Sensitivity Analysis of Wellbore Mud Pressure towards Anisotropic Shale Properties, Pore Fluid Pressure and Far Field Stresses

Abstract

The paper investigates the mud pressure to maintain the stability of wellbores drilled in transversely isotropic shale through sensitivity analyses, carried out with analytical and numerical modeling (FLAC). To this end, we interpreted the anisotropic strength of the Tournemire shale with the Weakness Plane Model (WPM) and the modified Hoek–Brown criterion (HBm). The sensitivity analyses of synthetic case studies indicated a different trend in mud pressure for the two criteria. In some cases, the WPM predicts mud pressures higher than those predicted by the HBm and vice versa. The mud pressures predicted by the HBm resulted in being more sensitive to the increase in the anisotropy of the far field stresses for all the inclinations of the weakness planes. In this context, the WPM predicts some anomalous low mud pressures in a wide range of inclinations of the weak planes. The change in the frictional component of strength decreases with an increase in the pore fluid pressure for both criteria. The mud pressure predicted by the WPM resulted in being more sensitive to the change in frictional strength. The change in trend of the two criteria with change in input data suggests caution in the “a priori” selection of the strength criterion. A simple solution is proposed to predict a safe and reliable mud pressure with a small number of lab tests.

1. Introduction

Wellbores drilled in sedimentary basins encounter different rocks with different mechanical responses.

Shale makes up approximately 50 percent of all sedimentary rocks in the stratigraphic record [1], and the majority of deep drilling operations come across this rock [2]. In the oil and gas industry, more than 75% of the world drilling is carried out in shales, which are responsible for the major source (90%) of instability [3].

Currently, the mitigation of the effects of anthropogenic greenhouse gas emissions should lead to oil and gas output being in gradual decline. However, depleted oil and gas fields are good targets for CO₂ and H₂ storage [4,5,6,7]. In these depleted fields, new wellbores must be drilled for gas injection operations. Consequently, drilling operation in shale is still a matter of concern [8,9,10,11,12,13,14,15,16,17].

The prediction of the mud pressure to avoid wellbore instability requires the knowledge of the mechanical properties of the rock. The determination of the shale mechanical properties at large depth is still a problem. An accurate assessment of rock mechanical properties is through laboratory tests; however, core retrieval at large depths can be a difficult task [18]. In fact, shale cores can be subject to alteration as they are retrieved from the subsurface due to changes in pressure, temperature and oxidation state [19]. Furthermore, the processes of specimen preparation can induce damage [20]. All these problems can make the results of tests uncertain. As the process of rock coring can damage the material, the mechanical properties can be obtained from laboratory tests carried out on outcropping formations with structures close to the rock at a large depth [21]. The laboratory tests for the geomechanical characterization of shales with anisotropic strength consists of uniaxial and triaxial tests carried out at five or more inclinations of the weakness planes. The tests should be repeated at least twice for each inclination of the weakness planes, to verify their reliability. Therefore, the extensive process of coring and testing can result in being expensive and time consuming.

More often, the mechanical properties of the rock at large depths are obtained from indirect measurements by using correlations with log data and microscopic models [22,23]. These empirical correlations are used to obtain a continuous profile with the depth of the elastic and strength properties [24] and, for a given wellbore, log-derived data, are correlated with the rock properties measured in laboratory. In general, the empirical relationships with log data are valid in the basin from where they were set up. A thorough review of published empirical relations is given in [25,26]. Furthermore, [27] presented new correlations for deep-water reservoirs and shale oil and shale gas formations.

Recently, the Machine Learning Approach and hybrid neural net based models were proposed for the indirect determination of the uniaxial compressive strength and internal friction angle of shale [28,29].

According to [22,23,24,30], the mechanical properties obtained by log and microscopic models are affected by relevant uncertainty and cannot provide a thoroughgoing characterization of the rock. Shale often exhibits variation in strength properties along and across the lamination planes. This variation must be taken into consideration when carrying out stability analyses of wellbores [17]. However, the transverse isotropy of shale is not specifically addressed in the indirect approaches.

The effect of water weakening in a variety of sedimentary rocks has been widely investigated (i.e., [17,31,32,33,34,35]). On the other hand, the low permeability of shale rock can induce undrained conditions, immediately after drilling (short term). The undrained pore fluid pressure evolution can enhance plasticity [36,37,38,39].

In this study, we investigated the mud pressure to maintain the stability of wellbores drilled in transversely isotropic shale through sensitivity analyses. These analyses were carried out with analytical and numerical modeling (FLAC 2D, ver.8.1, Itasca, Minneapolis, MN, USA). We determined the strength properties of the Tournemire shale from the regressions of the data of lab tests carried out by [40] with the Weakness Plane Model [41] (WPM) and the Hoek–Brown criterion (HB) [42,43] adapted to anisotropic rock (HBm). We performed sensitivity analyses of synthetic cases by varying the inclination of the weakness planes, the far field stress anisotropy, the pore fluid pressure and the frictional component of the rock strength.

The results of these analyses identified the pros and cons of the predictions of WPM and HBm. Finally, we propose a simple solution for predicting a safe and reliable mud pressure with a small number of lab tests.

The flowchart of this study is reported in Figure 1.

2. Materials and Methods

This section presents the strength criteria for isotropic and transversely isotropic rocks and the related mud pressure formulas for avoiding shear failure.

2.1. Strength Criteria

Sedimentary rocks are porous materials and can exhibit a significant strength decrease with the increase in the pore water pressure. Consequently, the pore water pressure must be taken into account in the mechanical analysis. The concept of effective stress was originally introduced by Terzaghi [44] for soils. According to [45], the theory of poroelasticity demonstrates that the deformation of the porous material is proportional to the effective stress:

$${\sigma }_{ij}^{\prime }={\sigma }_{ij}-\alpha p_f{\delta }_{ij},$$

(1)

where: ${\sigma }_{ij}^{\prime }$ and σ_ij are the effective and total stress tensors, respectively; α is the Biot coefficient, function of the bulk moduli of the rock skeleton, rock grains and fluid; p_f is the pore pressure in the porous medium; δ_ij is the Kronecker function.

Equation (1) assumes that the rock is linearly elastic. This assumption is not directly applicable to rocks subject to failure processes [46]. The Terzaghi’s principle of effective stress for soils (α = 1) seems to be the most appropriate definition to be used in rock failure analysis [46,47].

All the subsequent analyses are carried out with the effective stresses. The deviatoric stress (σ₁ − σ₃) is reported in terms of total stresses, because it is always an effective stress: [σ₁ − σ₃] = [(σ₁ − p_f) − (σ₃ − p_f)] = [${\sigma }_1^{\prime }-{\sigma }_3^{\prime }$].

Rock failure is generally approached in terms of isotropic behavior; however, the actual behavior of many rocks is anisotropic. Shales are frequently affected by parallel weakness planes, with reduced strength compared to intact material. The analysis of failure in such rocks needs a strength criterion which accounts for the strength anisotropy.

The most widely used criterion is WPM, that considers a set of parallel discontinuities on which failure occurs in the form of slip. WPM written in terms of principal stresses (σ₁ and σ₃) becomes

$${\left({\sigma }_1-{\sigma }_3\right)}_{slip}=\frac{2\left(c_w^{\prime }+{\sigma }_3^{\prime }\tan {\varphi }_w^{\prime }\right)}{\left(1-\frac{\tan {\varphi }_w^{\prime }}{\tan {\beta }_w}\right)\sin 2{\beta }_w}$$

(2a)

where β_w, $c_w^{\prime }$ and ${\varphi }_w^{\prime }$ are the inclination, the cohesion and the friction angle of the weakness planes, respectively.

Figure 2 shows the plot of Equation (2a): the minimum strength occurs at β_w = 45° + ${\varphi }_w^{\prime }$/2 and for values of β_w, close to 90° and in the range 0°–${\varphi }_w^{\prime }$, slip on the plane of weakness cannot occur. Within these ranges, shear failure occurs through the rock material in a direction not controlled by the plane of weakness [47]. The plateau of constant strength represents the resistance of the rock in the ranges of β_w that are not predicted by Equation (2a). The plateau of constant strength can be described by the Mohr–Coulomb criterion (M-C):

$${\sigma }_1^{\prime }=\frac{2c^{\prime }\cos {\varphi }^{\prime }}{\left(1-\sin {\varphi }^{\prime }\right)}+{\sigma }_3^{\prime }\frac{\left(1+\sin {\varphi }^{\prime }\right)}{\left(1-\sin {\varphi }^{\prime }\right)}={\sigma }_c+{\sigma }_3^{\prime }{N}_{\varphi },$$

(2b)

where $c^{\prime }$ is the cohesion, ${\varphi }^{\prime }$ is the friction angle, σ_c is the uniaxial compressive strength of the intact rock and N_ϕ is the slope of the strength envelope.

WPM needs the calculation of the strength with Equations (2a) and (2b) for a given β_w. The equation that predicts the lowest strength must be considered for failure. WPM is a simple criterion that does not take into account the contemporary development of slip and failure in the intact rock, observed in laboratory tests [48]. Furthermore, the plateau of constant strength is not always observed in the experimental tests.

The use of WPM needs the determination of the cohesion $c_w^{\prime }$ and the friction angle ${\varphi }_w^{\prime }$ of the weakness planes and the cohesion c’ and the friction angle ϕ′ of the rock matrix (plateau strength).

HB was specifically proposed for intact rock and rock masses and is widely used for a variety of rock engineering problems [42,43]. There are different versions of HBm adapted to transversely isotropic rocks (i.e., [48,49,50,51]). In this modified criterion, there is no distinction between failure along the weakness planes and failure in the rock matrix. A single continuous function describes the criterion.

Refs. [50,51] assumed that the rock is intact (s = 1), and the uniaxial compressive strength and the empirical dimensionless constants vary at every inclination β_w of the weakness planes (instantaneous isotropy). Under these conditions, HBm becomes

$${\left({\sigma }_1-{\sigma }_3\right)}_{\beta w}={\left(m_{\beta w}{\sigma }_{c\beta w}{\sigma }_3^{\prime }+{\sigma }_{c\beta w}^2\right)}^{0.5}$$

(3)

where σ_cβw and m_βw are the instantaneous uniaxial compressive strength of the rock and the instantaneous empirical dimensionless constant.

Figure 3a shows the strength envelops of a shale rock for different β_w. Equation (3) requires a high number of uniaxial/triaxial tests, for the determination of the strength parameters σ_cβw and m_βw at various inclinations of the weakness planes.

HBm generally matches the results of compression tests quite well, compared to WPM. The variation of the strength with β_w can assume different trends [52,53,54] and is not constrained by both the minimum at β_w = 45° + ${\varphi }_w^{\prime }$/2 and the plateau of constant strength. Figure 3b shows a typical interpolating function of this variation.

Finally, the prediction of the uniaxial tensile strength with WPM and HBm is affected by the overestimation of this property [52]. A thorough review of the strength criteria for the prediction of the uniaxial tensile strength in transversely isotropic rocks was presented by [55,56,57].

2.2. Calculation of the Mud Pressure

The calculation of the mud pressure of a wellbore, drilled along a principal direction, requires the Kirsch solution coupled with the strength criteria described in Section 2.1.

At the boundary of the hole, the Kirsch solution in plane strain conditions (Figure 4) becomes

$$\begin{array}{c}{{\sigma }^{\prime }}_{𝜗}=S-p_w-p_f,\\ {{\sigma }^{\prime }}_r=p_w-p_f,\\ {{\sigma }^{\prime }}_{axis}=S_z-p_f,\\ S={\sigma }_{MAX }+{\sigma }_{min }-2\left({\sigma }_{MAX }-{\sigma }_{min }\right)cos2𝜗,\\ S_z={\sigma }_z-2\nu \left({\sigma }_{MAX}-{\sigma }_{min}\right)\cos 2𝜗\end{array}$$

(4)

where σ_min, σ_MAX and σ_z are the far field stresses, S and S_z are the induced state of stress, 𝜗 is the borehole azimuth, p_w is the mud pressure and ν is the Poisson ratio.

The analysis focuses on the combination of the effective tangential stress ${\sigma }_{𝜗}^{\prime }$ and radial stress ${\sigma }_r^{\prime }$ which are both affected by the mud pressure p_w (we disregarded the occurrence of failure with ${{\sigma }^{\prime }}_{axis}$): σ_θ > σ_axis > σ_r.

The coupling between Equation (4) and WPM (Equations (2a) and (2b)) gives the limit mud pressures for the weakness planes and the rock material:

$$p_{w\_slip}=\frac{S\left(1-\frac{\tan {\varphi }_w^{\prime }}{\tan {\beta }_w}\right)\sin 2{\beta }_w-2c_w^{\prime }+2\tan {\varphi }_w^{\prime }{p}_f}{2\left[\tan {\varphi }_w^{\prime }+\left(1-\frac{\tan {\varphi }_w^{\prime }}{\tan {\beta }_w}\right)\sin 2{\beta }_w\right]},$$

(5a)

$$p_{w\_MC}=\frac{S-{\sigma }_c+p_f(N_{\varphi }-1)}{\left(N_{\varphi }+1\right)}$$

(5b)

The coupling between Equation (4) and HBm gives [52]:

$$p_{w\_H\&Bm}=\frac{4S+m_{\beta w}{\sigma }_{c\beta w}-{\left[{\sigma }_{c\beta w}^2(m_{\beta w}^2+16)+8m_{\beta w}{\sigma }_{c\beta w}\left(S-2p_f\right)\right]}^{0.5}}{8}$$

(6)

The angle β_w must be determined at any azimuth 𝜗 of the wellbore because it is one of the variables of Equation (5a) and is indirectly involved in Equation (6) for the calculation of the uniaxial compressive strength σ_cβw and the dimensionless constant m_βw. The authors of [52] found the equations that relate to β_w, the inclination of the weakness plane δ and the azimuth 𝜗. Figure 5 shows the change in the angle β_w with the inclination of the weakness plane δ, at different azimuth 𝜗.

Vertical weakness planes (δ = 90°) seem to be uncommon in real fields [58]. Figure 5c shows horizontal weakness planes with the reference system shown in Figure 5a and with a rotation of the maximum and minimum far field stresses. This new configuration is equivalent to δ = 90° and is here defined as δ* = 0°. In this new system, the mud pressure for avoiding slip along the horizontal weakness planes must be calculated according to the case with δ = 90° of the original reference system (vertical planes).

The mud pressures reported in Equations (5) and (6) take into account the pore fluid pressure p_f, according to the Terzaghi principle (Equation (1)). Shales have a low permeability and immediately after drilling, the condition close to the boundary of the borehole can become undrained (no fluid flow through the rock pores). The undrained conditions can induce a change in pore fluid pressure Δp_f ≠ 0 when R > 1 and the rock is soft. In other cases, there is no change in pore fluid pressure Δp_f = 0, with no fluid flow through the rock pores [59]. The evolution of the pore fluid pressure affects wellbore stability in the short term (undrained conditions) and in the long term (drained conditions).

3. Interpretation of Laboratory Tests on Tournemire Shale

The set-up of sensitivity analyses of wellbore stability requires the knowledge of the strength parameters. To this end, we interpreted laboratory tests carried out on Tournemire shale by [40].

3.1. Determination of the Strength Properties with WPM and HBm

Abdi et al. (2015) [40] carried out uniaxial, triaxial and Brazilian tests on Tournemire shale specimens at different inclinations of β_w of the weakness planes.

The strength parameters, necessary for the calculation of the mud pressures, are obtained with the regression of the data of these lab tests with WPM and HBm. We reduced the tensile strength of the Brazilian tests by a factor equal to 0.8, according to [60]. The tensile strength ranges from σ_tβw = −3.8 MPa at β_w = 90° to σ_tβw = −4.6 MPa at β_w = 0°.

The regressions carried out with HBm at different β_w were set up by using the complete data set (uniaxial, triaxial and Brazilian tests). The linear regression of the data [σ₃, (σ₁ − σ₃)²] enabled the calculation of the instantaneous uniaxial compressive strengths σ_cβw and the instantaneous constants m_βw.

Table 1. Strength parameters of the shale obtained with the HBm regression of lab data.

βw (°)	σcβw (MPa)	mβw (-)
0	25.97	3.90
30	20.09	4.69
45	17.40	4.39
60	18.12	5.60
90	29.60	7.16

reports the rock properties obtained through this process.

Figure 6a shows that the calculated uniaxial compressive strength matches well with the results of the lab tests. The figure indicates that the lower uniaxial compressive strength measured in lab tests (σ_cβw = 15.27 MPa) occurs at β_w = 60°. The minimum uniaxial compressive strength obtained from the regression (σ_cβw = 17.40 MPa) occurs at β_w = 45°.

Figure 6a also shows the interpolating function of the regression data of σ_cβw. The function is a second-order polynomial (parabola), which is used for the calculation of the mud pressure with Equation (6). The minimum of the parabola occurs close to β_w = 40°. The effect of this mismatch on the mud pressure will be discussed in the next section. Figure 6c shows the percentage increase in uniaxial compressive strength σ_cβw compared to the minimum at β_w = 40°. The maximum percentage increase occurs at β_w = 90° and β_w = 0°, as expected.

The instantaneous constants m_βw (Figure 6b and Table 1) change with β_w from m_βw = 3.90 to m_βw = 7.16, in agreement with the value of m = 6 ± 2, suggested by [61], for shale. Figure 6b also shows the interpolating function of the regression data of m_βw, which is a second-order polynomial (parabola).

It is worth to note that the instantaneous dimensionless constant m_βw does not exhibit a well-defined trend in several data regressions [52] and an average value must be assumed. The average value for our dataset is m_βw = 5.16. Figure 6d shows the percentage increase/decrease calculated with m_βw = 5.16 and the function of m_βw. The figure indicates that the variation is ≅ ±30% and in the range β_w = 0°–55° m_βw = 5.16 is higher than the m_βw of the function. We investigated the mud pressure sensitivity to the instantaneous constants m_βw in Section 4.

Figure 7a shows the comparison between the experimental results and the linear regression data obtained with WPM. The figure shows two regressions that were obtained with the complete experimental dataset (CD) and with the experimental data set without the Brazilian tests (NB). The figure indicates that the best matching of the experimental data is obtained with the regression without the Brazilian tests. The value of the uniaxial compressive strength at β_w = 0° and β_w = 90° for the CD regression are quite lower than the measured data. The figure also shows the WPM functions obtained with the two regressions. Both functions do not properly match the experimental data. However, the lowest value of the uniaxial compressive strength which occurs at β_w = 60° is equal for both cases and is in agreement with the experimental data. This fact indicates that the cohesion $c_w^{\prime }$ and the friction angle ${\varphi }_w^{\prime }$ of the weakness planes are very close in both cases.

In the end, the strength parameters of WPM were calculated with the NB regression data. The cohesion $c_w^{\prime }$ and the friction angle ${\varphi }_w^{\prime }$ of the weakness planes were calculated at β_w = 60° where the lowest value of the uniaxial compressive strength occurs. The plateau strength (c′ and ϕ′), which should correspond to the strength of the rock matrix, was calculated at β_w = 90°, with the M-C criterion.

Table 2. Strength parameters of the Tournemire shale obtained with the WPM regression of lab data.

	${\mathit{\varphi }}_{\mathit{w}}^{\prime }$(°)	${\mathit{c}}_{\mathit{w}}^{\prime }$(MPa)
Weakness planes	32	4.26
Plateau	33	8.37

reports these parameters. The plot of WPM in the diagram (β_w–σ_cβw) was obtained by finding the intersection of the plateau of constant strength with Equation (2a).

Figure 7b shows the comparison between the experimental uniaxial compressive strength σ_cβw and the interpolating function of the HBm and WPM. The figure indicates that the best data matching is obtained with HBm. Figure 6c shows the percentage change in uniaxial compressive strength between the functions and the experimental data. The figure indicates that WPM overestimates σ_cβw for all the inclinations of the weakness planes β_w with a maximum Δσ_cβw = +48.06% at β_w = 30°. HBm also overestimates σ_cβw for all the inclinations except at β_w = 30°, with a maximum Δσ_cβw= +27.72% at β_w = 60°. The overestimation of the uniaxial compressive strength is undoubtedly higher for WPM.

3.2. Variation of the Shale Properties with Wellbore Azimuth

WPM predicts the rock strength with two couples of constant strength parameters: the cohesion and friction angle of the plateau and weakness planes. The parameters remain constant and the variation of the strength is ruled by the change in β_w (Figure 5b).

HBm exhibits a continuous variation of the strength with the two instantaneous parameters, that can be calculated at any wellbore azimuth with the interpolating functions. Figure 8 shows the plots of the trend of m_βw and σ_cβw at the boundary of the wellbore with different δ. Wellbore azimuths with the lower strength do not necessarily correspond to failure zones. The induced state of stress S and the pore fluid pressure p_f can also influence the location of the failure zones.

4. Sensitivity Analyses: Results and Discussion

This section reports the results of the sensitivity analyses of mud pressure predicted by WPM and HBm, by varying the far field stresses and pore fluid pressure and the frictional component of strength. These parameters are often uncertain in real cases. The sensitivity analyses aim to investigate the responses of the two criteria and highlight the reliability of each model in different field contexts.

4.1. Synthetic Case Studies

We carried out a sensitivity analysis of synthetic cases. We assumed a wellbore drilled in Tournemire shale whose properties are reported in Table 1 and Table 2. In the majority of analyses, we assumed that the instantaneous dimensionless constant m_βw is fixed at the average value m_βw = 5.16. The effect of the variation in m_βw with wellbore azimuth on the magnitude of the mud weight is discussed in Section 4.2.

Table 3. Far field stresses and pore fluid pressures of synthetic cases.

R = σMAX/σmin	σMAX (MPa)	σmin (MPa)	pf (MPa)
1.0	20	20	11
1.0	33	33	11
1.0	33	33	15
1.1	20	22	0
1.1	20	22	8
1.1	20	22	11
1.25	20	25	8
1.25	20	25	11
1.5	20	30	0
1.5	20	30	11
1.5	20	30	15
1.75	20	35	11
2.0	20	40	11

reports the assumed far field stresses and pore fluid pressures. Finally, we assumed that the conditions are undrained with Δp_f = 0, immediately after drilling (see Section 2.2).

The geomechanical analysis of borehole stability is carried out by using stresses and pressures and is subsequently converted into mud weight (density units) [46]:

$${\rho }_{mud}=\frac{p_w}{gz},$$

(7)

where ρ_mud is the mud density, g is gravity and z is the depth.

All the subsequent sensitivity analyses are presented in terms of specific gravity (SG = ρ_mud/ρ_water), by assuming z = 1000 m, for the comparison of different cases.

4.2. Comparison between Mud Weight Patterns Calculated with WPM and HBm

Figure 9 shows the mud weight (SG) calculated at different inclinations of the weakness plane δ for some of the synthetic cases reported in Table 3. The figure also shows the mud weight of the rock matrix calculated with Equation (5b) for WPM.

A rough analysis of these results indicates an agreement of the trends of the mud weight calculated with WPM and HBm.

Figure 9a,b (case R = 1) show that the peaks of mud weight occur four times and have the same values for all the inclinations of the weakness plane δ.

Figure 9c–h (cases R > 1) show that the far field stress anisotropy influences the number of peaks of the mud pressures. The peaks occur twice, except for δ = 0° and δ = 90°. Figure 9 shows that the positions of the peaks calculated with the two criteria at a given δ have different azimuths. The magnitude of the mud weight peaks is generally different in the two predictions.

Figure 9e–h (cases R = 1.5) show an elongated pattern of the mud pressures towards 𝜗 = 90° for both criteria, because of the high far field stress anisotropy.

4.3. Influence of the Far Field Stresses and Pore Fluid Pressure on Mud Weight Prediction

The results reported in Section 4.2 evidenced some influence of the far field stresses and pore fluid pressure on the mud weight calculated with WPM and HBm. We investigated this influence with comparative analyses.

Figure 10 shows a quantitative comparison of SG shown in Figure 9 and Table 3. The figure shows that almost all mud weights with R > 1 are well approximated with second-order polynomial functions (solid and dotted lines).

Figure 10a (case R = 1) shows that SG calculated with WPM is slightly higher than the SG calculated with HBm, when the far field stresses are 20–20 MPa with p_f = 11 MPa and 33–33 MPa with p_f = 15 MPa. On the other hand, SG coincide when the far field stresses are 33–33 MPa with p_f = 11 MPa. The reason for the mismatch and coincidence of the two approaches can be attributed to the different sensitivity to the far field stresses and pore fluid pressure.

In all cases with R > 1, the SG calculated with the two criteria are alternatively higher/lower as a function of the inclination δ and the far field stress anisotropy R (Figure 9b–d). The trend for mud weight with δ is practically parallel in each approach. WPM predicts higher SG in the range δ = 0°–45° and lower SG in the range δ = 45°–90° at high R.

Figure 10b–d indicate that the difference between WPM and HBm predictions does not change with the magnitude of the pore fluid pressure. In fact, the change in trend (higher/lower SG) almost always occurs at the same δ, regardless of the variation of the pore fluid pressure.

The increase in pore fluid pressure (see Table 3) is Δp_f = +36% (case R = 1.5) and Δp_f = +37% (cases R = 1.1 and R = 1.25). This increase is practically equal. Figure 10e shows the percentage change in SG with the change in pore fluid pressure Δp_f. The figure indicates that the percentage increase in SG is higher at R = 1.1 for both criteria. This result indicates that SG is more sensitive to an increase in pore fluid pressure at low anisotropy of the far field stresses. The maximum SG always occurs at δ = 0° (HBm) and at δ = 90° (WPM). The difference between the percentage change in SG calculated with the two criteria becomes relevant when R > 1.1 and δ > 50°. The trends of the percentage change obtained with HBm are almost parallel for all cases (R = 1.1, R = 1.25 and R = 1.5). On the other hand, the trends of the percentage change obtained with WPM vary with R, with a remarkable rise from δ = 30°to 90°, when R = 1.5.

Figure 10f shows the comparison between the WPM and HBm predictions of SG, in terms of percentage change, when R = 1.1 and R = 1.5. The figure shows that the most significant decrease in SG of HBm (−9%), occurs when R = 1.1 at δ = 0°. The range δ = 75°–90° exhibits a lightly higher SG calculated with HBm (+2%).

On the other hand, the percentage decrease in HBm reduces when R = 1.5. In this case, the percentage increase in HBm is very high (+29–40%) from δ = 40° to δ = 90°. The reason for this high mismatch appears to be the remarkable decrease in WPM mud weight when R > 1.1 (Figure 10c,d). Finally, the figure shows that the change in trend (higher/lower SG) predicted by WPM and HBm is independent of the pore fluid pressure magnitude.

We performed a quantitative comparison between the predictions of WPM and HBm, by varying the far field stress anisotropy. Figure 11a,b show the comparison of the mud weight predicted by the two criteria at different R with constant pore fluid pressure. The figures indicate that the change in trend (higher/lower SG) occurs when R > 1.1. Furthermore, HBm predicts a definitely higher mud weight when R > 1.5 for all the inclinations δ of the weak planes. This outcome indicates that WPM provides questionable mud weights at high R, in the range δ = 50°–90°.

Figure 11c shows the percentage change in SG with the variation of the far field stresses ΔR (at constant pore fluid pressure p_f = 11 MPa). The change in R from 1.1 to 1.5 corresponds to ΔR = +36% and the change in R from 1.1 to 1.25 corresponds to ΔR = +14%. The figure indicates that the variation of SG calculated with HBm is more sensitive to the increase in far field stress anisotropy, especially from δ = 30° to δ = 90°. The change in SG calculated with HBm is overall higher, except for a few inclinations of the weakness planes with ΔR = +14%, and shows a maximum at δ = 45°. On the other hand, the change in SG calculated with WPM after the peak exhibits a substantial decrease from δ = 30° to δ = 90°. This prediction seems to be anomalous, particularly close to δ = 90°.

Figure 11d shows the comparison between the SG of the weakness planes and rock matrix when R = 1.5. In general, the SG of the rock matrix is expected to be lower than the SG of the weak planes. Nevertheless, the figure clearly shows that the SG of the rock matrix is higher than the SG of the weakness planes in the range δ = 75°–90°. On the other hand, the peaks calculated with HBm (Figure 11a–c) are higher than the peaks of the rock matrix in the range δ = 45°–90°.

The change in trend (higher/lower SG) of the predictions with WPM and HBm occurs with increasing R. This change depends on the interpolating functions of the strength. WPM provides anomalous results when R > 1.1. These outcomes are related to the constrain of the minimum strength at β_w = 45° + ${\varphi }_w^{\prime }$/2 and the fast growth of the interpolating function.

Figure 10 and Figure 11 indicate that the maximum mud weight occurs at a specific inclination δ = δ_crit of the weakness planes, irrespective of R and p_f. The maximum SG predicted by WPM always occurs at δ_crit = 45° − ${\varphi }_w^{\prime }$/2, as found by [52] and in this paper (δ_crit = 30°). On the other hand, the maximum SG predicted by HBm occurs at δ_crit = 50° with β_w = 40°, which corresponds to the minimum strength of the interpolating function.

The analysis of the location of the mud weight peaks at different δ evidenced that the correspondent angles β_w have different distribution. Figure 12 shows the frequency distribution of the angles β_w at which the peaks of the mud weight occur. The distribution of the angles β_w was calculated at different far field stresses (R = 1–1.5), pore fluid pressures (8 MPa, 11 MPa and 15 MPa) and inclination of the weakness planes (δ = 0°–15°–30°–45°–50°–60°–90°).

The peaks calculated with HBm occur in a wide range of β_w = 15°–75°. The range of β_w is affected by the far field stress anisotropy R. Furthermore, the pore fluid pressure magnitude does not seem to influence the location of the peaks, although the high variation Δp_f = +37%.

The peaks calculated with WPM occur in the range β_w = 50°–70°. Most peaks occur at β_w = 60°, which corresponds to the minimum strength β_w = 45° + ${\varphi }_w^{\prime }$2. This result suggests that the mud weight is lightly influenced by the anisotropy of the far field stresses R and heavily depends on the minimum strength. Furthermore, the pore fluid pressure magnitude negligibly affects the location of peaks.

These outcomes can explain why SG calculated with WPM are higher/lower than the peaks calculated with HBm. Finally, HBm is more sensitive to the far field stress anisotropy than WPM, in terms of β_w.

4.4. Influence of the Frictional Component of Strength on the Mud Weight Prediction

We investigated the mud pressure sensitivity to the frictional component of strength. We increased the friction angle of the weakness planes from ${\varphi }_w^{\prime }$ = 32° (Table 2) to ${\varphi }_w^{\prime }$ = 38°. The percentage increase is Δ${\varphi }_w^{\prime }$ = +19%. Figure 13a shows the percentage decrease in SG at different R and pore fluid pressures p_f. The figure indicates that the maximum change occurs close to δ = 50°–60° for all cases. The minimum change occurs at δ = 0°. The maximum SG change occurs when R = 1.5 and p_f = 11 MPa. The SG change increases with R and is mitigated by high pore fluid pressure.

The instantaneous dimensionless constant m_βw of HBm can be considered as a frictional component of the rock strength [62]. Figure 6d shows the percentage increase/decrease in frictional strength calculated with m_βw = 5.16 and the function of m_βw. The figure indicates that the variation is ≅ ±30%. The constant m_βw = 5.16 is higher than m_βw of the function, in the range β_w = 0°–55°. Figure 13b shows the correspondent change in SG and indicates that SG is lightly underestimated with the average value m_βw = 5.16, when R = 1.1, with a maximum decrease of −1.9%. Furthermore, an overestimation of SG (+2.5%) with the average value m_βw = 5.16 occurs in the range of inclinations δ = 0°–30° when R = 1.5. The underestimation of SG occurs in the range δ = 25°–90°. The more remarkable underestimations occur at δ = 90°, in all cases. The more substantial underestimation of SG occurs at high far field stress anisotropy, but an increase in pore fluid pressure reduces the magnitude of the underestimation. This outcome indicates a mitigation effect of the pore fluid pressure on the overestimation of the frictional strength, as found with WPM. The previous discussion indicates that the change in frictional strength component lightly influences the peak of mud pressure. In fact, the influence of a high change in frictional strength produces a small underestimation of the mud weight (−3.6%), in the worst case (Δm_βw = +27%). The mud weight calculated with WPM results in being more sensitive to the change in frictional strength.

Finally, the assumed constant value m_βw = 5.16 is in agreement with m = 6 ± 2 suggested by [61] for shale, and the majority of m_βw calculated from lab tests by [52,53,54].

4.5. Comparison of the Results with Other Studies

There is a wide literature on wellbore stability in shales. The influence of thermo/hydro/mechanical coupling and multi-planes on mud pressure prediction is generally carried out with WPM (i.e., [14,36,63,64,65,66,67]). The majority of studies reports predictions of mud weight carried out with WPM in synthetic cases or real cases by systematically varying wellbore azimuth and inclination. Generally, the friction angle and the cohesion of the weakness planes are directly estimated/assumed and not measured in lab.

The reliability of the SG prediction with measured rock properties with WPM is not specifically and quantitatively investigated in other studies. Furthermore, to our knowledge, the prediction of the mud pressure with HBm is just reported in [52,53,54] and in [68]. The results of our study are in agreement with the results reported in [52,53,54]. The study reported in [68] analyzes a synthetic case with assumed rock properties and a very low frictional component of strength, that makes the comparison difficult.

These observations highlight the need for further studies related to the analysis of the reliability of mud pressure predictions, based on strength properties of shale from laboratory measurements.

5. Numerical Modeling of Wellbore Stability with WPM and HBm

We carried out numerical simulations of wellbore stability with the code FLAC (Fast Lagrangian Analysis of Continua, ver 8.10, Itasca, Minneapolis, MN, USA). FLAC is a two-dimensional explicit finite difference program for engineering mechanic computations.

The computational domain is a radial grid (Fish DONUT, implemented in FLAC) with 60 zones in the radial direction and 60 zones on the circumference. The wellbore diameter is 1 m and the diameter of the grid is 40 m (Figure 14). The state of stress within the model is initialized by imposing σ_MAX in the horizontal and σ_min in the vertical direction. The same horizontal and vertical stresses are applied at the external boundary of the grid. The grid is configured for fluid flow analysis. The undrained conditions used in the analytical solution (constant pore fluid pressure Δp_f = 0) are obtained by (a) imposing and fixing the pore fluid pressure (p_f); (b) imposing the bulk modulus of the fluid equal to zero; (c) setting of the mechanical calculations and turning off the fluid flow calculations.

A mechanical pressure, equal to the mud pressure calculated with the analytical solution, is applied at the internal boundary (wall of the borehole). A higher mechanical pressure avoids slip or failure in the rock matrix. A lower mechanical pressure results in an enlargement of the failed zone and possibly in borehole collapse.

We carried out numerical simulations with the Ubiquitous Joint Model (UBI) and HB criterion for isotropic rock. Both constitutive laws are a continuum based model.

UBI accounts for joint implicitly: the weakness planes are smeared across the rock mass (ubiquitous) with a given inclination (δ). At first the code detects general failure with the M-C in the rock matrix and applies plastic corrections. The corrected stresses are then analyzed for failure on the weakness plane with the WPM, and updated accordingly.

Figure 15 shows the results of the numerical simulations carried out with R = 1.5 with p_f = 11 MPa at δ = 0°–30°–45°–90° carried out with UBI. The pattern of failure location at δ = 0°, δ = 30° and δ = 45° shown in Figure 15a–c is in agreement with the peaks shown in Figure 9e. Figure 15d shows the results of the numerical simulations carried out at δ* = 0°, with the new reference system (see Figure 5c). The figure clearly shows that there is the occurrence of plasticity in the rock matrix (at 𝜗 = 90°) and, surprisingly, slip does not occur along the weakness planes, in agreement with the anomalous predictions shown in Figure 9e and Figure 11d. The agreement between the results obtained with the WPM and UBI does not indicate that the prediction of the mud pressure at δ* = 0° is reliable. Instead, the results indicate that WPM under some field conditions cannot correctly predict mud pressures.

Numerical simulations with HBm, by using the HB isotropic model implemented in FLAC, can be obtained by programming a routine or simply dividing the area around the well into slices with different σ_cβw.

The trend of σ_cβw with wellbore azimuth 𝜗 are shown in Figure 8a. We identified the slices with constant averaged σ_cβw according to Figure 8a. Figure 16a,b show the variation of σ_cβw for the inclination δ = 45°, δ = 0° and δ* = 0°.

The results of the numerical simulations with R = 1.5 with p_f = 11 MPa are shown in Figure 17. The numerical results shown in Figure 17a,b are in agreement with the analytical predictions. Furthermore, the cases with δ = 0° and δ* = 0° also show plasticity close to 𝜗 = 90° (Figure 17c,d). This result is in agreement with Figure 9f, where a very small drop of mud pressure occurs between the peaks. Finally, Figure 17d shows the results of the numerical run carried out with the low mud pressure predicted by the WPM (Figure 9e and Figure 11d). The figure shows a wide plasticity area, indicating that the WPM prediction underestimates the mud pressure of the weakness planes.

The results of all numerical simulations are in agreement with the analytical solutions: the mechanical pressure that induces slip or rock matrix failure is almost equal to the mud pressure calculated analytically. The numerical results obtained with UBI and HB criterion confirm the results obtained with the analytical solutions, also for the anomalous cases. The task is now to identify a reasonable solution for a safe, reliable and simple analysis for the stability of wellbores drilled in anisotropic shale. In the next section, the pros and cons of the WPM and HBm are discussed.

6. Pros and Cons of the WPM and HBm: Conclusions

The results presented in Section 4 and Section 5 evidenced that the SG predicted by the HBm is lower than SG predicted by the WPM at least in the range δ = 0°–40°. The highest gap is −9% and occurs when the far field stress anisotropy is R = 1.1. On the other hand, the SG predicted by the WPM is substantially lower than SG predicted by the HBm in the range δ = 40°–90°, when R > 1.1. Furthermore, SG calculated with WPM has a sudden drop, in the range δ = 50°–90°, which appears anomalous. The highest gap is −37% and occurs at δ = 90° when the far field stress anisotropy is R = 1.5. The inclination δ = 90° seems uncommon, as discussed in Section 2.2. However, this case becomes δ* = 0° (rotated reference system) and can often occur in real fields.

We calculated the coefficient of variation of SG [CV = (Standard Deviation/Mean) × 100]. Figure 18 shows the comparison between the coefficient of variation of SG calculated with the two criteria.

The coefficient of variation related to HBm is CV ≤ 3% when R = 1.1 and CV ≤ 5% at higher values of R. The coefficient of variation related to WPM is CV ≤ 4% when R = 1.1 and shows an increasing extent of variability with R. This result confirms that the SG prediction with WPM should be avoided in the range δ = 60°–90° when R > 1.1.

This analysis suggests that a conservative estimate of SG can be obtained with WPM at low far field stress anisotropy and with HBm at high far field stress anisotropy. This conclusion needs further investigations, always based on lab measurements of shale properties. In fact, the HBm function fitted the experimental data of the Tournemire shale quite well, even though the minimum strength is predicted at β_w = 40° and the experimental one is at β_w = 60°. On the other hand, the WPM function fitted the experimental data rather poorly, even though the predicted and experimental minimum strengths occur at the same β_w = 60°.

Figure 12 indicates how many inclinations of the weakness planes must be tested in uniaxial/triaxial compression with the two criteria. The WPM predicts the majority of mud pressure peaks in correspondence of β_w = 50°–60° and therefore needs a lower number of lab tests compared to HBm. Nevertheless, WPM provides anomalous results in some conditions. On the other hand, the WPM is implemented in the majority of the commercial software of geomechanics.

The prediction of SG with WPM and HBm, in a real field, must also take into account the uncertainties in the estimation of the variables that affect the mud pressure: inclination of the weak planes; magnitude of the far field stresses and pore fluid pressure. All these issues suggest that the limit mud weight could be higher than that predicted at a specific δ.

The higher value of SG, which corresponds to δ_crit, could be considered the reference SG at least with small δ and low R for HBm and with large δ and high R for WPM. The results of the analyses reported in Section 4 indicated that δ_crit = 30° for WPM and δ_crit = 60° for HBm.

Figure 19 shows the percentage change in SG between the value calculated at δ_crit and the other δ. The percentage change has the highest values at δ = 0° when SG is calculated with HBm and at δ = 90° (which corresponds to δ* = 0°) when SG is calculated with WPM.

As the prediction with HBm seems to “underestimate” (compared to WPM) the mud weight at low R, the use of SG which corresponds to δ_crit for R = 1.1 results in a maximum percentage decrease equal to −8% that occurs at δ = 0°. In the range δ = 15°–75°, the maximum percentage decrease is −4% (Figure 19a) The percentage decrease at high R when δ = 0° becomes high (−13%) and the occurrence of tensile failure (hydraulic fracturing) must be carefully investigated in order to find a properly safe mud weight window. According to [52], the lower SG for hydraulic fracturing occurs at δ = 0°.

The maximum percentage decrease calculated with the WPM in the range δ = 15°–45° is −4% for all the far field stress anisotropy R and pore fluid pressures p_f. We highlight that the percentage change in Figure 19b has been truncated at δ = 50° when R = 1.5, because of the anomalous drop that was discussed previously.

Based on the outcomes of this paper and those reported in [52,53,54], the critical inclination for shales is expected to occur in the range δ_crit = 30°–60°. This range of δ_crit corresponds to β_w = 40°–60° at the boundary of the borehole. Consequently, the use of δ_crit, for the prediction of the mud pressure in cases with other inclinations δ, results in a reduction of the number of compression tests to be carried out for determining the strength parameters, especially with the HBm criterion. The shale specimens can be tested with compression tests at three inclinations within the range β_w = 40°–60°. Finally, the use of the HBm, with the assumption of the dimensional constant fixed at m_βw = 6 ± 2 for shale rock [61], and the direct use of the uniaxial compressive strength σ_cβw measured (and not obtained from the data regression) with uniaxial compressive tests can avoid carrying out triaxial compressive tests which are costly and challenging.

Future research developments will focus on the same sensitivity analyses of this study coupled with the interpretation of lab tests carried out on shale rocks with WPM and HBm. A further development will be the analysis of a real case of wellbore instability, by using reference mud weights based on the post-drilling experience coupled with the strength properties of shale rock obtained by lab tests.