to the Mohr–Coulomb failure criterion, the concept of failure ratio is introduced to judge the instability risk of the rock around the well. The formation failure ratio is defined as follows:

$$Fr=\frac{{\sigma }_1-\beta P_p(r)}{\left({\sigma }_3-\beta P_p(r)\right)ct{g}^2\left({45}^0-\phi /2\right)+2C\cdot ctg\left({45}^0-\phi /2\right)}$$

(23)

where ${\sigma }_1$ and ${\sigma }_3$ represent maximum and minimum principal stress, respectively; $C$ represents cohesive strength; $\varphi$ represents the internal friction angle; $\beta$ represents effective stress coefficient; $P_p(r)$ represents pore pressure at different locations around the well.

According to the above definition, failure occurs when the failure ratio of rock surrounding a well is greater than 1. The greater the formation failure ratio, the greater the risk of wellbore instability

7. Patterns of Change in Wellbore Instability Risk under Fluctuating Pressure

On the one hand, the fluctuation of bottom hole pressure causes the pressure, velocity, etc., of drilling fluid in the internal seepage process of the formation to change with time, affecting the pore pressure and effective stress state around the borehole; on the other hand, under the action of cyclic loading and unloading of dynamic load in the bottom hole, the fractures in the mud shale gradually open and close, and fatigue damage occurs in the original mud shale, resulting in the decay of formation strength and making the risk of wellbore destabilization show significant temporal correlation characteristics.

For an overall consideration of the effective stress and strength changes caused by the infiltration of drilling fluid into the formation and the fatigue damage of the muddy shale under cyclic loading, the derivation process of the maximum failure ratio of the formation around the borehole with time under different pressure fluctuation amplitudes and different pressure fluctuation cycles was analyzed and calculated; the simulated bottom hole pressure fluctuations during the calculation are shown in Figure 5, and the calculation results are shown in Figure 6 and Figure 7, respectively.

According to the calculation results, when overbalance drilling:

(1)

A fluctuating increase pattern of the maximum failure ratio of the formation around the borehole is propelled with time due to the seepage of drilling fluid into the formation and cyclic loading and unloading under fluctuating bottom hole pressure conditions;

(2)

When the period of pressure fluctuation at the bottom of the well is certain, the greater the fluctuation range, and the faster the maximum failure ratio of the formation around the borehole increases, the shorter the corresponding borehole drilling open time. When the maximum failure ratio of the formation around the borehole reaches 0.9 and 1, the shorter the period of wellbore stability;

(3)

When the extent of pressure fluctuation at the bottom of the well is certain, the smaller the fluctuation period, the faster the maximum failure ratio of the formation around the borehole increases; when the maximum failure ratio of the formation around the borehole reaches 0.9, the shorter the corresponding borehole drilling open time, the borehole moves into a high-risk period more quickly; however, the correlation between the borehole drilling open time and the fluctuation period corresponding to a maximum failure ratio of 1 in the formation surrounding the borehole changes the shortest period of fluctuation (10 h) and the shortest period of borehole drilling open time, corresponding to a maximum failure ratio of 1 in the formation around the borehole and the shortest period of wellbore stabilization. When the fluctuation period is in the middle (20 h), the maximum failure ratio of the formation around the borehole reaches 1, which corresponds to the longest borehole drilling time and the longest period of wellbore stability. In contrast, during the maximum fluctuation period (30 h), the maximum failure ratio of the formation around the borehole reaches 1, which corresponds to the middle of the borehole drilling open time and the middle of the wellbore stabilization period.

8. Discussion

In this paper, a quantitative evaluation method for the risk of wellbore instability under fluctuating pressure is established and the changing pattern of wellbore instability risk is analyzed. The advantages of this method are that the static hydration damage and dynamic mechanical damage of shale are comprehensively considered, and the influence of bottomhole pressure and pore pressure fluctuation around well is quantitatively included. Meanwhile, the dynamic evolution law of borehole instability risk is characterized by the index value of the formation failure ratio, which has certain guiding significance for prediction and early warning of borehole instability. This method is easy to solve and convenient for quick evaluation on site. However, this method also has its disadvantages; it does not consider the full dynamic mechanical–chemical seepage coupling of wellbore pressure fluctuation with the formation and cannot reflect the real law exactly. In fact, the calculation of full coupling is relatively complex, with more basic parameters input, and the calculation speed is slow, which does not take advantage of the fast and effective field application. Based on the fully coupled model, developing algorithm optimization is the main direction for the future of this research.

9. Conclusions

The fluctuating pressure at the bottom of a well caused by the movement of the tubular column in a fluid-filled borehole has a large impact on the collapse and destabilization of the wellbore. The mechanism of influence mainly includes the change in pore pressure and effective stress caused by the percolation of drilling fluid into the formation under fluctuating pressure and the strength damage of the formation under fluctuating pressure. Based on theoretical and experimental test analysis, a calculation method for the variation law of pore pressure and strength around the well under fluctuating pressure was established, and on this basis, a quantitative evaluation method for the risk of wellbore instability under fluctuating pressure was established and the pattern of change in the risk of wellbore instability under fluctuating pressures was also analyzed.

(1)

The seepage of drilling fluid into the interior of the formation under fluctuating bottom pressure conditions causes the pore pressure of the formation around the well to increase in a fluctuating trend overall as time advances, and the fluctuation of the pore pressure inside the formation has a certain lag; the greater the magnitude of the fluctuation of bottom pressure, the larger the area of the fluctuation of the formation pressure around the well; and the shorter the fluctuation period, the smaller the area of the fluctuation of the formation pressure around the well, but the fluctuation frequency of the formation near the wellbore is also faster;

(2)

Seepage of drilling fluid into the formation interior at fluctuating pressures can cause static hydration damage and dynamic strength damage to the strength of mud shale formations. For static hydration, the longer the mud shale is immersed in the drilling fluid, the greater the reduction in strength, and the reduction in strength is greater at the beginning of immersion, while the reduction in strength is greater in water-based drilling fluids than in oil-based drilling fluids. For dynamic damage, the damage variable and compressive strength gradually increased and decreased with the number of cycles at the beginning of the pressure cycle; after several cycles, the change in damage variable and compressive strength gradually decreased with the number of cycles;

(3)

We quantitatively evaluate wellbore destabilization risk by introducing the formation failure ratio in the Moore–Coulomb criterion. When the fluctuation period of bottom pressure is certain, the greater the range of the fluctuation and the shorter the period of wellbore stability; when the fluctuation period of bottom pressure is certain, the smaller the fluctuation period and the shorter the drilling open time, corresponding to the maximum failure ratio of the formation around the borehole reaching 0.9. The borehole enters the high-risk period faster, but the correlation between the drilling open time, corresponding to the maximum failure ratio of the formation around the borehole, and the fluctuation period reaches 1. The correlation between the borehole drilling start time and fluctuation period changes, with the shortest period of wellbore stability occurring at the lowest fluctuation period and the longest period of wellbore stability occurring at the middle of the fluctuation period.