Wellbore collapse caused by thermo-chemical coupling effects drilling in shale formation

Abstract

Wellbore instability in shale formations has always been a concern in oil exploration field. According to the Mohr–Coulomb criterion, a new model for evaluating and classifying the collapse of horizontal wellbore in shale formation is established. In this model, the temperature field and the pore pressure field affected by hydraulic pressure, thermal potential, and chemical potential difference between the drilling fluid and formation fluid are considered. Using the model, the influences of the temperature and solute concentration of drilling fluid on the failure degree of horizontal wellbore in shale formation with weak planes are simulated and analyzed. Based on the research results, weak planes that exist in shale formation will cause a significant wellbore collapse and using cool drilling fluid with high solute concentration can reduce the collapse degree of horizontal wellbore. The model established may help drilling engineers in practice to take effective measures for reducing wellbore instability and avoiding possible wellbore collapse during drilling in shale formations.

Introduction

It is almost inevitable that a drilling process will encounter a shale formation, because shale formations are ubiquitous in sedimentary basins worldwide (Speight 2016). With the great success of shale oil and gas exploration, wellbore stability of shale formation has been an important research area within drilling engineering in recent years. Shale rock contains many weak planes that will cause shale rock failure under a lower stress conditions (Ding et al. 2018). As to the effects of weak planes on the strength of material such as shale, Jaeger (1960) used two Mohr–Coulomb envelopes to describe the failure of the material itself and the failure along weak planes respectively. Since then, many authors have considered failure along the weak plane in their research on the wellbore stability of shale formations (Aadnoy and Chenevert 1987; Narayanasamy et al. 2010; Lee et al. 2012; Liang et al. 2014; He et al. 2015; Ma and Chen 2015; Gao et al. 2017a, b; Ding et al. 2018; Zhou et al. 2018). In addition, the variation of pore pressure and temperature around the wellbore may also have a great influence on wellbore stability during drilling shale formations (Mody and Hale 1993; Wang and Papamichos 1994; Chen et al. 2003; Rahman et al. 2003; Chen and Ewy 2005; Ghassemi et al. 2009). Mody and Hale (1993) confirmed that hydraulic pressure and chemical potential of drilling fluid will change the pore pressure near the wellbore. Both Chen and Ewy (2005) and Ghassemi et al. (2009) established thermo-chemical coupling models and analyzed the effects of drilling fluid solute concentration and temperature on the stress distribution around the wellbore in shale formation. Ma and Chen (2015) established a model that considers the influence of drilling fluid solute concentration and weak planes on wellbore stability of shale formation, but neglected the temperature changes. The influence of temperature difference on wellbore failure can be significant because temperature diffusion is much faster than solute and hydraulic diffusions in shale rock. Gao et al. (2017a, b) simulated the effects of temperature and weak shale on formation wellbore failure, but neglected the influence of solute concentration, which is not negligible during drilling a shale formation. Drilling a shale formation, both the weak planes existing in shale formation and the drilling fluid performance, including the temperature and the solute concentration, has a significant influence on wellbore stability.

In order to accurately predict the wellbore stability of shale formation, this paper will present a new model that will consider the effects of both weak planes and drilling fluid performance. Using this new model, the influence of the weak planes and the temperature and solute concentration of drilling fluid on the failure degree of wellbore of shale formation will be simulated.

Stresses acting on the near-wellbore rock

Shale rock is generally considered as an anisotropic material with transversely isotropic properties (Li and Weijermars 2019; Wang et al. 2020; Dubey et al. 2021). Jin et al. (2012) analyzed the effect of this property of shale rock on the stress distribution and showed that the effect is relatively small. Therefore, Ma and Chen (2015) ignored this effect in the process of deriving the wellbore stress distribution. Usually, the stress distribution of wellbore is considered a time-independent function of in situ stress, wellbore orientation, wellbore pressure, and pore pressure, given by the well-known Kirsch equation (Bradley 1979; Lee et al. 2012). However, because there are more or less difference between the drilling fluid and the formation liquid in temperature, pressure, and solute concentration, both the pore pressure and the formation temperature near the wellbore will change with time, altering the stress distribution around the wellbore. Assuming the variation of pore pressure and formation temperature with time is axisymmetric, Tang et al. (2018) derived the instantaneous stress distribution around the horizontal wellbore, as shown in Eq. (1).

$$\left\{\begin{array}{c}{\sigma }_{rr}=\frac{{\sigma }_{11}+{\sigma }_{22}}{2}(1-\frac{{r}_{\text{w}}^{2}}{{r}^{2}})+\frac{{\sigma }_{11}-{\sigma }_{22}}{2}(1+3\frac{{r}_{\text{w}}^{4}}{{r}^{4}}-4\frac{{r}_{\text{w}}^{2}}{{r}^{2}})\mathrm{cos}2\theta +{\sigma }_{12}(1+3\frac{{r}_{\text{w}}^{4}}{{r}^{4}}-4\frac{{r}_{\text{w}}^{2}}{{r}^{2}})\mathrm{sin}2\theta +\frac{{r}_{\text{w}}^{2}}{{r}^{2}}{P}_{\text{w}}\text{+}{\sigma }_{rr,{\text{pore}}}\text{+}{\sigma }_{rr,{\text{thero}}}\\ {\sigma }_{\theta \theta }=\frac{{\sigma }_{11}+{\sigma }_{22}}{2}(1+\frac{{r}_{\text{w}}^{2}}{{r}^{2}})-\frac{{\sigma }_{11}-{\sigma }_{22}}{2}(1+3\frac{{r}_{\text{w}}^{4}}{{r}^{4}})\mathrm{cos}2\theta -{\sigma }_{12}(1+3\frac{{r}_{\text{w}}^{4}}{{r}^{4}})\mathrm{sin}2\theta -\frac{{r}_{\text{w}}^{2}}{{r}^{2}}{P}_{\text{w}}\text{+}{\sigma }_{\theta \theta ,{\text{pore}}}\text{+}{\sigma }_{\theta \theta ,{\text{thero}}}\\ \begin{array}{c}{\sigma }_{zz}={\sigma }_{33}-2\nu ({\sigma }_{11}-{\sigma }_{22})\frac{{r}_{\text{w}}^{2}}{{r}^{2}}\mathrm{cos}2\theta -4\nu {\sigma }_{12}\frac{{r}_{\text{w}}^{2}}{{r}^{2}}\mathrm{sin}2\theta \text{+}{\sigma }_{zz,{\text{pore}}}\text{+}{\sigma }_{zz,{\text{thero}}}\\ {\tau }_{r\theta }=(\frac{{\sigma }_{22}-{\sigma }_{11}}{2}\mathrm{sin}2\theta +{\sigma }_{12}\mathrm{cos}2\theta )(1+\frac{2{r}_{\text{w}}^{2}}{{r}^{2}}-3\frac{{r}_{\text{w}}^{4}}{{r}^{4}})\\ \begin{array}{c}{\tau }_{rz}=({\sigma }_{23}\mathrm{sin}\theta +{\sigma }_{13}\mathrm{cos}\theta )(1-\frac{{r}_{\text{w}}^{2}}{{r}^{2}})\\ {\tau }_{\theta z}=({\sigma }_{23}\mathrm{cos}\theta -{\sigma }_{13}\mathrm{sin}\theta )(1+\frac{{r}_{\text{w}}^{2}}{{r}^{2}})\end{array}\end{array}\end{array}\right.$$

(1)

where $\theta$ is the well round angle, ${r}_{\text{w}}$ is the wellbore radius, $r$ is the radial distance from wellbore axis, ${P}_{\text{w}}$ is the internal wellbore pressure, and $\nu$ is the Poisson’s ratio. The stresses caused by pore pressure and temperature changes are given by (Chen et al. 2003; Wang and Papamichos 1994):

$$\left\{\begin{array}{c}{\sigma }_{rr,{\text{pore}}}=\frac{\eta (1-2\nu )}{(1-\nu ){r}^{2}}{\int }_{{r}_{\text{w}}}^{r}[P(\rho ,t)-{P}_{0}]\rho d\rho -\eta P(r,t)\\ {\sigma }_{\theta \theta ,{\text{pore}}}=-\frac{\eta (1-2\nu )}{(1-\nu )}\{\frac{1}{{r}^{2}}{\int }_{{r}_{\text{w}}}^{r}[P(\rho ,t)-{P}_{0}]\rho d\rho -[P(r,t)-{P}_{0}]\}-\eta P(r,t)\\ {\sigma }_{zz,{\text{pore}}}=\frac{\eta (1-2\nu )}{1-\nu }[P(r,t)-{P}_{0}]-\eta P(r,t)\end{array}\right.$$

(2)

$$\left\{\begin{array}{c}{\sigma }_{rr,{\text{thero}}}=\frac{E{\alpha }_{\text{m}}^{\text{T}}}{3(1-\nu ){r}^{2}}{\int }_{{r}_{\text{w}}}^{r}[T(\rho ,t)-{T}_{0}]\rho d\rho \\ {\sigma }_{\theta \theta ,{\text{thero}}}=-\frac{E{\alpha }_{\text{m}}^{\text{T}}}{3(1-\nu )}\{\frac{1}{{r}^{2}}{\int }_{{r}_{\text{w}}}^{r}[T(\rho ,t)-{T}_{0}]\rho d\rho -[T(r,t)-{T}_{0}]\}\\ {\sigma }_{zz,{\text{thero}}}=\frac{E{\alpha }_{\text{m}}^{\text{T}}}{3(1-\nu )}[T(r,t)-{T}_{0}]\end{array}\right.$$

(3)

where ${P}_{0}$ is the initial pore pressure, $E$ is the Young’s modulus, ${\alpha }_{\text{m}}^{\text{T}}$ is the thermal expansion coefficient of rock, ${T}_{0}$ is the initial formation temperature, $t$ is the time, $\eta$ is the Biot’s coefficient, and $T(r,t)$ and $P(r,t)$ are the instantaneous temperature field and the pore pressure field respectively. The mathematical description of the temperature field is shown in the Appendix.

In Eq. (1), ${\sigma }_{ij}$($i,j=\mathrm{1,2},3$) is the stress components of in situ stress in the Cartesian coordinate system of wellbore, given by Eq. (4).

$$\left\{\begin{array}{c}{\sigma }_{11}={\mathrm{cos}}^{2}\beta [{S}_{\text{H}}{\mathrm{cos}}^{2}(\alpha -{\alpha }_{\text{H}})+{S}_{\text{h}}{\mathrm{sin}}^{2}(\alpha -{\alpha }_{\text{H}})]+{S}_{\text{V}}{\mathrm{sin}}^{2}\beta \\ {\sigma }_{22}={S}_{\text{H}}{\mathrm{sin}}^{2}(\alpha -{\alpha }_{\text{H}})+{S}_{\text{h}}{\mathrm{cos}}^{2}(\alpha -{\alpha }_{\text{H}})\\ \begin{array}{c}{\sigma }_{33}={\mathrm{sin}}^{2}\beta [{S}_{\text{H}}{\mathrm{cos}}^{2}(\alpha -{\alpha }_{\text{H}})+{S}_{\text{h}}{\mathrm{sin}}^{2}(\alpha -{\alpha }_{\text{H}})]+{S}_{\text{V}}\mathrm{cos}{}^{2}\beta \\ {\sigma }_{12}=({S}_{\text{h}}-{S}_{\text{H}})\mathrm{cos}\beta \mathrm{cos}(\alpha -{\alpha }_{\text{H}})\mathrm{sin}(\alpha -{\alpha }_{\text{H}})\\ \begin{array}{c}{\sigma }_{13}=[{S}_{\text{H}}{\mathrm{cos}}^{2}(\alpha -{\alpha }_{\text{H}})+{S}_{\text{h}}{\mathrm{sin}}^{2}(\alpha -{\alpha }_{\text{H}})-{S}_{\text{V}}]\mathrm{sin}\beta \mathrm{cos}\beta \\ {\sigma }_{23}=({S}_{\text{h}}-{S}_{\text{H}})\mathrm{sin}\beta \mathrm{cos}(\alpha -{\alpha }_{\text{H}})\mathrm{sin}(\alpha -{\alpha }_{\text{H}})\end{array}\end{array}\end{array}\right.$$

(4)

where ${S}_{\text{H}}$ is the maximum horizontal stress, ${S}_{\text{h}}$ is the minimum horizontal stress, ${S}_{\text{V}}$ is the vertical stress, ${\alpha }_{\text{H}}$ is the maximum horizontal stress direction angle from the north direction, $\alpha$ is the wellbore azimuth angle from the north direction, and $\beta$ is the wellbore inclination angle.

When the failure across intact rock around the wellbore occurs, the radial stress ${\sigma }_{rr}$ is one of the three principal stresses acting on the rock, and the other two principal stresses (${\sigma }_{t\mathrm{max}}$ and ${\sigma }_{t\mathrm{min}}$) are as follows (Peška and Zoback 1995; Zoback et al. 2003):

$$\left\{\begin{array}{c}{\sigma }_{t\mathrm{max}}=\frac{1}{2}[{\sigma }_{zz}+{\sigma }_{\theta \theta }+\sqrt{{({\sigma }_{zz}-{\sigma }_{\theta \theta })}^{2}+4{\tau }_{\theta z}^{2}}]\\ {\sigma }_{t\mathrm{min}}=\frac{1}{2}[{\sigma }_{zz}+{\sigma }_{\theta \theta }-\sqrt{{({\sigma }_{zz}-{\sigma }_{\theta \theta })}^{2}+4{\tau }_{\theta z}^{2}}]\end{array}\right.$$

(5)

Stresses acting on weak planes

The principal stresses acting on the tangent plane of wellbore wall ${\sigma }_{t\mathrm{max}}$, ${\sigma }_{t\mathrm{min}}$, and ${\sigma }_{rr}$ in Eq. (5) are in the Cartesian coordinate system. Because the parameters of this Cartesian coordinate system are unknown, it is difficult to convert these three principal stresses to the weak plane. Since knowing the stress components in the wellbore cylinder coordinate system Eq. (1), we can convert the stress components to the wellbore Cartesian coordinate system. Figure 1 shows the conversion relationship between the two coordinate systems.

Fig. 1

Conversion relationship between Cartesian and cylindrical coordinate systems

The coordinate transformation matrix between Cartesian and cylindrical coordinate systems is given by:

$$C=\begin{bmatrix}\cos\theta&\sin\theta&0\\-\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}$$

(6)

Set up a global coordinate system O-XYZ, in which the O-X axis points to true north and O-Y axis points to true East. Rotating the O-Z axis with a rotation angle equaling to wellbore azimuth angle (α) and rotating the O-Y axis with a rotation angle equaling to wellbore inclination angle (β), the Cartesian coordinate system of wellbore ($o-{x}_{\text{b}}{y}_{\text{b}}{z}_{\text{b}}$) can be obtained, as shown in Fig. 2. Therefore, the conversion matrix between global and wellbore coordinate systems is given by Eq. (7).

Fig. 2

Conversion relationship between global and wellbore coordinate systems

$$\begin{array}{c}B=\begin{bmatrix}\cos\;\beta&0&-\sin\;\beta\\0&1&0\\\sin\;\beta&0&\cos\;\beta\end{bmatrix}\begin{bmatrix}\cos\;\alpha&\sin\;\alpha&0\\-\sin\;\alpha&\cos\;\alpha&0\\0&0&1\end{bmatrix}\\=\begin{bmatrix}\cos\;\alpha\;\cos\;\beta&\sin\;\alpha\;\cos\;\beta&-\sin\;\beta\\-\sin\;\alpha&\cos\;\alpha&0\\\cos\;\alpha\;\sin\;\beta&\sin\;\alpha\;\sin\;\beta&\cos\;\beta\end{bmatrix}\end{array}$$

(7)

Similarly, rotating the O-Z axis with a rotation angle equaling to the dip direction of weak plane (${\alpha }_{\text{w}}$) and rotating the O-Y axis with a rotation angle equaling to dip angle of weak plane (${\beta }_{\text{w}}$), the Cartesian coordinate system of weak plane ($o-{x}_{\text{w}}{y}_{\text{w}}{z}_{\text{w}}$) can be obtained, as shown in Fig. 3.

Fig. 3

Conversion relationship between global and weak plane coordinate systems

The conversion matrix between global and weak plane coordinate systems is given by Eq. (8).

$$\begin{bmatrix}\cos\;\alpha_\text{w}\cos\;\beta_\text{w}&\sin\;\alpha_\text{w}\cos\;\beta_\text{w}&-\sin\;\beta_\text{w}\\-\sin\;\alpha_\text{w}&\cos\;\alpha_\text{w}&0\\\cos\;\alpha_\text{w}\sin\;\beta_\text{w}&\sin\;\alpha_\text{w}\sin\;\beta_\text{w}&\cos\;\beta_\text{w}\end{bmatrix}$$

(8)

Lee et al. (2012) put forward the process that how to convert the stress distribution from wellbore cylindrical coordinate system (${\sigma }_{\text{CCS}}$) to the Cartesian coordinate system of weak plane, given by:

$$\left[\begin{array}{ccc}{\sigma }_{xx}^{\text{w}}& {\tau }_{xy}^{\text{w}}& {\tau }_{xz}^{\text{w}}\\ {\tau }_{xy}^{\text{w}}& {\sigma }_{yy}^{\text{w}}& {\sigma }_{yy}^{\text{w}}\\ {\tau }_{xz}^{\text{w}}& {\tau }_{yz}^{\text{w}}& {\sigma }_{zz}^{\text{w}}\end{array}\right]=W{B}^{T}{C}^{T}{\sigma }_{\text{CCS}}CB{W}^{T}$$

(9)

where ${\sigma }_{\text{CCS}}=\{{\sigma }_{rr},{\tau }_{r\theta },{\tau }_{rz};{\tau }_{r\theta },{\sigma }_{\theta \theta },{\tau }_{\theta z};{\tau }_{r\theta },{\tau }_{\theta z},{\sigma }_{zz}\}$ is the stress distribution in wellbore cylindrical coordinate system, given by Eq. (1).

Then, we can obtain the normal stress (${\sigma }_{\text{n}}$) and shear stress ($\tau$) acting on the plane of weakness (Fig. 4), given by:

Fig. 4

The stresses acting on the weak plane

$$\left\{\begin{array}{c}{\sigma }_{\text{n}}={\sigma }_{zz}^{\text{w}}\\ \tau =\sqrt{{({\tau }_{zx}^{\text{w}})}^{2}+{({\tau }_{zy}^{\text{w}})}^{2}}\end{array}\right.$$

(10)

Wellbore failure of shale formation

Once the stress distribution around wellbore has been obtained, whether wellbore will fail and the failure degree of wellbore can be determined by using the rock failure criterion. Unlike conventional formations, shear failures of wellbore in shale formations include the failure of rock body and the failure along weak planes. Since the strength (cohesion and internal friction angle) of weak plane is much lower than that of rock body, even a low shear stress can easily cause the sliding of weak plane. Jaeger (1960) used the Mohr–Coulomb criterion to analyze the failure of material containing weak planes. For the failure across intact rock, we can use the Mohr–Coulomb criterion presented in terms of principal stress (Jaeger et al. 2009):

$${\sigma }_{1}=UCS+{\sigma }_{3}{\mathrm{tan}}^{2}(45^\circ +\varphi /2)$$

(11)

where ${\sigma }_{1}$ and ${\sigma }_{3}$ are major and minor effective principal stress respectively, $UCS$ is the uniaxial compressive strength of rock, and $\varphi$ is the internal friction angle of rock.

Shear failure of wellbore is usually due to a low wellbore pressure. Therefore, if shear failure occurs on the wellbore wall, the effective stresses are usually as follows:

$${\sigma }_{1}={\sigma }_{t\mathrm{max}};{\sigma }_{3}={\sigma }_{rr}$$

(12)

Substituting the normal stress and shear stress acting on the plane of weakness (Eq. 10) into the Mohr–Coulomb criterion presented in terms of shear stress, we can obtain the following equation:

$$\tau ={S}_{\text{w}}+{\sigma }_{\text{n}}\mathrm{tan}{\varphi }_{\text{w}}$$

(13)

where ${S}_{\text{w}}$ is the cohesion of weak plane and ${\varphi }_{\text{w}}$ is internal friction angle of weak plane.

Using Eq. (11) to Eq. (13), we can calculate the values of r at the point where the failure of rock and weak planes occurs in all directions around the wellbore, and then determine the degree of wellbore failure.

Example for analysis

In order to study the effects of drilling fluid performance and weak plane on the stability of wellbore in shale formation, we assume a horizontal wellbore of shale formation and the relevant data, including parameters of rock mechanics and parameters of drilling fluid performance, are given in Table 1.

Table 1 Parameters for analysis (Tang et al. 2018; Ghassemi et al. 2009)

Using these parameters and the rock failure criterion, we can determine the shear failure radius of the wellbore in all directions. Through the C# program, we can simulate and draw the area of wellbore failure and intuitively understand the failure degree of wellbore in shale rock. The failures of wellbore under different drilling conditions are shown in Figs. 5, 6, and 7. The blue lines show the area of failure across intact rock around the wellbore and the red lines show the area of failure along weak planes. Compared to the two conventional shear failures that symmetrically occur on the wellbore wall, the sliding of the weak planes will cause four failures, symmetrically appearing on the wellbore wall.

Fig. 5

The wellbore failure at high drilling fluid concentrations

Fig. 6

The wellbore failure at high drilling fluid concentrations and temperature

Fig. 7

The wellbore failure at low drilling fluid concentrations

Assuming that ${T}_{\text{w}}=300\text{K, }{T}_{0}=350\text{K, }{C}_{\text{w}}=0.25,{C}_{0}=0.05$, the wellbore failures at 2 h, 5 h, and 10 h are simulated as shown in Fig. 5. When the solute concentration of drilling fluid is greater than that of formation fluid, the degree of wellbore failure is relatively small. The results show the wellbore expansion is mainly caused by the failure of weak plane, and the wellbore expansion caused by the failure of rock body is not obvious. The shear failure occurs near the upper and lower parts of the wellbore cross-section, and the maximum diameter expansion direction is about 40° away from the vertical axis. As the time increases, the degree of wellbore expansion will first decrease and then increase, but the change is not obvious. The maximum expansion rates of the wellbore are 8.95%, 6.74%, and 8.81%, respectively.

If that ${T}_{\text{w}}=350\text{K, }{T}_{0}=350\text{K, }{C}_{\text{w}}=0.25,{C}_{0}=0.05$, the wellbore failures at 2 h, 5 h, and 10 h are shown as in Fig. 6. Compared to Fig. 5, increasing the drilling fluid temperature will slightly increase the wellbore failure, including conventional wellbore failure and failure along weak plane. The maximum expansion rates of the wellbore are 7.25%, 9.85%, and 10.88%, respectively. Therefore, the cooling effect of drilling fluid will be constructive to the stability of horizontal wellbore in shale formation. When conditions permit, a drilling fluid cooling device should be added to the wellhead to cool the circulating hot drilling fluid.

Figure 7 shows the failure degree of wellbore drilled for 2 h, 5 h, and 10 h, and the temperature and solute concentration are ${T}_{\text{w}}=300\text{K, }{T}_{0}=350\text{K, }{C}_{\text{w}}=0.05,{C}_{0}=0.25$. The comparison of Figs. 5 and 7 indicates that the wellbore will experience a significant failure especially failure along weak plane if the solute concentration of drilling fluid is less than formation fluid. The maximum expansion rates of the wellbore are 44.67%, 41.33%, and 31.21%, respectively. Although the figures show that the failure degree will decrease with time, it is meaningless once a significant failure occurs. Therefore, drilling fluid with a high solute concentration is more important for the wellbore stability during drilling a shale formation.

Conclusion

It is found in this research that the failure degree of a horizontal wellbore in shale formation is significantly affected by drilling fluid performance and drilling operation, in addition to the effects of wellbore pressure and drilling direction. The wellbore failure in shale formation may be deteriorated under the coupling effects of hydraulic, chemical, and thermal potential gradients between the drilling fluid and formation fluid. A model is established in this research for analyzing the wellbore failure during drilling processes in shale formations. The model shows efficiency in analyzing the effects of pore pressure changes, temperature, and stress on wellbore failure in shale formation with weak planes. It is found in the research that the presence of weak planes, such as stratum bedding plane existing in shale formations, may have a significant impact on wellborn failure. Also, a low solute concentration of drilling fluid may cause a serious wellbore failure, especially the failure along the weak plane, when drilling a shale formation. Additionally, the cooling effect of drilling fluid can be constructive to the stability of horizontal wellbore in shale formation. Based on the research results, some effective measures can be developed to improve wellbore stability in the operation practices.

Appendix

The temperature field around a wellbore can be expressed as (Tang et al. 2018)

$$T(r,t)-{T}_{0}=\frac{({T}_{\text{w}}-{T}_{0}){\text{Ei}}(-u(r,t))}{Ei\left(-u\left({r}_{w},t\right)\right)}$$

(A1)

where

$$Ei\left(-x\right)= \gamma +\mathrm{ln}x+\sum_{k=1}^{\infty }\frac{{\left(-1\right)}^{k}{x}^{k}}{k!k}$$

(A2)

$$u(r,t)=\frac{{r}^{2}}{4{c}^{\text{T}}t}$$

(A3)

and $\gamma$ is the Euler constant, $\gamma \approx 0.5772$; ${c}^{\text{T}}$ is the thermal diffusivity.

Tang et al. (2018) also presented an analytic expression to calculate the pore pressure considering the thermo-chemical coupling effect, given by

$$P\left(r,t\right)={K}^{P}\frac{Ei\left(-f\left(r,t\right)\right)}{Ei\left(-f\left({r}_{w},t\right)\right)}+{K}^{TP}\frac{Ei\left(-u\left(r,t\right)\right)}{Ei\left(-u\left({r}_{w},t\right)\right)}+{K}^{CP}\frac{Ei\left(-g\left(r,t\right)\right)}{Ei\left(-g\left({r}_{w},t\right)\right)}+{P}_{0}$$

(A4)

where

$$f(r,t)=\frac{{r}^{2}}{4{C}^{\text{F}}t}$$

(A5)

$$g(r,t)=\frac{\phi {r}^{2}}{4{D}^{\text{S}}t}$$

(A6)

$$\begin{array}{c}{K}^{\text{P}}={P}_{\text{w}}-{P}_{0}-\frac{{D}^{\text{S}}{C}^{\text{CF}}}{{D}^{\text{S}}-\phi {C}^{\text{F}}}({C}_{\text{w}}-{C}_{0})\\ -[\frac{{c}^{\text{T}}{C}^{\text{TF}}}{{c}^{\text{T}}-{C}^{\text{F}}}+(\frac{{c}^{\text{T}}{C}^{\text{CF}}}{{c}^{\text{T}}-{C}^{\text{F}}}-\frac{{D}^{\text{S}}{C}^{\text{CF}}}{{D}^{\text{S}}-\phi {C}^{\text{F}}})\frac{{\overline{C} }^{\text{S}}{D}^{\text{T}}}{{c}^{\text{T}}\phi -{D}^{\text{S}}}]({T}_{\text{w}}-{T}_{0})\end{array}$$

(A7)

$${K}^{\text{TP}}=(\frac{{c}^{\text{T}}{C}^{\text{TF}}}{{c}^{\text{T}}-{C}^{\text{F}}}+\frac{{c}^{\text{T}}{C}^{\text{CF}}}{{c}^{\text{T}}-{C}^{\text{F}}}\frac{{\overline{C} }^{\text{S}}{D}^{\text{T}}}{{c}^{\text{T}}\phi -{D}^{\text{S}}})({T}_{\text{w}}-{T}_{0})$$

(A8)

$${K}^{\text{CP}}=\frac{{D}^{\text{S}}{C}^{\text{CF}}}{{D}^{\text{S}}-\phi {C}^{\text{F}}}[{C}_{\text{w}}-{C}_{0}-\frac{{\overline{C} }^{\text{S}}{D}^{\text{T}}}{{c}^{\text{T}}\phi -{D}^{\text{S}}}({T}_{\text{w}}-{T}_{0})]$$

(A9)

$${C}^{\text{F}}=\kappa \frac{3BK(1-\nu )}{3\eta (1-\nu )-2B{\eta }^{2}(1-2\nu )}$$

(A10)

$${C}^{\text{CF}}=-{C}^{\text{F}}[\frac{\mathfrak{R}{\rho }_{\text{f}}R\overline{T}}{{M }^{\text{S}}{\overline{C} }^{\text{S}}(1-{\overline{C} }^{\text{S}})}\frac{\phi }{{D}^{\text{S}}}-\frac{1}{\kappa }\frac{\eta \omega (1-2\nu )}{2G(1-\nu )}]$$

(A11)

$${C}^{\text{TF}}=\frac{{C}^{\text{F}}}{\kappa }[\kappa \frac{\mathfrak{R}{\rho }_{\text{f}}R\overline{T}}{{M }^{\text{S}}(1-{\overline{C} }^{\text{S}})}\frac{{D}^{\text{T}}}{{D}^{\text{S}}{c}^{\text{T}}}-\frac{\eta (1+\nu )}{3(1-\nu )}{\alpha }_{\text{m}}^{\text{T}}+(\eta {\alpha }_{\text{m}}^{\text{T}}-\phi {\alpha }_{\text{m}}^{\text{T}}+\phi {\alpha }_{\text{f}}^{\text{T}})+\frac{{K}^{\text{T}}}{{c}^{\text{T}}}]$$

(A12)

In Eq. (A5) to Eq. (A12), ϕ is the porosity, ${D}^{\mathrm{S}}$ is the solute diffusivity, ${C}_{0}$ is the solute mass fraction of formation fluid, ${C}_{\mathrm{w}}$ is the solute mass fraction of drilling fluid, ${\overline{C} }^{\mathrm{S}}$ is the mean solute mass faction of drilling and formation fluids, ${D}^{\mathrm{T}}$ is the thermal diffusion coefficient, $G$ is the shear modulus of rock, $K$ is the bulk modulus of rock, ${\rho }_{\mathrm{f}}$ is the fluid density, $\omega$ is the chemical swelling coefficient, $B$ is the Skempton coefficient, ${T}_{\mathrm{w}}$ is the temperature of drilling fluid, $\overline{T }$ is the mean temperature of drilling fluid and formation, $\kappa$ is the permeability coefficient of rock, $\mathfrak{R}$ is the reflection coefficient, ${M}^{\mathrm{S}}$ is the solute molar mass, $R$ is the ideal gas constant, ${\alpha }_{\mathrm{f}}^{\mathrm{T}}$ is the thermal expansion coefficient of fluid, ${\alpha }_{\mathrm{m}}^{\mathrm{T}}$ is the thermal expansion coefficient of rock matrix, and ${K}^{\mathrm{T}}$ is the thermal osmotic coefficient.