1 Introduction

Wellbore instability has been a long-term issue in the drilling industry. It increases non-productive time, causing billions of dollars to be lost every year. In the past decades, researchers have studied wellbore stability with aspects to consolidation effect, temperature effect, chemical effect, etc. No matter which kind of effects that need to be studied, accurate wellbore stress distribution is the prerequisite for wellbore stability analysis. Nearly, all the innovative theoretical methods have to be utilized through the changes of stress distribution using the superposition principle. In 1898, the Kirsch solution was published to calculate stress distribution around a circular hole in an infinite plate. Later, Bradley (1979) showed the elastic analytical solution of stresses around the wellbore by taking advantage of the Kirsch solution together with the solution by Fairhurst (1967). It was then regarded as the general solution to calculate stress distribution by many researchers (Aadnoy 1987; Fjar et al. 2008; Chen 2008; Jaeger et al. 2009; Zoback 2010; Meng 2019b). Although this method is commonly used by most researchers, one important point to be noted is this method is mostly used for the steady state. It is different from the solution strategy provided by Detournay and Cheng (1988), which was used for wellbore stress distribution under instantaneous drilling or wellbore excavation. They decomposed the calculation of wellbore stresses into three modes including far-field isotropic stress, virgin pore pressure, and a far-field stress deviator. The far-field stress deviator could cause induced pore pressure. The solution is more complex than the general Kirsch solution and requires the Laplace transform and numerical Laplace inverse methods to get the final results.

The tripping operation is one important procedure in the drilling process due to the necessities of changing the drill bit, downhole assembly, subsea equipment, etc. During tripping operations, the drill string movement generates a dynamic force resulting in a longitudinal wave. The wave propagates along the drill string and causes a variance in the wellbore pressure. The reduced pressure caused by tripping the drill string outside the wellbore is called swab pressure and the increased pressure caused by tripping the drill string inside the wellbore is called surge pressure. The surge/swab pressure is one major reason for severe drilling problems, including wellbore instability, lost circulation, fluid kick, or even blowouts (Cannon 1934; Lubinski 1977; Mitchell 1988). Tripping velocity is commonly regarded as one major factor that is positively related to the values of surge/swab pressure. If the tripping velocity is slow, the surge/swab pressure is low enough to guarantee safe drilling process. However, a slow tripping velocity makes drilling very time consuming, and the daily well drilling cost is extremely high, especially in the offshore drilling. If the tripping velocity is high, safety issues, especially wellbore instability, will occur. Therefore, it is highly necessary to analyze the wellbore stability under tripping operation. To realize this goal, there are three major steps: the prediction of surge/swab pressure, the calculation of dynamic stress, and the application of failure criteria.

The prediction of surge/swab pressure has gained the interest of drilling engineers for a long time (Crespo 2012; Mme 2012; Wang 2013; Karlsen 2014). Cannon (1934) first proposed that the swab pressure could cause fluid influx, lost circulation, or even blowouts. Then, it was realized that the prediction of the surge/swab pressure is significant. Early researchers (Burkhardt 1961; Schuh 1964; Fontenot 1974) used “steady flow” models to predict surge/swab pressure. The results were not realistic, because it neglected fluid inertia and compressibility of the fluid and wellbore. Later, Lubinski (1977) initially proposed one fully dynamic surge-pressure model used for Bingham plastic fluids. This model was based on transient wave propagation theory which considered fluid compressibility and wellbore expansion. Mitchell (1988) further improved the dynamic model by coupling pipe and annulus pressures through pipe elasticity. Based on the former studies, Zamanipour (2015, 2016) developed the program to predict dynamic surge/swab pressure for both Bingham plastic fluids and power-law fluids.

Although the conventional Kirsch method and Detournay and Cheng’s method are the widely used methods, neither of them is appropriate for dynamic stress distribution of the wellbore under tripping operations. The Kirsch method is used for steady state, but dynamic surge/swab is transient. Detournay and Cheng’s method is used for excavation, but tripping operations occur after the wellbore has already been excavated. Therefore, new methods should be proposed to calculate dynamic stress around the wellbore under tripping operations. Zamanipour (2016) studied stress distribution around a directional wellbore under tripping conditions. It took advantage of the method proposed by Wang (2000) which regarded the surge/swab pressure effect as the sudden pressurization of a borehole. This method neglected the inertia effect in the equilibrium equation, which is one significant difference between dynamic and static loading. Zhang et al. (2016) used the instantaneous drilling method to study transient stress distribution around the wellbore. Detournay and Cheng’s method is usually used for wellbore excavation, or instantaneously drilling process, but tripping operations occur right after the wellbore has already been drilled. The influence of deviatoric loading on the tangential stress only dissipates slowly when the formation permeability is very low. Therefore, a new method should be proposed to calculate dynamic stress distribution under tripping operations.

The last step of wellbore stability analysis is the application of failure criteria. During tripping operations, both shear failure and tensile failure may happen. Shear failure, also called wellbore collapse, could happen during tripping out. Tensile failure, also called wellbore fracturing, could happen during tripping in. For the wellbore collapse, there are many failure criteria, including Mohr–Coulomb failure criterion, Mogi–Coulomb failure criterion (Al-Ajmi 2006; Islam et al. 2010), Modified Lade failure criterion (Ewy 1999), Hoek–Brown failure criterion, 3D Hoek–Brown failure criterion (Zhang 2007, 2008), and Drucker–Prager failure criterion. For wellbore fracturing, the pressurization rate is important, because it can influence the breakdown pressure of a wellbore. Overall, a reasonable failure criterion should be used for analyzing wellbore stability under dynamic loading conditions (Zhao 2000; Zhou and Zhao 2011) and shear failure is usually focused when researchers work on wellbore stability analysis (Zoback 2010). To the best of our knowledge, whether the static failure criteria could be used for wellbore stability analysis under tripping operations and which is the best option has not been studied by past researchers.

In this paper, a new poroelastodynamics model is established to calculate dynamic stress distribution under tripping operations. It is solved numerically by the implicit finite-difference method. The dynamic surge/swab pressure is predicted based on Lubinski’s method. They are implemented into the numerical solving procedure of the poroelastodynamics model to calculate dynamic stress distribution. Then, different failure criteria are applied to the dynamic wellbore collapse analysis under tripping operations. Furthermore, the triaxial compression test has been performed on Bedford limestone under different confining pressure and strain rate. The results are used to check the accuracy of different failure criteria and verify they can be used for wellbore stability analysis under tripping operations.

2 Modeling and Solution Strategy

2.1 Prediction of Dynamic Surge/Swab Pressure

The prediction of surge/swab pressure is calculated using Lubinski’s method (1977, 1988) and the procedures are introduced clearly by Zamanipour (2015, 2016) and Zhang et al. (2016, 2018). Only the basic procedures will be introduced here. To calculate the surge/swab pressure, the wellbore needs to be divided by a discrete number of small sections. The connection between two adjacent small sections is called “junctions with artificial orfices” by Lubinski, which is shown in Fig. 1. Mechanisms of energy dissipation (irreversible drops of pressure) are concentrated at these junctions.

Fig. 1
figure 1

Schematic diagram of junction with an artificial orifice

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Consider that at time t − 1, flow rates Q and pressure p are known at the junctions, and it is now desired to calculate Q and p at time t at junction B. The following two equations of propagation are the algebraic equivalents of the corresponding Bergeron graphical construction:

$$ p_{{t,{\text{Bb}}}} - p_{{t - 1,{\text{Aa}}}} = - S_{{{\text{AB}}}} \left( {Q_{{t,{\text{Bb}}}} - Q_{{t - 1,{\text{Aa}}}} } \right), $$
(1)
$$ p_{{t,{\text{Ba}}}} - p_{{t - 1,{\text{Cb}}}} = + S_{{{\text{BC}}}} \left( {Q_{{t,{\text{Ba}}}} - Q_{{t - 1,{\text{Cb}}}} } \right), $$
(2)

in which the first subscript pertains to time and the second to location. \({S}_{\text{AB}}\) and \({S}_{\text{BC}}\) are characteristic impedances of sections \(\text{AB}\) and \(\text{BC}\), respectively. \({p}_{t,\text{Bb}}\) and \({p}_{t,\text{Ba}}\) are pressure of position \({\text{B}}_{\text{b}}\) and \({\text{B}}_{\text{a}}\) at time \(t\). \({p}_{t-1,\text{Aa}}\) and \({p}_{t-1,\text{Cb}}\) are pressure of position \({\text{A}}_{\text{a}}\) and \({\text{C}}_{\text{b}}\) at time \(t-1\), respectively. \({Q}_{t,\text{Bb}}\) and \({Q}_{t,\text{Ba}}\) are flow rate of position \({\text{B}}_{\text{b}}\) and \({\text{B}}_{\text{a}}\) at time \(t\). \({p}_{t-1,\text{Aa}}\) and \({p}_{t-1,\text{Cb}}\) are flow rate of position \({\text{A}}_{\text{a}}\) and \({\text{C}}_{\text{b}}\) at time \( t - 1 \), respectively. Upward flow is considered positive, which dictates the signs of \({S}_{\text{AB}}\) and \({S}_{\text{BC}}\).

In Eqs. (1) and (2), there are four unknowns at time \(t\) at the junction \(B\), namely: \({p}_{t,\text{Bb}}\), \({p}_{t,\text{Ba}}\), \({Q}_{t,\text{Bb}}\), \({Q}_{t,\text{Ba}}\). Therefore, we need two more equations to find these unknown values. These equations will pertain to changes in values of \(Q\) and \(p\) between \({\text{B}}_{\text{b}}\) and \({\text{B}}_{\text{a}}\).

Assume the downward as the positive direction of pipe velocity, the continuity equation is

$$ Q_{{t,{\text{Ba}}}} - Q_{{t,{\text{Bb}}}} = V_{t} \Delta {\text{A}}_{{\text{B}}} , $$
(3)

in which \(\Delta {\text{A}}_{\text{B}}\) is the change in pipe cross-sectional area at the junction \(B\), and \({V}_{t}\) is pipe velocity at time \(t\).

The continuity equation applies to all junctions except at the surface, at the bottom of the hole, and at the lower end of the pipe string. There is no continuity equation at the surface. At the bottom of the hole, \(Q=0\). The continuity equation at the lower end of the pipe string is written in Fig. 2. If there is no cross-sectional area change, then \({Q}_{t,\text{Ba}}={Q}_{t,\text{Bb}}\).

Fig. 2
figure 2

Continuity and pressure loss equations at the lower end of the pipe string

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The fourth equation is

$${p}_{t,\text{Ba}}-{p}_{t,\text{Bb}}=\varDelta p,$$
(4)

in which \(\Delta p\) is the loss of pressure in the artificial orifice.

For the pressure boundary condition, the pressure loss Eq. (4) applies to all junctions except at the surface and at the lower end of the pipe string. At the surface, \({p}_{s}=0\). At the lower end of the pipe string, the boundary conditions are shown in Fig. 2.

2.2 Dynamic Stress Distribution

2.2.1 Governing Equations

To obtain the governing equations for stress distribution around the wellbore under dynamic loading, fundamental knowledge of poroelasticity should be used. The normal method is to combine strain–displacement relation, equations of motion in poroelastic medium, and Hooke’s law together. Then, the governing equations for the poroelastodynamics model could be generated. The major assumptions are: (1) the rock is continuous and isotropic; (2) the plane strain condition works for this problem; (3) the fluid inertia can be neglected; and (4) it is a cylindrically symmetrical displacement field.

For cylindrical coordinates, the equilibrium equations are (Sadd 2009)

$${\sigma }_{ij,j}=\rho \frac{{\partial }^{2}{u}_{i}}{\partial {t}^{2}},$$
(5)

where \({\sigma }_{ij,j}\) is the stress gradient tensor, \(\rho\) is density, \({u}_{i}\) is the displacement vector.

The constitutive equations are

$${\sigma }_{ij}=2G {\sigma }_{ij}+2G\frac{v}{1-2v}{\epsilon }_{kk}{\delta }_{ij}-\alpha p{\delta }_{ij}\rho \frac{{\partial }^{2}{u}_{i}}{\partial {t}^{2}},$$
(6)

where \({\sigma }_{ij}\) is the stress tensor, \(G\) is Lame parameter, \(v\) is Poisson’s ratio, \({\epsilon }_{kk}\) is the volumetric strain, \({\delta }_{ij}\) is the Kronecker delta, and \(\alpha\) is the Biot coefficient.

The strain–displacement relation is

$$ \varepsilon _{r} = \frac{{\partial u_{r} }}{{\partial r}},\varepsilon _{\theta } = \frac{1}{r}\left( {u_{r} + \frac{{\partial u_{\theta } }}{{\partial \theta }}} \right),\varepsilon _{z} = \frac{{\partial u_{z} }}{{\partial z}}, $$
(7)

where \(\varepsilon _{r} \), \(\varepsilon _{\theta } , \) and \(\varepsilon _{z} \) are the radial, tangential, and vertical normal strain, respectively. \(r,\theta ,{\text{and}}\;z \) are radial, tangential, and vertical coordinates, respectively.

Through combining Eqs. (5) to (7), the governing equation for poroelastodynamics becomes

$$\left( {2G + \lambda } \right)\left( {\frac{{\partial ^{2} u_{r} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial u_{r} }}{{\partial r}} - \frac{{u_{r} }}{{r^{2} }}} \right) - \frac{\alpha }{G}\frac{{\partial p}}{{\partial r}} = \rho \frac{{\partial ^{2} u_{r} }}{{\partial t^{2} }}, $$
(8)

where \(\lambda\) is Lame parameter.

For the constitutive equations, either pore pressure or the increment of fluid content can be chosen as the fluid variable. A nonhomogeneous diffusion equation for pore pressure is

$$\alpha \frac{{\partial \varepsilon _{{kk}} }}{{\partial t}} + S_{\varepsilon } \frac{{\partial p}}{{\partial t}} = \frac{k}{\mu }\nabla ^{2} p + Q_{{fs}} , $$
(9)

where \(S_{\varepsilon } \) is the specific storage at constant strain, \(k\) is permeability, \(\text{and}\; \mu\) is fluid viscosity:

$$S_{\varepsilon } = \frac{{\alpha ^{2} }}{{K_{u} - K}}, $$
(10)

where \(\alpha\) is the Biot’s coefficient, \({K}_{u}\) is the undrained bulk modulus, and \(K\) is the drained bulk modulus.

For Eq. (9), the pore pressure field is coupled with the time rate of change of the volumetric strain. This equation can be mathematically uncoupled from the mechanical equilibrium equations for an irrotational displacement field in an infinite domain without body forces (Wang 2000):

$${\nabla }^{2}p=\frac{1}{{c}_{v}}\frac{\partial p}{\partial t},$$
(11)

where \({c}_{v}\) is the hydraulic diffusivity.

Assuming cylindrically symmetrical displacement field and no shear stresses acting upon the element, then the normal stresses \({\sigma }_{r}\) and tangential stress \({\sigma }_{\theta }\) are independent of the tangential coordinate \(\theta\). Also assuming plane strain and neglecting in-situ stress, the governing equation could be achieved by combining Eqs. (5) to (11):

$$\left\{\begin{array}{c}\left(2G+\lambda \right)\left(\frac{{\partial }^{2}{u}_{r}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {u}_{r}}{\partial r}-\frac{{u}_{r}}{{r}^{2}}\right)-\frac{\alpha }{G}\frac{\partial p}{\partial r}=\rho \frac{{\partial }^{2}{u}_{r}}{\partial {t}^{2}}\\ \frac{{\partial }^{2}p}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial p}{\partial r}=\frac{1}{{c}_{v}}\frac{\partial p}{\partial t},\end{array}\right.$$
(12)

The solving strategy of Eq. (12) is explained in detail in "Appendix A1". In this model, the fluid inertia term is neglected. This is called up formulation by Zienkiewicz (1980), Schanz (2009), and Wrana (2013). This approximation is economical and convenient in the transient numerical analysis, and it will not influence the result when the loading frequency is low. For the influence of surge/swab pressure, neglecting fluid inertia is reasonable, since the loading frequency is low. Furthermore, even if we have a significantly high loading frequency inside the wellbore, fluid inertia could also be neglected when the rock is relatively impermeable, such as shale rock, or the wellbore surface is sealed by mud cake.

2.2.2 Wellbore Stress Distribution

To get the total wellbore stress distribution, reasonable boundary conditions should be used. For the diffusion equation in Eq. (12), the internal boundary is wellbore pressure plus dynamic surge/swab pressure, and the external boundary is the formation pressure. For the displacement equation, the internal boundary is wellbore pressure plus dynamic surge/swab pressure, the external boundary is zero rather than far-field stress. Taking advantage of the superposition principle, the total stress is the stress calculated by Eq. (12) plus the far-field stress. Here, the maximum horizontal stress is assumed to exist and is not the same as the minimum horizontal stress:

$${\sigma }_{r}={\sigma }_{r}^{1}+{\sigma }_{r}^{2}, {\sigma }_{\theta }={\sigma }_{\theta }^{1}+{\sigma }_{\theta }^{2}, {\sigma }_{z}={\sigma }_{z}^{1}+{\sigma }_{z}^{2},$$
(13)

where \({\sigma }_{r}^{1}\), \({\sigma }_{\theta }^{1}\), and \({\sigma }_{z}^{1}\) are the solution of Eq. (12), and \({\sigma }_{r}^{2}\), \({\sigma }_{\theta }^{2}\), and \({\sigma }_{z}^{2}\) are the far-field stress terms for radial, tangential, and vertical stresses. The far-field stress terms are

$${\sigma }_{r}^{2}=\left(\frac{{\sigma }_{x}^{0}+{\sigma }_{y}^{0}}{2}\right)\left(1-\frac{{r}_{w}^{2}}{{r}^{2}}\right)+\left(\frac{{\sigma }_{x}^{0}-{\sigma }_{y}^{0}}{2}\right)\left(1+\frac{{3r}_{w}^{4}}{{r}^{4}}-\frac{{4r}_{w}^{2}}{{r}^{2}}\right)cos2\theta +{\sigma }_{xy}\left(1+\frac{{3r}_{w}^{4}}{{r}^{4}}-\frac{{4r}_{w}^{2}}{{r}^{2}}\right)sin2\theta$$
$${\sigma }_{\theta }^{2}=\left(\frac{{\sigma }_{x}^{0}+{\sigma }_{y}^{0}}{2}\right)\left(1+\frac{{r}_{w}^{2}}{{r}^{2}}\right)-\left(\frac{{\sigma }_{x}^{0}-{\sigma }_{y}^{0}}{2}\right)\left(1+\frac{{3r}_{w}^{4}}{{r}^{4}}\right)cos2\theta -{\sigma }_{xy}\left(1+\frac{{3r}_{w}^{4}}{{r}^{4}}\right)sin2\theta ,$$
(14)
$${\sigma }_{z}^{2}={\sigma }_{z}^{0}-v\left[\frac{2{r}_{w}^{2}}{{r}^{2}}\left({\sigma }_{H}-{\sigma }_{h}\right)cos2\theta +4{\sigma }_{xy}\frac{{r}_{w}^{2}}{{r}^{2}}sin2\theta \right],$$

where \({\sigma }_{H}\) and \({\sigma }_{h}\) are maximum and minimum horizontal stresses, respectively. \({\sigma }_{x}^{0}\), \({\sigma }_{y}^{0}\), \({\sigma }_{z}^{0}\),\({\sigma }_{xy}\), \({\sigma }_{xz}\), \(\text{and}\, {\sigma }_{yz}\) are in-situ stress terms in the Cartesian coordinate. \(\theta\) is the circumferential angle. \({r}_{w}\) is the wellbore radius.

Here, the detailed derivation of Eq. (14) is shown in the "Appendix A2". The conventional Kirsch solution provided by Chen (2008) considering seepage effect is also shown in the "Appendix A2".

2.3 Failure Criteria

Within the last decades, a number of failure criteria have been proposed. Many of them could be used to describe rock failure under static and quasi-static loading conditions. Under dynamic and cyclic loading, the rock failure is controlled by different parameters and relations. However, there is no agreement on the divisions for static, quasi-static, and dynamic loading conditions. According to several articles (Cai et al. 2007; Liang 2015), when the strain rate is lower than \(5\times {10}^{-4} /s\), it is static loading. When the strain rate is between \(5\times {10}^{-4} /s\) and \(1\times {10}^{2} /s\), it is quasi-static or quasi-dynamic loading. When the strain rate is larger than \(1\times {10}^{2} /s\), it is dynamic loading. The divisions of strain rate with respect to engineering activities are shown in Fig. 3. The magnitude of the strain rate of the wellbore caused by tripping operations is less than \({10}^{-3} /s\) and mostly less than \({10}^{-4} /s\), which lies in the static loading domain. Therefore, the commonly used failure criteria of rock could be studied and selected to describe wellbore stability during tripping operations. This will also be confirmed by the triaxial compression test which is described in Sect. 3.

Fig. 3
figure 3

Divisions of strain rate with respect to engineering activities

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Failure of rocks can be described by stress criteria, energy criteria or strain criteria. Stress criteria are the most popular one in rock mechanics. Some criteria only consider the minimum principal stress \({\sigma }_{3}\) and the maximum principal stress \({\sigma }_{1}\), such as Mohr–Coulomb, Hoek–Brown, and Drucker–Prager failure criteria. However, much evidence has shown that the intermediate principal stress does indeed influence rock strength (Handin et al. 1967; Fjær 2002; Al-Ajmi 2006). Then, more advanced criteria were proposed and they include the influence of intermediate principal stress \({\sigma }_{2}\), such as Mogi–Coulomb criterion, Modified Lade criterion, and 3-D Hoek–Brown criterion. Finally, six failure criteria have been applied to analyze wellbore stability under tripping operations. The detailed explanations of these six failure criteria have been introduced by many researchers (Fjar et al. 2008; Zoback 2010; Ulusay 2014), so it will not be repeated in this paper.

2.4 Calculation flow chart

A flowchart shown in Fig. 4 has been made to present the logical sequence to analyze dynamic wellbore stability under tripping conditions.

Fig. 4
figure 4

Flowchart for analyzing wellbore stability under tripping operations

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3 Experiments

The tested rock is Bedford limestone located in northeast Oklahoma, US. Standard cylindrical specimens of 50 mm in diameter and 100 mm in length are drilled from the same block. The sample preparation and triaxial compression test are performed based on ASTM standards.

The GCTS electric hydraulic servo-controlled rock mechanics facility was used for the triaxial compression test of limestone under different confining pressures and strain rates. The confining pressure is 0, 6.9 MPa, 13.8 MPa, and 20.7 MPa. The strain rate of the monotonic loading is \(1\times {10}^{-6} /{\text{s}}\), \(1\times {10}^{-5}/{\text{s}},\) and \(1\times {10}^{-4} /{\text{s}}\). The purpose of experimental work is to check whether well-known static failure criteria could be used for wellbore stability analysis under dynamic tripping operations. The basic procedures are: (1) using the data of different confining pressures with the slowest strain rate of \(1\times {10}^{-6} /{\text{s}}\) to generate static rock cohesion and internal angle; (2) plot constitutive failure line for different failure criteria; and (3) check the accuracy of different failure criteria by comparing experimental data of higher strain rate with the constitutive failure line. All the results are presented in Sect. 4.4.

4 Results and Discussion

4.1 Dynamic Surge/Swab Pressure

Figure 5 presents the tripping velocity profile and transient swab pressure profile during tripping out. They are calculated using the parameters, as listed in Table 1. The velocity is negative, because the downward movement of drill pipe is assumed to be positive. The pressure is negative, because the swab pressure reduces the wellbore pressure. It shows that the swab pressure is transient, and at about 5.5 s, the absolute transient swab pressure is at a maximum, which is 4.34 MPa. The wellbore stress will be calculated at this moment. If the wellbore collapse can be avoided at this point, other moments are also considered to be safe.

Fig. 5
figure 5

Tripping velocity, dynamic surge/swab pressure versus time

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Table 1 Input parameters for calculation in case I
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4.2 Distribution of Dynamic Wellbore Stress

To calculate the distribution of dynamic wellbore stress, the wellbore azimuth angle is assumed to be zero, which also means that the maximum horizontal stress direction is north/south. The wellbore inclination angle is also assumed to be zero. Using the inputs listed in Table 2, the wellbore stress calculated by the conventional method and the poroelastodynamics method will be compared.

Table 2 Input parameters for calculation
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The comparison of stress distribution between the conventional model and the poroelastodynamics model is shown in Fig. 6. In each of these four figures, the maximum horizontal stress is in the \(N30^\circ E\) direction, and the minimum horizontal stress is in the \(N60^\circ W\) direction. The degree 0, 30,… 330 represents the circumferential angle of the wellbore. The yellow dotted circles with a different radius of 1, 1.5, 2, 2.5, and 3 represent dimensionless distance \(\xi (r/{r}_{w})\). In the color bar, the unit of the number is MPa. Figure 6a, b shows, respectively, the radial stress, tangential stress calculated by the conventional method. Figure 6c, d shows, respectively, the radial stress and tangential stress calculated by the poroelastodynamics model. From these figures, it is found that radial stress increases with the radial distance, while tangential stress decreases with radial distance. Both radial stress and tangential stress are influenced by the circumferential angle due to the influence of non-uniform hydrostatic stress. After comparing Fig. 6a–d, it is found that the conventional Kirsch solution is distinguishable from the poroelastodynamics solution. For the conventional Kirsch solution, the radial stress is between 30 and 61 MPa and the tangential stress is between 59 and 120 MPa, whereas for the poroelastodynamics solution, the radial stress is between 18 and 58 MPa and the tangential stress is between 60 and 131 MPa. The significant difference in the stress distribution could indicate the difference in wellbore stability analysis.

Fig. 6
figure 6

Wellbore stress distribution under tripping out condition, a is radial stress calculated by the conventional method; b is tangential stress calculated by the conventional method; c is radial stress calculated by the poroelastodynamics model; d is tangential stress calculated by the poroelastodynamics model

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4.3 Dynamic Wellbore Stability

The calculation of dynamic wellbore stability is performed according to the flowchart shown in Fig. 4. For wellbore stability analysis, the choice of stress regime, failure criterion, the permeable property of the wellbore, and wellbore depth is significant. Table 3 presents the parameters of in-situ stress regimes. Five stress regimes are considered, including normal fault (\({\sigma }_{v}>{\sigma }_{H}>{\sigma }_{h}\)), the transition between normal fault and strike-slip (\({\sigma }_{v}={\sigma }_{H}>{\sigma }_{h}\)), strike-slip (\({\sigma }_{H}>{\sigma }_{v}>{\sigma }_{h}\)), the transition between strike-slip and thrust fault (\({\sigma }_{H}>{\sigma }_{v}={\sigma }_{h}\)), and thrust fault (\({\sigma }_{H}>{\sigma }_{h}>{\sigma }_{v}\)).

Table 3 Parameters of in-situ stress
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The results of the wellbore stability analysis are presented by the hemisphere plot, as shown in Fig. 7. The direction of maximum horizontal stress is \(N30^\circ E\), and the direction of minimum horizontal stress is \(N60^\circ W\). The circles with different radii represent different inclination angles, such as 15°, 30°,… 90°. For each circle, there are 360° which represent the wellbore azimuth angles. The color bar on the right side shows the different colors that indicate different values of the minimum mud density (MMD) with the unity of g/cm3. The color will be distributed on the left figure. Figure 7 shows one simple example, where all the values are set to be 1 g/cm3. In this example, the red triangular point means that at a certain depth, for a wellbore with an inclination degree of 45° and an azimuth angle of N30°E, the minimum mud density is 1 g/cm3. One point to be noted is that in the hemisphere plot of the minimum mud density (MMD), all the values are the minimum mud density to avoid wellbore collapse (shear failure). As we know, kick is also a serious drilling problem that needs to be avoided; therefore, the wellbore pressure should also be higher than the formation pore pressure. As mentioned by Zoback (2010), if wellbore stability is not a concern in a given area, the minimum mud weight is usually taken to be the pore pressure, so that a well does not flow while drilling. When wellbore stability is a consideration, the lower bound of the mud window is the minimum mud weight required to achieve the desired degree of wellbore stability. In this paper, since the normal pore pressure (1.01 g/cm3 ) is considered, if the value in the hemisphere plot is less than 1.01 g/cm3, the lower bound of the safe mud window should be the pore pressure plus the maximum swab pressure to avoid the kick, rather than the MMD to avoid wellbore collapse. However, in the following analysis, the results are focused on wellbore instability (shear failure) problems.

Fig. 7
figure 7

Hemisphere plot of the minimum mud density (MMD) for all inclined wellbore

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(1) The influence of failure criteria

Figure 8 is the distribution of minimum mud density (MMD) of different failure criteria for a normal fault. It shows that although different failure criteria could generate different values of MMD, for a certain in-situ stress regime, the distribution of MMD in all these six hemisphere plots are similar, showing that the highest MMD occurs at the maximum horizontal stress direction with the inclination angle of 80°–90°, whereas the lowest MMD occurs at the minimum horizontal stress direction with the inclination angle about 30°–45°. Mohr–Coulomb and Hoek–Brown always generate the highest MMD. Many researchers (Zhang 2010; Gholami 2014; Maleki 2014; Ma 2015; Rahimi 2015; Meng 2018, 2019a) have pointed out that these two criteria usually overestimate the rock breakout and provide the highest MMD, because they do not consider the influence of intermediate principal stresses. Instead, Drucker–Prager underestimates the rock breakout and provides the lowest MMD. Leandro Alejano (2012) conducted thorough research on Drucker–Prager criterion and pointed out that although it is a three-dimensional pressure-dependent model, it consistently overestimates the strength of intact rock, or underestimates the MMD. It is only appropriate for a narrow range of stress in the vicinity of the intermediate and minor principal stress from which the parameters of the criterion are obtained. For the other failure criteria, such as Mogi–Coulomb, Modified Lade, and 3D Hoek–brown, they provide the MMD neither as high as Mohr–Coulomb, nor as low as Drucker–Prager. These three failure criteria seem to be more dependable than Mohr–Coulomb and Hoek–Brown, because they are three-dimensional failure criteria that consider the influence of intermediate principal stresses, which is also proved by Reza Rahimi (2015), Zhang (2010), Bahrami et al. (2017), etc.

Fig. 8
figure 8

Distribution of MMD of different failure criteria for normal fault

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However, there are also some minor differences among these three dependable failure criteria. In our analysis, their relationship is MMD3DHoek–Bown > MMDMogi−Coulomb > MMDModified Lade. Researchers do not have uniform opinions for the reliability of these three failure criteria. Since their results are similar, Reza Rahimi (2015) mentioned that they could be used interchangeably. However, other researchers, such as Zhang (2010), mentioned that 3D Heok-brown and Mogi–Coulomb are better, whereas Modified Lade could generate significant over-prediction. Bahrami (2017) pointed out that the Modified Lade is the best after comparing the failure criteria with polyaxial data. Considering the complexity of the field formation and early researchers’ inconsistent conclusions, more cases should be conducted to verify the reliability of these three failure criteria. It is difficult and even impossible to generate the direct relationship between any certain formation and the best choice of failure criteria. To better understand which failure criterion is the best, in Sect. 4.4 of this paper, triaxial compression tests have been performed under dynamic loading conditions to verify the accuracy of different failure criteria. The experimental data could be used to demonstrate whether these failure criteria could be used under dynamic loading conditions and which one is the best.


(2) The influence of in-situ stress regimes

Figure 9 shows the distribution of MMD for different in-situ stress regimes using Modified Lade failure criterion. For different in-situ stress regimes, the distribution of MMD in the hemisphere plot is very different. This demonstrates that the in-situ stress regimes determine the distribution of MMD under various inclination and azimuth angles. The optimal and worst wellbore trajectory under different stress regimes could also be generated.

Fig. 9
figure 9

Distribution of MMD of different in-situ stress regimes using Modified Lade failure criteria

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For the normal fault (NF) stress regime, the worst case, or in other words, the highest required MMD occurs in the maximum horizontal stress direction with a high inclination angle of 75°–90°. They represent horizontal wellbores drilled along the maximum stress direction. If any inclined wellbores are drilled in this stress regime condition, the maximum horizontal stress direction should be avoided. Otherwise, a narrow safe mud window will be encountered, and it increases the difficulty of safe drilling operations. The lowest required MMD occurs in the minimum horizontal stress direction with the inclination angle of about 40°. It is suggested to design the inclined wellbore in the minimum horizontal stress direction from the wellbore stability point of view.

For the transition of the normal fault and strike-slip (NF-SS) stress regime, the worst case also occurs in the maximum horizontal stress direction. The lowest required MMD occurs in the minimum horizontal stress direction. However, different from the NF regime, the lowest required MMD occurs in high inclination angles of 90° instead of 40°. Meanwhile, if the maximum horizontal stress direction was designed according to the reservoir simulation, the highest MMD is required for all inclined wellbore, because the inclination angle has little effect on the MMD at this azimuth angle.

For the strike-slip (SS) stress regime, the distribution of MMD is different from all other four stress regimes, because the highest required MMD always occurs in the low inclination angles between 0° and 10°, which indicate the vertical or low inclined wellbores. Within the inclination angle of 15°, the azimuth has little influence on the distribution of MMD. Different from the NF and NF-SS regimes, the MMD in the maximum horizontal stress direction is smaller than the minimum horizontal stress. The lowest MMD does not occur in the maximum horizontal stress direction. It occurs at four nearly symmetrical azimuth angles of N70°E, S10°E, S70°W, and N10°W with an inclination angle of 90°, which indicate horizontal wellbores.

For the transition of the strike-slip and thrust fault (SS-TF) stress regime, the distribution of the highest and lowest MMD is opposite of the NF-SS regime. The worst case appears in the minimum horizontal stress direction, while the optimal case appears in the maximum horizontal stress direction. It is recommended to design the inclined wellbore in or close to the maximum horizontal stress direction. Meanwhile, one point to be noted is in the minimum horizontal stress direction, the influence of the inclination angle is trivial; whereas in the maximum horizontal stress direction, the influence of the inclination angle is significant and the MMD decreases as the inclination angle increases.

For the thrust fault (TF) stress regime, as with the SS-TF, the highest MMD occurs in the minimum horizontal stress direction, while the lowest MMD occurs in the maximum horizontal stress direction. In addition, the variance of MMD in the maximum horizontal stress direction is greater than that of the minimum horizontal stress direction. However, unlike the SS-TF, the inclination angle is significant in all azimuth angles for TF. Finally, the summarized optimal and worst drilling direction for these stress regimes are shown in Table 4.

Table 4 Optimal and worst wellbore trajectory for different in-situ stress regimes
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(3) The influence of permeable property of the wellbore

The permeable property of the wellbore could influence the MMD under different inclination and azimuth angles. From Fig. 10, it is found that, compared with the impermeable wellbore, the ranges of the MMD are wider, and the lower bound becomes even lower for the fully permeable wellbore. In the field, mud cake could buildup on the wellbore surface during drilling because of the residue deposited on the permeable wellbore when the drilling fluid is forced against the wellbore under the wellbore pressure. If the mud cake does not build up or the quality of mud cake is very poor, or the wellbore is during underbalanced drilling, the wellbore could be assumed to be fully permeable. Instead, if the mud cake has built up already and possesses a good sealing effect, the wellbore could be assumed to be impermeable. However, if the mud cake has some permeability and thickness, then its influence on wellbore stability cannot be ignored (Tran 2010; Feng 2018). Fortunately, impermeable and fully permeable wellbores are the two boundaries of the mud cake effect. Before a detailed study of the mud cake effect is provided, it is meaningful to consider these two boundaries at the beginning.

Fig. 10
figure 10

Distribution of MMD of the different permeable property of the wellbore using Modified Lade criteria

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(4) Comparison between the conventional solution and poroelastodynamics solution

The conventional Kirsch solution is widely used to analyze wellbore stability. It is meaningful to compare the wellbore stability analysis based on the newly proposed poroelastodynamics model and the conventional method. Figure 11 demonstrates a comparison between these two methods and the Modified Lade failure criteria are applied to both methods. Figure 11a presents the conventional solution, Fig. 11b presents the poroelastodynamics solution, and Fig. 11c presents the difference between these two solutions. Here, the difference is the MMD calculated by the conventional method minus the MMD calculated by the poroelastodynamics model. As shown in this figure, the difference is between − 0.08 and 0.11 g/cm3. In the maximum horizontal stress direction with inclination angles between 30° and 90°, the difference is negative, which indicates that compared with the new poroelastodynamics model, the conventional method underestimates the MMD. In the minimum horizontal stress direction with inclination angles between 15° and 90°, the difference is positive, which indicates that compared with the new poroelastodynamics model, the conventional method overestimates the MMD. Therefore, if the conventional solution was used to analyze wellbore stability under tripping operations, both underestimation and overestimation of MMD will occur. However, one case is not enough to generate conclusions; therefore, another case with a lower depth will be compared.

Fig. 11
figure 11

Comparison between two solutions for Modified Lade under NS stress regime (difference is the conventional solution minus the poroelastodynamics solution)

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The second case of a 1000 m depth wellbore is calculated and compared. Table 5 lists the input parameters of the second case. Substituting these parameters into the method presented in Sect. 2.1, the tripping velocity profile and transient swab pressure profile during tripping out could be generated (Fig. 12). The negative value represents that the swab pressure reduces the wellbore pressure. At about 3 s, the absolute transient swab pressure is at a maximum of 2.7 MPa. The wellbore stress will be calculated at this moment.

Table 5 Input parameters for calculation of case II
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Fig. 12
figure 12

Tripping velocity, dynamic surge/swab pressure versus time

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Figure 13 shows the comparison between the conventional solution and the poroelastodynamics solution for case II. Such as in case I, compared with the new poroelastodynamics model, the conventional method could both overestimate and underestimate the MMD. The difference is between − 0.1 and 0.08 g/cm3. The overestimation occurs in the minimum horizontal stress direction with the inclination angle of 15° and 90°, whereas the underestimation occurs in the maximum horizontal stress direction with the inclination angle of 30°–90°.

Fig. 13
figure 13

Comparison between two solutions for Modified Lade under NS stress regime (difference is the conventional solution minus the poroelastodynamics solution)

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Through these two cases, it is found that the MMD predicted by the poroelastodynamics model can be significantly different from the conventional solution. The difference can be higher than 0.1 g/cm3. Compared with the poroelastodynamics solution, the conventional solution could both overestimate and underestimate the MMD. In this paper, the poroelastodynamics model is strictly derived from the basic poroelastic equations. It can not only simulate the coupling effect between deformation and fluid seepage, but also consider the inertia effect under dynamic loading conditions. The conventional solution based on the Kirsch solution provided by Chen (2008), also used by Ma (2015) and many other researchers is a simplified calculation considering the fluid seepage effect. Therefore, it is reasonable to assume that the solution of the poroelastodynamics model is more accurate than the conventional solution provided by Chen (2008). In both cases, overestimation occurs in the minimum horizontal stress direction with the inclination angle between 15° and 90°, whereas underestimation occurs in the maximum horizontal stress direction with the inclination angle between 30° and 90°. Therefore, to get a more accurate MMD during tripping operations, the poroelastodynamics model should be used.

4.4 Experimental Validation

Table 6 shows the triaxial compression strength of Bedford limestone under different strain rates and confining pressure. The stress–strain curves are presented from Figs. 14, 15 and 16. It shows that the rock strength increases with confining pressure at the same strain rate. Meanwhile, at the same confining pressure, the rock strength increases with the strain rate. This is consistent with the experimental results provided by many other researchers (Lajtai et al. 1991; Li et al. 1999; Mahmutoğlu 2006; Liang et al. 2015). However, none of these researchers provided enough experimental data or used the data to check the accuracy of different static failure criteria under dynamic loading conditions.

Table 6 Strength of limestone under different strain rates and confining pressures
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Fig. 14
figure 14

Stress–strain curves under 6.9 MPa confining pressure and different strain rates

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Fig. 15
figure 15

Stress–strain curves under 13.8 MPa confining pressure and different strain rates

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Fig. 16
figure 16

Stress–strain curves under 20.7 MPa confining pressure and different strain rates

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The experimental data with different confining pressures under the strain rate of 10− 6/s were chosen to generate rock cohesion and internal angle, which are presented in Fig. 17. The rock cohesion is optimized to be 14.3 MPa and the internal angle is 17.3°. The reason that the data under the strain rate of 10− 6/s were used is that they are regarded as the static rock properties. After the rock cohesion and internal angle are calculated, the constitutive relationship of different failure criteria and experimental data could be plotted together, as shown in Figs. 18, 19, 20, 21, 22 and 23. The constitutive relationship of each failure criteria is different, therefore, the x-axis and y-axis are varied for different plots (Ulusay 2014). It is not quite reliable to compare the accuracy of different failure criteria by only looking at these graphs. The reasonable method should be proposed to evaluate the accuracy of prediction.

Fig. 17
figure 17

Generation of rock cohesion and internal angle

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Fig. 18
figure 18

Mohr–Coulomb failure criterion (solid line), fitted to triaxial test data

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Fig. 19
figure 19

Circumscribed Drucker–Prager failure criterion (solid line), fitted to triaxial test data

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Fig. 20
figure 20

Inscribed Drucker–Prager failure criterion (solid line), fitted to triaxial test data

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Fig. 21
figure 21

Modified Lade failure criterion (solid line), fitted to triaxial test data

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Fig. 22
figure 22

Mogi–Coulomb failure criterion (solid line), fitted to triaxial test data

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Fig. 23
figure 23

3D Hoek–Brown failure criterion (solid line), fitted to triaxial test data

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Fig. 24
figure 24

Diagram of directional well

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Zhang (2010) has introduced prediction errors to check the accuracy of different failure criteria with polyaxial compression data under static loading conditions. Based on Zhang’s work, two modified non-dimensional numbers are introduced to calculate the prediction errors:

$${\text{Diff}}_{{{\text{ave}}}} = 100\frac{{\sum\limits_{{i = 1}}^{n} {\left\{ {\left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{pre}}}} } \right]_{i} - \left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{exp}}}} } \right]_{i} } \right\}} }}{{n\cdot{\text{UCS}}}}, $$
(15)
$${\text{Diff}}_{{{\text{abs}}}} = 100\frac{{\sum\limits_{{i = 1}}^{n} {\left\{ {\left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{pre}}}} } \right]_{i} - \left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{exp}}}} } \right]_{i} } \right\}} }}{{n \cdot {\text{UCS}}}}, $$
(16)

where \(\left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{pre}}}} } \right]_{i} \) and \(\left[ {\left( {\sigma _{1}^{\prime } } \right)_{{{\text{exp}}}} } \right]_{i} \) are, respectively, the predicted and experimental major principal stresses at failure for all data points; \(n\) represents the total number of data points; \(\text{UCS}\) is the uni-axial rock strength, and it is used to make the prediction difference to be a non-dimensional number; \({\text{Diff}}_{{{\text{ave}}}} \) is a non-dimensional number that represents the average prediction difference which can be used to indicate whether these failure criteria overestimate or underestimate the rock strength, the positive number of \( {\text{Diff}}_{{{\text{ave}}}} \) represents overestimation; \( {\text{Diff}}_{{{\text{abs}}}} \) is a non-dimensional number that represents the average absolute difference which can be used to indicate how close the predicted and experimental strength values are with each other. The smaller number of \( {\text{Diff}}_{{{\text{abs}}}} \) and \( {\text{Diff}}_{{{\text{ave}}}} \), the more accurate of the prediction.

According to average prediction errors in Table 7, Mohr–Coulomb significantly underestimates the rock strength, because \( {\text{Diff}}_{{{\text{ave}}}} \) is a relatively small negative number. Inscribed Drucker–Prager significantly overestimates the rock strength, since \( {\text{Diff}}_{{{\text{ave}}}} \) is a relatively large positive number. The circumscribed Drucker–Prager underestimates the rock strength, but it is not so significant as Mohr–Coulomb. The Mogi–Coulomb only slightly underestimates the rock strength, whereas the Modified Lade and the 3D Hoek–Brown slightly overestimates the rock strength. For the absolute errors in Table 7, Modified Lade has the highest accuracy, whereas the inscribed Drucker–Prager has the lowest accuracy. The accuracy sequence is Modified Lade > 3D Hoek–Brown > Mogi–Coulomb > Circumscribed Drucker–Prager > Mohr–Coulomb > Inscribed Drucker–Prager. If we look back to Figs. 18, 19, 20, 21, 22 and 23, the experimental points lie very close to the constitutive failure line for Modified Lade. Therefore, it shows that when the strain rate is less than 10− 4/s, which is the range of strain rate caused by tripping operations, the static failure criteria, especially the modified Lade, could be used to describe rock failure. It verifies that our theoretical calculation of using Modified Lade failure criteria to predict wellbore stability under tripping operations is valid. Due to the time and materials’ limit, only Bedford limestone was used in our test. The same tests are needed to be repeated on many different types of rocks to prove the accuracy of the Modified Lade failure criteria in the future.

Table 7 Strength parameters and prediction errors for limestone
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5 Concluding Remarks

Based on the above analysis, several interesting conclusions can be made. A new poroelastodynamics model has been established to analyze wellbore stability under dynamic loading conditions. Compared with the conventional Kirsch solution, this model can simulate the coupled effect of deformation–diffusion in the porous medium under dynamic loading conditions. The model is solved by the finite-difference method with the specific swab pressure caused by tripping operations as the dynamic loading source. The hemisphere plots of the distribution of MMD are provided to present the wellbore stability results.

The results of wellbore stability analysis show that failure criterion, in-situ stress regime, and the permeable property of the wellbore could influence the distribution of MMD. The in-situ stress regime could determine the distribution of MMD under various inclination and azimuth angles. The optimal and worst wellbore trajectory for different in-situ stress regimes is also generated. The three-stress dependent failure criteria, Modified Lade, Mogi–Coulomb, and 3D Hoek–Brown are better than the two-stress dependent failure criteria, Mohr–Coulomb, and Hoek–Brown, to get reasonable MMD. For the fully permeable wellbore, the range of MMD in the hemisphere plot is broader than that of the impermeable wellbore.

For the experimental part, the triaxial compression tests under different strain rate and confining pressure have been performed on Bedford limestone. The results have been used to check the applicability and accuracy of different conventional failure criteria under dynamic loading conditions. This analysis proves that when the strain rate is less than 10−4/s, which is the range of strain rate caused by tripping operations, the static failure criteria (especially the modified Lade) could be used to describe rock failure.

Two specific cases are used to compare the results of the conventional method and the poroelastodynamics model. Using Modified Lade failure criteria, it shows that compared with the newly developed poroelastodynamics model, the conventional method could either overestimate or underestimate the MMD and the difference can be as significant as 0.11 g/cm3. In both cases, overestimation occurs in the minimum horizontal stress direction with the inclination angle of 15°–90°, whereas underestimation occurs in the maximum horizontal stress direction with the inclination angle of 30°–90°.

In future studies, there are several aspects to be improved. First, experimental data of the other kinds of rocks should be provided to check the accuracy and applicability of different failure criteria under different strain rates. Second, apart from the surge/swab pressure caused by tripping operations, there are many sources of dynamic loading within the wellbore, such as the pressure fluctuations caused by hydraulic fracturing, seismic waves caused by earthquakes, etc. The behavior of wellbores under these dynamic loading conditions is also highly interested. Third, the poroelastodynamics model should be modified to consider fluid inertia. Under high-frequency dynamic loading conditions, both solid inertia and fluid inertia could influence the mechanical behavior of a porous medium. Finally, a more advanced and accurate surge/swab pressure prediction model is expected to be applied in the dynamic wellbore stability analysis under tripping operations.