Introduction

Geomechanical analysis plays an important role in the various stages of exploration and exploitation of hydrocarbon reservoirs (Hui and Deli 2005). In the exploratory stage, geomechanical models are very important in predicting pore pressure and estimating in situ stresses (Hui and Deli 2005). Likewise, in drilling stage, geomechanical properties have critical impact on bit selection, optimization of the drilling path, casing design, and determining the safe mud weight window (Zhang et al. 2010; Karakus and Perez 2014).

The main elements of a geomechanical model are rock mechanical properties, pore pressure, and in situ stress state (Plumb 1994). Rock mechanical tests are prerequisite for any geomechanical investigation (Plumb 1994). In recent years, a variety of rock mechanics’ tests developed and used worldwide in petroleum industry. However, because of limited availability of core samples for each project, carefully selected lab tests are of great importance. Unconfined compressive strength (UCS) is used to evaluate the wellbore stability, probability of sand production (Santarelli et al. 1989), and determination of the range of in situ stresses (Zoback et al. 2003). It is measured directly using uniaxial compressive test on core samples. Also, there are some empirical correlations to estimate this parameter (Chang et al. 2006). Determining the magnitudes and directions of stresses is the key component of a geomechanical model. Image logs are essential tools for determining in situ stresses direction (Ezati et al. 2014; Ezati et al. 2019b).

The vertical and azimuthal resolution of FMI image log is 0.5 mm which makes it able to detect very small and narrow events like tiny laminations, stylolites, solution seams, and fractures.

Cohesion (C) and angle of internal friction (Φ) are critical parameters to evaluate the wellbore stability and determine the optimal drilling mud weight (Zhang et al. 2010). These parameters are the intrinsic characteristics of rocks and could be estimated by several empirical relationships (e.g., Plumb 1994; Jaeger et al. 2009).

Pore pressure is required to evaluate in situ stresses and the safe window of drilling mud (Zoback 2010). The formation pore pressure can be measured directly (in separate points) using MDT (Modular Dynamics Tester), RFT (Repeat Formation Tester), and DST (Drill Stem Test) tools (Zoback 2010). In addition, the pore pressure is estimated using empirical relationships based on petrophysical logs in a continuous manner (Zhang 2011). Then, the estimated values can be calibrated using the directly measured values.

The stability of the wellbores has been studied in different literature, worldwide (Roegiers 2002; Awal et al. 2001; Chen et al. 2003; Al-Ajmi and Zimmerman 2006; Akhtarmanesh et al. 2013; Li et al. 2017., Zhou et al. 2018). However, challenges such as complete mud losses and large washouts still happen. Geomechanical studies in Iran were mainly performed for the Asmari Formation (Yaghoubi and Zeinali 2009; Fattahpour et al. 2012; Darvish et al. 2015; Ghafoori et al. 2018; Lashkaripour et al. 2018), which is the main Iranian oil reservoir (Motiei 1993). In addition, various geomechanical aspects of Dalan and Kangan formations have been studied. The stress state (Haghi et al. 2013; Aghajanpour et al. 2017), pore pressure (Azadpour et al. 2015), and rock mechanical properties (Mehrgini et al. 2016) were analyzed for these formations in one of the gas fields of Southern Iran.

Najibi et al. (2015) presented a number of relationships to estimate UCS and ES for the Asmari and Sarvak Formations commonly in the Dezful Embayment. In their study, VP (compressional wave velocity) and VS (shear wave velocity) in ambient conditions were used for the elastic modules calculation. Najibi et al. (2017), using 1D geomechanical modeling, estimated the principal stresses in one of the Dezful Embayment wells for the Bangestan reservoir. Rajabi et al. (2010) determined the horizontal stress directions in two wells in the Abadan Plain.

The Sarvak Formation (Albian to Turonian) contains more than 20% of the oil reserves of Iran (Motiei 1993). It is one of the main reservoirs in the Abadan Plain, SW Iran (Abdollahie-Fard et al. 2006). However, Sarvak reservoir has not been accurately evaluated from a geomechanical point of view, and in situ stress state is almost unknown in this region. In this study, a comprehensive data set from three wells was used to evaluate the rock mechanical parameters, in situ stresses, and wellbore stability in the Sarvak Formation in one of the oil fields in Abadan Plain. The final results can be considered for the future development plans and field management.

Geological setting

The studied field is an N-S–trending anticline, which is located in the Abadan Plain, SW Iran. The positions of the three studied wells are shown in Fig. 1a.

Fig. 1
figure 1

a The three studied wells (shown as A, B, and C) in Abadan Plain, SW Iran. The SHmax directions in the studied wells are indicated by arrows. b The Cretaceous stratigraphic column of the Abadan Plain

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The Abadan Plain shows three main structural trends: N-S, NE-SW, and NW-SE which are related to the deep-seated faults (Abdollahie-Fard et al. 2006). The associated structures to these trends are relevant to reactivation of basement-rooted normal faults which caused forced folding in overlaying sediments (Abdollahie-Fard et al. 2006).

The generation of the studied anticline associated with a complex horst system. Seismic data of this anticline show steep faulting in the core of the structure (Abdollahie-Fard et al. 2006). Arabian N-S–trending, basement-involved horst systems have been named differently, according to their geographical location, such as Kuwait High, Burgan High, Khurais-Burgan anticline, and Basrah High in Iraq (Abdollahie-Fard et al. 2006). Several oil fields are located along this regional trend in Saudi Arabia, Kuwait, and Iraq. In SW Iran, the N-S trend is identified in the Abadan Plain (Abdollahie-Fard et al. 2006).

The Sarvak Formation was deposited in upper Cretaceous during late Albian to early Turonian (Omidvar et al. 2014). In Abadan Plain, the Sarvak Formation is overlain by the Lafan shale with an unconformity, and its lower contact with Kazhdumi Formation is gradual. The Sarvak Formation comprises thick limestone and dolomitic limestone, with minor clayey and argillaceous intervals, reaching totally 600–700 m in thickness (Assadi et al. 2016, Fig. 1b).

In the Abadan Plain, natural fractures have little expansion in the Sarvak Formation (Ezati et al. 2018; Ezati et al. 2019a). Diagenetic studies of the Sarvak Formation in Abteymour field (one of the Southwestern Iranian fields) show that the reservoir quality of this reservoir is enhanced both by dissolution and dolomitization (Rahimpour-Bonab et al. 2012). Dissolution features in the Sarvak Formation are often caused by outcrops of this formation due to tectonic activities (Rahimpour-Bonab et al. 2013). Results of facies analysis in some SW Iranian fields indicate a homoclinal ramp-type carbonate platform for the Sarvak Formation (Mehrabi and Rahimpour-Bonab 2014). Also, Mehrabi et al. 2015 showed that the large-scale variations in facies and thicknesses of the Sarvak Formation are controlled by regional tectonic evolution and sea-level changes during the Upper Cretaceous.

Available data

The studied wells are situated in northern (well-A), central (well-B), and southern (well-C) parts of the studied field. The top and base depths of the Sarvak Formation in well-A, well-B, and well-C are 3708–4331, 3720–4371, and 3631–4269 m, respectively. A summary of the data resource of this study is presented in Table 1.

Table 1 Summary of data used in this study
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Fullset logs were used for petrophysical evaluation, pore pressure estimation, overburden stress measurement, and the elastic modulus calculation. FMI (Fullbore Formation Microimager), UBI (Ultrasonic Borehole Imager), and OBMI (Oil-Base Microimager) image logs were utilized to detect the wellbore wall failures. DSI data was used to extract VS for the elastic modulus measurement. MDT pressure points were considered to calibrate the pore pressure curves. Also, 27 plug samples were taken from the Sarvak reservoir for the geomechanical tests.

Methods

1D geomechanical modeling

A Mechanical Earth Model (MEM) is a numerical depiction of the geomechanical properties of a formation (Ali et al. 2003). In addition to property distribution (e.g., lithology, density, porosity), the MEM model includes pore pressures, stress state, and rock mechanical characteristics.

1D geomechanical model is constructed based on the well data. The well logging data such as shear and compression waves velocity, density, caliper, porosity, and gamma ray are used to present mechanical properties and stress state near wellbore (Ali et al. 2003). This model can be used to investigate the mechanical properties of rocks and in situ stresses near the wellbore (Ali et al. 2003).

Dynamic rock elastic properties

The dynamic Young’s modulus (Ed) and dynamic Poisson’s ratio (νd) were calculated using VP, VS, and density of rock (ρ) as follows (Fjaer et al. 2008):

$$ {\mathrm{E}}_{\mathrm{d}}=\uprho {\mathrm{V}}_{\mathrm{S}}^2\frac{3{\mathrm{V}}_{\mathrm{P}}^2-4{\mathrm{V}}_{\mathrm{S}}^2}{{\mathrm{V}}_{\mathrm{P}}^2-{\mathrm{V}}_{\mathrm{S}}^2} $$
(1)
$$ {\upnu}_{\mathrm{d}}=\frac{{\mathrm{V}}_{\mathrm{P}}^2-2{\mathrm{V}}_{\mathrm{S}}^2}{2\left({\mathrm{V}}_{\mathrm{P}}^2-{\mathrm{V}}_{\mathrm{S}}^2\right)} $$
(2)

Typically, high elastic values are obtained in the dynamic condition, so dynamic values should be converted to static values by performing mechanical tests (Zoback 2010).

The rock mechanical tests

In this study, 27 vertical plug samples were taken from the Sarvak reservoir cores of well-A (from the 3710.25 to 3950.57-m depths) for the rock mechanical tests. The diameters of the plugs are 1.5 in., and their lengths are twice the diameter. Sixteen plugs were selected for uniaxial compression test, and 11 plugs were used for multistage triaxial compression test.

The ASTM D3148–93 standard was followed for the uniaxial test. For the uniaxial test, the plugs were placed in a loading frame. Axial load was increased on the specimen at a constant and continuous rate, while axial and lateral deformations were monitored as a function of load. Figure 2a shows the stress-strain diagram for one of the plugs which was failed with a shear fracture. The ES and UCS were determined for all 16 plugs. The results of uniaxial test are shown in Table 2.

Fig. 2
figure 2

a S28 plug sample before and after uniaxial testing and its stress-strain diagram. b S11 plug sample before and after multistage triaxial test. The shear fractures of these tests are highlighted by the yellow dashed lines

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Table 2 The results of uniaxial test on the 16 plug samples
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The ASTM 2664-95a standard was considered for the triaxial test. For the multistage triaxial test, the plugs were placed in a triaxial loading chamber (cell), subjected to confining pressure. At least, three triaxial compression tests (with different confining pressure) were accomplished on the plugs. In Fig. 2b, triaxial test is shown along with its Mohr circles. A shear fracture has been developed in the plug specimen (Fig. 2b).

The C and Φ were determined using the triaxial tests. The results of the triaxial test are shown in Table 3.

Table 3 The results of multistage triaxial tests on the 11 plug samples
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Log data calibration with the rock mechanical tests

Throughout the cored section of well-A, ES and UCS were determined using uniaxial test on 16 plug samples from Sarvak reservoir. In the next step, the Ed was calculated using the VP, VS (from DSI data), and density logs. The relationship between the static and dynamic Young’s modulus is as follows (also, see Fig. 3a):

$$ {E}_s=0.0173{E_d}^2-0.9715{E}_d+17.274 $$
(3)
Fig. 3
figure 3

a Static Young’s modulus vs. dynamic Young’s modulus. b UCS vs. dynamic Young’s modulus. c Cohesion vs. effective porosity. d Angle of internal friction vs. effective porosity

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In order to estimate UCS continuously, UCS values were compared with some of the rock properties (such as porosity, ES, and Ed), and it was observed that UCS has the best correlation with the Ed (Fig. 3b):

$$ UCS=0.0635{E_d}^2-3.2185{E}_d+55.949 $$
(4)

The C and Φ parameters were determined for the 11 plug samples using the multistage triaxial test. It was observed that these parameters had the best correlation with the effective porosity (PHIE), which is derived from the petrophysical evaluation (Fig. 3c). Therefore, the relationships obtained in Fig. 3c and d were used to estimate the C and Φ values in a continuous manner.

$$ C=233.35{PHIE}^2-95.31 PHIE+14.523 $$
(5)
$$ \varPhi =672.49{PHIE}^2-339.81 PHIE+68.473 $$
(6)

In this study, the Ed was converted to ES values using the Eq. 3. UCS was estimated continuously, using the Eq. 4. The C and Φ curves were estimated using the Eqs. 5 and 6, respectively.

Pore pressure

The Eaton equation (Eaton 1975) for estimating pore pressure, using petrophysical logs, is as follows:

$$ {\mathrm{P}}_{\mathrm{P}\mathrm{g}}={\mathrm{S}}_{\mathrm{V}}-\left({\mathrm{S}}_{\mathrm{V}}-{\mathrm{P}}_{\mathrm{P}\mathrm{Hyd}}\right){\left(\frac{\mathrm{NCT}}{\mathrm{DT}}\right)}^3 $$
(7)

where PPg is pore pressure gradient, SV is vertical stress magnitude, PPHyd is hydrostatic pore pressure gradient, DT is slowness of compressional wave, and NCT is normal compaction trend of the sediments. Zhang (2011) modified Eaton’s equation for estimation of pore pressure as follows:

$$ {\mathrm{P}}_{\mathrm{P}\mathrm{g}}={\mathrm{S}}_{\mathrm{V}}-\left({\mathrm{S}}_{\mathrm{V}}-{\mathrm{P}}_{\mathrm{P}\mathrm{Hyd}}\right){\left(\frac{{\mathrm{DT}}_{\mathrm{m}}+\left({\mathrm{DT}}_{\mathrm{m}\mathrm{l}}-{\mathrm{DT}}_{\mathrm{m}}\right){\mathrm{e}}^{-\mathrm{cZ}}}{\mathrm{DT}}\right)}^3 $$
(8)

where DTm is P wave slowness in shale lithology with zero porosity, DTml is P wave slowness in drilling mud, c is an experimental constant, and Z is the depth.

In this study, it was observed that by considering c = 0.0005 and the coefficient power equal to 0.44, a good correlation between MDT points and estimated pore pressure curve will be obtained.

Estimation of the principal stresses

The magnitude of vertical stress (SV) is determined using the overburden weight (Jaeger and Cook 1979):

$$ {S}_V={\int}_0^z\uprho \left(\mathrm{z}\right)\mathrm{g}\kern0.33em \mathrm{dz}\approx \overline{\uprho}\mathrm{gz} $$
(9)

where z is depth, ρ(z) is rock density as a function of depth, g is Earth’s gravity acceleration, and ͞ρ is overburden average density.

The poroelastic equations are used to estimate the maximum (SHmax) and minimum (Shmin) horizontal stresses (Fjaer et al. 1992):

$$ {\mathrm{S}}_{hmin}=\frac{\nu }{\left(1-\nu \right)}\left({S}_V-\alpha PP\right)+\alpha PP+\frac{E_S}{\left(1-{\nu}^2\right)}\ {\upvarepsilon}_x+\frac{\nu \times {E}_S}{\left(1-{\nu}^2\right)}\ {\upvarepsilon}_y $$
(10)
$$ {\mathrm{S}}_{Hmax}=\frac{\nu }{\left(1-\nu \right)}\left({S}_V-\alpha PP\right)+\alpha PP+\frac{E_S}{\left(1-{\nu}^2\right)}\ {\upvarepsilon}_y+\frac{\nu \times {E}_S}{\left(1-{\nu}^2\right)}\ {\upvarepsilon}_x $$
(11)

where ν is Poisson’s ratio, SV is vertical stress, α is Biot’s coefficient (in the absence of data, is conventionally considered as 1), PP is pore pressure, and ES is static Young’s modulus. Moreover, the tectonic strain on x (ɛx) and y (ɛy) axes are determined by the following equations (Kidambi and Kumar 2016):

$$ {\upvarepsilon}_x=\frac{S_V\times \nu }{E_S}\left(\frac{1}{1-\nu }-1\right) $$
(12)
$$ {\upvarepsilon}_y=\frac{S_V\times \nu }{E_S}\left(1-\frac{\nu^2}{1-\nu}\right) $$
(13)

Wellbore stability

Generally, if the drilling fluid pressure is lower than the breakout pressure, shear failure occurs. On the other hand, if the pressure of the drilling mud reaches the minimum principal stress, mud loss happens. One of the main applications of geomechanical model is to determine the optimum drilling mud weight.

The distance between the pore pressure and minimum principal stress is called safe mud window, in which case we may have breakouts. Also, the distance between the breakout and breakdown pressures is called stable mud window (Nabaei et al. 2011). At best, the weight of the drilling mud should be determined in such a way to avoid the formation of borehole breakouts and mud loss (safe/stable mud window).

Wellbore wall stresses

The stresses affecting the wellbore wall can be determined using Kirsch equations (Zoback 2010). Based on these equations, effective radial (σθ), tangential (σθ), and axial (σz) stresses are calculated around a vertical well with radius R as follows:

$$ {\sigma}_r=\frac{1}{2}\left({S}_{Hmax}+{S}_{hmin}-2\mathrm{PP}\right)\left(1-\frac{{\mathrm{R}}^2}{{\mathrm{r}}^2}\right)+\frac{1}{2}\left({S}_{Hmax}-{S}_{hmin}\right)\times \left(1-\frac{4{\mathrm{R}}^2}{{\mathrm{r}}^2}+\frac{3{\mathrm{R}}^4}{{\mathrm{r}}^4}\right)\cos 2\uptheta +\frac{{\mathrm{PPR}}^2}{{\mathrm{r}}^2} $$
(14)
$$ {\sigma}_{\theta }=\frac{1}{2}\left({S}_{Hmax}+{S}_{hmin}-2\mathrm{PP}\right)\left(1+\frac{{\mathrm{R}}^2}{{\mathrm{r}}^2}\right)-\frac{1}{2}\left({S}_{Hmax}-{S}_{hmin}\right)\times \left(1+\frac{3{\mathrm{R}}^4}{{\mathrm{r}}^4}\right)\cos 2\uptheta -\frac{{\mathrm{PPR}}^2}{{\mathrm{r}}^2}-{\upsigma}^{\Delta \mathrm{T}} $$
(15)
$$ {\sigma}_z={S}_V-2\nu \left({S}_{Hmax}-{S}_{hmin}\right)\mathit{\cos}2\theta - PP-{\sigma}^{\Delta T} $$
(16)

where θ is the angle from the SHmax direction azimuth, r is desired distance from wellbore wall, ΔP is the difference between the drilling mud pressure and the pore pressure (PP), and σΔT is the thermal stress caused by the temperature difference between the drilling mud and the formation. In case of r = R (in the wellbore wall), the σθ and σr on the wellbore wall are:

$$ {\sigma}_{\theta }={S}_{hmin}+{S}_{Hmax}-2\left(\ {S}_{Hmax}-{S}_{hmin}\right)\mathit{\cos}2\theta -2 PP-\Delta P-{\sigma}^{\Delta T} $$
(17)
$$ {\sigma}_r=\Delta P $$
(18)

In this study, the stresses around the wellbore were calculated using Eqs. 16 to 18.

Mohr–Coulomb failure criterion

After calculating the stresses and mechanical properties of the rocks, stability of wells can be investigated using failure criteria. According to Mohr–Coulomb criterion, normal stress (Sn) and shear stress (τ) at the failure can be expressed by (Mohr 1900):

$$ \tau =C+\mu {S}_n\kern0.5em $$
(19)

where C and μ are cohesion and coefficient of internal friction, respectively. Also, the linearized form of the Mohr-Coulomb criterion is relating the maximum (S1) and minimum (S3) principal stresses (Al-Ajmi and Zimmerman 2006):

$$ {S}_1= UCS+q{S}_3 $$
(20)

where:

$$ q={\mathit{\tan}}^2+\left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.+\raisebox{1ex}{$\Phi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) $$
(21)

Regarding to effective stress concept, the Mohr–Coulomb criterion can be rewritten as:

$$ {S}_1=c+q{S}_3 $$
(22)

where

$$ c= UCS+ PP\left(q-1\right) $$
(23)

The breakout pressure and fracture pressure from Mohr–Coulomb criterion are shown in Tables 4 and 5. In these tables, Pw is actual drilling mud pressure, Pwb is drilling mud pressure which causes borehole breakouts, and Pwf is drilling mud pressure which causes induced fractures.

Table 4 Breakout Pw pressure according to Mohr-Coulomb failure criterion (Al-Ajmi and Zimmerman, 2006)
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Table 5 Fracture Pw pressure according to Mohr-Coulomb failure criterion (Al-Ajmi and Zimmerman, 2006)
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Mogi–Coulomb failure criterion

It was concluded by Mogi (1971) that the mean normal stress which opposes the creation of the fracture plane will be Sm,2, and therefore, a new failure criterion was suggested:

$$ {\tau}_{oct}=f\left({S}_{m,2}\right) $$
(24)

where Sm,2 and τoct are the effective mean stress and octahedral shear stress, respectively. The Sm,2 and octahedral shear stress are defined as follows:

$$ {S}_{m,2}=\frac{S_1+{S}_3}{2} $$
(25)
$$ {\tau}_{oct}=\frac{1}{3}\sqrt{{\left({S}_1-{S}_2\right)}^2+{\left({S}_2-{S}_3\right)}^2+{\left({S}_3-{S}_1\right)}^2} $$
(26)

Al-Ajmi and Zimmerman (2006) used Mogi-Coulomb criterion and suggested a function to relate parameters of this criterion to the Mohr–Coulomb strength parameters, C and Φ. Sm,2 and τoct are the effective mean stress and octahedral shear stress which are formulated as follows (Al-Ajmi and Zimmerman 2006):

$$ {\tau}_{oct}=a+b{S}_{m,2} $$
(27)

where:

$$ a=\frac{2\sqrt{2}}{3}\mathrm{C}\ \mathrm{cos}\Phi $$
(28)
$$ b=\frac{2\sqrt{2}}{3}\mathrm{C}\ \mathrm{sin}\Phi $$
(29)

There are determined invariants related to the stress tensor, which their values are independent of the chosen coordinate system. With regard to the first and second stress invariants, I1 and I2 will be:

$$ {I}_1={S}_1+{S}_2+{S}_3 $$
(30)
$$ {I}_2={S}_1{S}_2+{S}_2{S}_3+{S}_3{S}_1 $$
(31)

where S1, S2, and S3 are maximum, intermediate, and minimum principal stresses, respectively. Using the effective stress concept (Al-Ajmi and Zimmerman 2006):

$$ \sqrt{\left({I}_1^2-3{I}_2\right)}={a}^{\prime }+{b}^{\prime}\left({I}_1-{S}_2-2 PP\right) $$
(32)

where

$$ {a}^{\prime }=2\mathrm{C}\ \mathrm{cos}\Phi $$
(33)
$$ {b}^{\prime }=\mathrm{sin}\Phi $$
(34)

The borehole breakout and fracture pressures from Mogi–Coulomb criterion are shown in Tables 6 and 7, respectively.

Table 6 Breakout Pw pressure according to Mogi-Coulomb failure criterion (Al-Ajmi and Zimmerman, 2006)
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Table 7 Fracture Pw pressure according to Mogi-Coulomb failure criterion (Al-Ajmi and Zimmerman, 2006)
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Results

Geomechanical models

Figures 4, 5, and 6 show the 1D geomechanical models of the three studied wells. In these figures, the tracks from left to right are depth, caliper-bit size, lithology, dynamic and static Young’s modulus, dynamic Poisson’s ratio, UCS, cohesion, angle of internal friction and stresses/pressures.

Fig. 4
figure 4

The 1D geomechanical model of well-A. The SHmax is approximately equal to the SV, which generally shows a strike-slip normal stress state. Moreover, the pore pressure is close to hydrostatic pore pressure

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Fig. 5
figure 5

The 1D geomechanical model of well-B. The stress state is generally strike-slip, and pore pressure is higher than well-A

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Fig. 6
figure 6

The 1D geomechanical model of well-C. The stress state is generally strike-slip reverse, but horizontal stresses from 4053 to 4072 m are very low, which could be related to the shale lithology

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In Fig. 4 (well-A), generally, the SHmax is approximately equal to the SV, and these two stresses are larger than Shmin, indicating that the stress regime in this well for the Sarvak reservoir is strike-slip normal (SHmax = SV > Shmin). However, in some zones (e.g., Sar-4 to Sar-7), the SHmax is higher than the SV, which represents a strike-slip stress regime.

In the Sarvak Formation of well-B, SHmax is generally greater than SV, and Shmin is less than SV (Fig. 5). Therefore, the stress state is strike-slip (SHmax > SV > Shmin). Generally, the pore pressure of this well is larger than well-A. The pore pressure of this well is partly separated from the hydrostatic pore pressure (Fig. 5). In well-C, SV and Shmin are mostly equal, and both are smaller than SHmax (Fig 6), which indicates a strike-slip reverse stress state (SHmax > SV = Shmin).

Table 8 shows the statistical parameters of principal stresses and mechanical properties of the rocks in three studied wells (based on the 1D models). In all wells, the Ed, νd, C, and Φ have similar average values. Moreover, based on the average values of the stresses, the in situ stress regimes in well-A, well-B, and well-C are strike-slip normal, strike-slip, and strike-slip reverse, respectively. However, the average pore pressure in well-B and well-C is greater than well-A.

Table 8 Average values of geomechanical parameters in the studied wells
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Wellbore stability analysis

In the studied wells, the relationship among the wellbore wall stresses is σθ ≥ σz ≥ σr. Hence, the second equations in Tables 4 and 6 were used to calculate the breakout pressure. The usual condition of wellbore wall stresses relationship to create induced fractures is σr ≥ σz ≥ σθ (Al-Ajmi and Zimmerman 2006; Maleki et al. 2014). Therefore, the second equations in Tables 5 and 7 were used for the fractures pressure calculation in the studied wells.

In Figs. 7, 8 and 9, the tracks from left to right are showing depth, lithology, image log (static image), wellbore stability analysis using Mohr-Coulomb criterion, wellbore stability analysis using Mogi-Coulomb criterion, caliper-bit size and the recognized borehole breakouts based on the image logs. In wellbore stability analysis, the tracks with red, orange, green, and blue colors indicate the kick, breakouts, loss, and breakdown limits, respectively. Also, the actual mud pressure (PW) is shown with a purple line.

Fig. 7
figure 7

The wellbore stability analysis for well-A. At most depths, the mud pressure line (PW) situated in the breakout zone (orange color), but Mogi-Coulomb criterion breakout pressure is more acceptable regarding to the recognized breakouts

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Fig. 8
figure 8

The wellbore stability analysis for well-B. Like well-A, breakout pressure is overestimated by Mohr-Coulomb criterion, but major breakout part in Sar_Intra well-recognized by the both failure criteria

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Fig. 9
figure 9

The wellbore stability analysis for well-C. In the depth 4053 to 4072 m, the upper limit of mud weight is very low

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In all three wells, the pressure of the selected drilling mud (PW) is close to the pore pressure (kick pressure). Table 9 shows the drilling mud weight and the volume of mud loss in the Sarvak Formation of the studied wells. Based on Table 9, mud loss is very low and can be considered as seepage for the wells (Figs. 7, 8 and 9).

Table 9 Drilling mud weight and mud loss volume in the studied wells
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The Mohr-Coulomb breakout pressures are higher than Mogi–Coulomb pressures (Figs. 7, 8 and 9). However, considering the borehole failures (caliper-bit size and image logs), it seems that Mogi–Coulomb breakout pressure is more compatible with the wellbore failures. In well-B, from 4175 to 4200-m depth, there is a major breakout zone which is well-recognized by both failure criteria (Fig. 8). In well-C, from 4053 to 4072-m depth, there is a breakout interval; and also the minimum principal stress is very low in this zone (Fig. 9).

The mud weight windows were determined for the three wells using geomechanical models and failure criteria (Table 10). According to the analysis, in all the wells, the Mohr-Coulomb breakout pressures are higher than Mogi-Coulomb breakout pressure (Table 10).

Table 10 Mud weight windows in the studied wells
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The orientations of the horizontal stresses in the studied wells are shown in Table 11. The SHmax orientation in all the three wells is NE-SW. In this study, Shmin and SHmax directions were determined using a large number of breakout observations in the image logs. The formations which used to identify horizontal stresses direction are mentioned in Table 11. In these formations, the horizontal stresses direction is uniform. Based on the quality ranking system for stress indicators (Zoback 2010), the quality of these directions are considered as “A.”

Table 11 The horizontal stress orientations in the studied wells
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The direction of horizontal stresses has a great influence on the stability of the deviated wells. In directional wells, the direction where the stress difference on the wellbore wall is smaller would be more stable.

The effect of horizontal stress orientation on the stability of wells was evaluated for the three wells (Fig. 10). Near the well-A, due to the strike-slip normal stress regime, higher mud weight is needed to prevent borehole breakouts in vertical and NE-SW–directed wells. In this region, the most stable drilling is to the Shmin direction, i.e., NW-SE (Fig. 10). Near the well-B, as a strike-slip stress state, the most stable directions are toward N10 °E and N80 °W (Fig. 10). Near the well-C, as a strike-slip reverse stress state, the most stable direction is toward SHmax, i.e., NE-SW (Fig. 10).

Fig. 10
figure 10

Analysis of wellbore stability in different drilling directions. Note the minimum mud weight required to prevent borehole breakouts in different drilling directions

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Discussion

Apparently, porosity has a great influence on Young’s modulus and UCS. Same as ES and UCS, porosity also has a significant effect on the C and Φ. Also, the reduction of the angle of internal friction with porosity increase for sandstone samples has also been reported (Chang et al. 2006). The mean of ES is equal to 7.01 GPa, while the mean of Ed is 40.27 GPa. Therefore, there is a large difference between the values of the static and dynamic Young’s modulus in the Sarvak reservoir.

The pore pressure gradient of Sar-1 is relatively higher than the other zones in well-A. Sar-1 situated between two shaly layers (i.e., Lafan formation and Sar-2), in this condition, the vertical displacement of the fluids in Sar-1 is restricted and pore pressure increased. Although Sar-3 and Sar-8 which are considered as good reservoir zones (due to high porosity and low water saturation), have a relatively low pore pressure which is close to the hydrostatic pore pressure (Fig. 4).

Generally, in all of the three wells, the pore pressure of Sarvak Formation is close to the hydrostatic pore pressure. Therefore, in the future, production will not be possible without improved/enhanced oil recovery technologies. Also, if the pore pressure is reduced below the hydrostatic pressure, a lighter mud weight is needed for Sarvak reservoir, and a new casing may be needed for this reservoir.

The difference between pore pressure and minimum principal stress of the studied wells is often relatively high. Therefore, the probability of induced fracturing in the Sarvak Formation is very low; this is in line with the fact that no induced fractures could be detected in the image logs. Also, according to Figs. 7, 8 and 9, it is clear that the difference between the mud pressures and the estimated minimum stresses in all studied wells is high.

The Mohr-Coulomb mud weight results are more conservative than Mogi–Coulomb criterion. For instance, in the lower part of the Sar_intra zone of well-A (Fig. 7), the wellbore wall is in stable condition regarding to Mogi–Coulomb criterion, and caliper bit-size track confirms this, but Mohr-Coulomb criterion indicates that such breakouts should happen in this interval (Fig. 7).

In the well-C, Mohr-Coulomb breakout pressure gradient is higher than the fracture pressure gradient, because the breakout pressure is overestimated at 3950 to 4000-m and 4053 to 4072-m depth intervals (Fig. 9), and the fracture pressure is very low. Hence, based on the Mohr-Coulomb criterion, the mud pressure gradient that avoids breakouts at 3950 to 4000 m causes fracturing at the 4053 to 4072-m depth. Also, in this interval, the safe mud weight window is very narrow, due to the presence of a shale layer. Minimum horizontal stress and tensile strength of this zone are low; so there is a risk of total mud loss in this interval.

Conclusion

In this study, 1D geomechanical models were constructed for three wells in the north, center, and south parts of an oil field in the Abadan Plain. In the north well, ES and UCS were determined using uniaxial test. Also, C and Φ were measured using triaxial tests. Furthermore, several relationships were presented for converting Ed to ES and estimating UCS, C, and Φ based on the geomechanical tests and well logs integration.

The pore pressure in the Sarvak reservoir of the studied oil field is low and close to the hydrostatic state. Therefore, for the future production, enhanced oil recovery programs should be considered. Also, with decreasing pore pressure, the Sarvak Formation may need a different mud weight for drilling and subsequently a new casing.

According to the 1D geomechanical models, the in situ stress regimes in the north, center, and south wells are strike-slip normal, strike-slip, and strike-slip reverse, respectively. Based on the picked borehole breakouts, the SHmax orientations in the three wells are toward NE-SW. In all wells, the failure was of breakout type, and induced fracture type was not observed. By comparing the wellbore stability models with the failures detected using image logs and caliper bit-size, it was found that the Mogi-Coulomb criterion performs a better stability model than Mohr-Coulomb criterion.

According to the in situ stress magnitudes and orientation of horizontal stresses, it is concluded that in the north, center, and south wells, the more stable drilling directions are NW-SE, N10 °E and N80°W, and NE-SW, respectively.

This paper is a part of an ongoing research, which includes the development of the 3-D geomechanical model; however, in current manuscript, the results of the application of 1D geomechanical for wellbore stability purposes are addressed. The assumptions for developing the 1D geomechanical model were thoroughly outlined in the paper.