Introduction

A comprehensive understanding of the stress state and fluid-rock interaction conditions in porous media prior to various scenarios such as drilling, stimulation, and production of hydrocarbon reservoirs plays a very important role in predicting their safe and economical operation. Subsurface stresses and their interaction with reservoir pore pressure affect a variety of operational aspects such as wellbore stability, wellbore integrity, caprock integrity, fault reactivation, early water-cut, sand production, pore collapse, reservoir compression, surface subsidence, and water/gas flooding during an oil/gas field life as well as stimulation techniques such as hydraulic fracture and acid fracturing (Zoback et al. 1985; Koutsabeloulis et al. 2009; Herwanger 2011; Fischer and Henk 2013; Sanei et al. 2017; Duran et al. 2020; Sanei et al. 2021; Sanei et al. 2022).

In addition, as a result of improved living standards and the advancement of technology, the demand for energy in the world has increased. This has also led to many challenges related to the discovery of new fields (Osinowo et al. 2018). Hydrocarbon exploration is moving toward reservoirs with more complex geologies, thus systematic operations to increase and optimize oil production to meet energy demand are essential. Therefore, the development of an integrated approach that requires various sciences such as geology, geophysics, petrophysics, geostatistics, geomechanics, and reservoir engineering to describe the reservoirs and their properties is essential (Ma 2011; Yu et al. 2011; Osinowo et al. 2018; Ashraf et al. 2019; Jiang et al. 2021; Anees et al. 2022a; Mangi et al. 2022; Safaei et al. 2022).

The advances in computational and computer technology have made it possible for modern reservoir models to incorporate more accurate spatial distributions of reservoir properties. Subsurface reservoir characterization generally includes well log data augmented with seismic data to establish the geological model of the reservoir (Patrick et al. 2002). Reservoir characterization is a technique for quantifying the rock and fluid properties of a reservoir. This technique requires all the information needed to describe a reservoir (Chopra and Michelena 2011; Yu et al. 2011; Ashraf et al. 2020b; Anees et al. 2022b). Reservoir modeling is an essential step in field development because it provides a place to integrate all available data and geologic concepts (Branets et al. 2009).

The 3D static geological modeling of the reservoir contains a 3D structure of its volume (Chambers et al. 2000; Thanh et al. 2019; Thanh et al. 2022). A three-dimensional static model of the reservoir is a definition of the structure, stratigraphy, and rock properties at a special time (Thanh et al. 2020; Ashraf et al. 2021). One of the key challenges in reservoir modeling is the appropriate representation of reservoir geometry, including the structural framework and detailed stratigraphic layers (Novak et al. 2014). Various data such as well logging, cores, well testing, and production data are utilized to build the static reservoir model (Davis 2002). The built reservoir model comprises two stages of construction modeling and petrophysical modeling, in which the reservoir properties are distributed by the geostatistical methods in the structure of the reservoir (Flugel 2004). The geostatistical methods such as deterministic methods (such as Kriging) and stochastic (such as Sequential Gaussian Simulation) methods are generally applied in order to construct the reservoir model (Thanh et al. 2020; Thanh and Sugai 2021). The static reservoir geomechanical modeling consists of a reservoir model and a geomechanics model. As expressed, the static reservoir model can be constructed using geostatistical methods. In addition, the geomechanical properties model can be built similar to the static reservoir model by applying the geostatistical methods. Numerical modeling, especially the reservoir geomechanical model is a reliable tool that considers a variety of data such as geological, geophysical, and engineering data to be capable to analyze various scenarios throughout the life of the reservoir (Fischer et al. 2013; Henk 2009; Khaksar et al. 2012; Tenthorey et al. 2013; Guerra et al. 2019).

In conventional reservoir simulation, the only geomechanical parameter involved is rock compressibility. This parameter is not enough to indicate different rock behaviors such as stress path dependency, pore collapse, dilatancy (Gutierrex, M., and R.W. Lewis 1998; Pereira et al. 2008). To fundamentally represent these behaviors, especially in complex, heterogeneous, and unconventional reservoirs, a realistic 3D model of reservoir geomechanics is essential, which is very difficult to construct. When the numerical model of reservoir geomechanics has been built, it can be used as a prior tool for various future operational scenarios such as wellbore stability, optimal trajectory of drilling, caprock integrity, fault reactivation, pore collapse, surface subsidence, water/gas flooding in order to improve the safety and economics of the projects (Bachmann et al. 1987; Khaksar et al. 2012; Sanei et al. 2021, 2022).

In this study, considering the importance of geomechanics in reservoirs, the numerical workflow of static reservoir geomechanical modeling is presented based on the data of the Volve oil field, which is a complex geological reservoir. This workflow consists of 1D geomechanical and reservoir modeling which are computed using well logging data. The 3D static reservoir geomechanical modeling comprises the 3D geomechanical properties, e.g., pore pressure, elastic properties, vertical stress, horizontal stresses, and 3D reservoir properties, e.g., porosity, permeability, and hydrocarbon saturation, which are constructed using geostatistical methods such as Kriging and Sequential Gaussian Simulation method. This three-dimensional static reservoir geomechanical modeling provides a fundamental spatial distribution for reservoir and geomechanics properties. In addition, this model will make an opportunity to generate a hydromechanical coupled dynamic numerical model which is a prior tool for analyzing various scenarios e.g., drilling, injection, production, etc. by considering the geomechanics effects in the future.

The structure of the manuscript is disposed of as follows. Firstly, the 1D geomechanical modeling is calculated at well locations by using log data. Secondly, the 1D reservoir modeling is computed at well locations by using well log data. Thirdly, the 3D geomechanical properties are distributed using geostatistical methods. Fourthly, the 3D reservoir properties are presented using geostatistical algorithms. Fifthly, the 3D static reservoir geomechanical modeling is built comprised of the 3D geomechanical and reservoir modeling which provides an opportunity for future dynamic scenarios. Finally, a summary and conclusions are provided.

Case study: Volve oil field, Norway

Description of the study area

This study is done based on the data from the Volve oil field, located in the central part of the North Sea, as shown in Fig. 1. The stratigraphic chart of the Volve oil field is given in Table 1. This oil field was discovered in 1993, and the plan for development was approved in 2005 (Sen and Ganguli 2019). Field production was launched in early 2008, achieving 56,000 bbl/day of peak oil rate. The main drainage strategy was pressure maintenance by water injection, with production wells placed high on the structure and water injectors at the flanks. The Volve field is described as a fault block structure with an initial estimate of 173 million bbl of oil in place (Equinor 2021). The reservoir is a small dome-shaped structure and is believed to be formed due to the collapse of adjacent salt ridges during the Middle Jurassic age (Equinor 2021; Szydlik et al. 2006). Oil was produced from the sandstone of the Middle Jurassic age in the Hugin formation at an average depth of \(2700 to 3100 m\) true vertical depth (TVD) below sea level. The recovery rate of Volve field was about \(54\%\) and it was shut down in 2016 after eight years of field operation.

Fig. 1
figure 1

The geographic location of the Volve field (Wang et al. 2021)

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Table 1 The stratigraphic chart of the Volve oil field (Sen and Ganguli, 2019)
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Equinor company together with other operators of Volve filed have released all datasets. Released data consist of subsurface geology, geophysics, borehole logs, and various drilling and production reports (Equinor 2021). For this study, nine wells are selected that are included F-1A, F-1B, F-4, F-5, F-11T2, F-14, F-15A, F-15B, and F-15C.

Geological setting

The petrophysical and geomechanical information applied in this study is related to nine wells in the Volve oil field. The reservoir part of this field is related to Hugin and Heather formations and the caprock of this field is related to the Draupne formation. This field is composed of formations (zones) such as Hordaland, Grid, Balder, Sele, Lista, Ty, Ekofisk, Tor, Hod, Blodoks, Hidra, Rodby, Asgard, Draupne, Heather, Hugin, Sleipner, Mime, Skade, Skagerrak, Utsira, Smith Bank, Tryggvason, and Heimdal. Geological and petrophysical studies show that the rocks of the field for each formation are mainly sandstone, limestone, siltstone, claystone, calcite, marl, tuff, and coal as the lithology of each formation is shown in Fig. 2. The Volve oil field is geologically complex and requires precise knowledge of geology, tectonic structures, geophysical-seismic, and petrophysical section. One of these complexities can be mentioned in faults in the field (3D shape of the reservoir is shown in Fig. 3). Therefore, accurate estimation of petrophysical and geomechanical parameters with this volume of anisotropy is a very complex task and requires appropriate experimental knowledge in the field of geology, petrophysics, and geomechanics.

Fig. 2
figure 2

Stratigraphic column of the studied formations in the Volve oil field, Norway

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Fig. 3
figure 3

Three-dimensional shape of Volve oil field, Norway

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Methodology

In this study, all available data in the Volve oil field are collected, including geological, geophysical, petrophysical, and geomechanical data. The workflow adopted for the static reservoir geomechanical model is indicated in Fig. 4. The structural interpretation of this study is performed using seismic data. The workflow starts with 1D modeling of each well to determine petrophysical and geomechanical properties. 1D petrophysical parameters are evaluated using well log data from nine wells. 1D geomechanical properties are calculated using well log data from nine wells. The main focus of this article is on petrophysical and geomechanical modeling for porosity, permeability, hydrocarbon saturation, pore pressure, elastic properties, vertical stress, and horizontal stresses. Core data have been used to evaluate the accuracy of petrophysical and geomechanical well data.

Fig. 4
figure 4

Flowchart adopted for the static reservoir geomechanical model

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Static modeling consists of two main parts. The first part of structural modeling is the process of making a skeleton or a 3D reservoir geomechanical network (Avseth et al. 2005; Cannon 2018). The second part is petrophysical and geomechanical data analysis and static reservoir geomechanical modeling. The process of petrophysical and geomechanical data analysis consists of (1) upscaling the well log, (2) analyzing the data, (3) geocell modeling, and (4) petrophysical modeling. The 3D reservoir geomechanical model is built based on a structural/geological model which is generally done by using data of seismic and borehole information. Previous studies expressed that the upscaling of geomechanical and petrophysical properties from various 1D models to the 3D models can be done successfully through geostatistical methods such as Gaussian and Kriging interpolation schemes (Kim et al. 2002; Amanipoor et al. 2019; Adewunmi et al. 2019; Zain et al. 2020). Then, the upscaling of geomechanical properties such as pore pressure, horizontal stresses, vertical stresses, Young modulus, Poisson ratio, and unconfined compressive strength (UCS) from 1D models to 3D models are done using the geostatistical methods. Moreover, the upscaling of reservoir characterizations such as porosity, permeability, and hydrocarbon saturation are done from 1D models to 3D models using geostatistical methods.

The 3D static reservoir geomechanical model which is composed of the combination of the reservoir and geomechanical model can be used to make a coupled geomechanics and fluid flow model (dynamic reservoir). Generally, Geomechanics and dynamic reservoir model are coupled iteratively through pore pressure and effective stresses that cause rock deformation, which affects the porosity and permeability of the reservoir. Then, the coupled dynamic reservoir geomechanics can be applied to future studies, such as production, injection, and history matching. The workflow adopted for the future dynamics reservoir geomechanical scenarios is shown in Fig. 5.

Fig. 5
figure 5

Flowchart adopted for the future dynamics reservoir geomechanical scenarios

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The 1D reservoir geomechanics modeling

The one-dimensional 1D modeling of reservoir geomechanics comprises the 1D geomechanical modeling and the 1D petrophysical (reservoir) properties modeling that are presented as follows:

The 1D geomechanical modeling

1D geomechanical modeling is a continuous numerical representation of geomechanical properties, pore pressure, and the in situ stresses along a borehole (Plumb et al. 2000). To represent the geomechanical modeling, well log data are used to estimate various material properties, pore pressure, and in situ stresses along a wellbore. The quality of a 1D geomechanical model depends on the quality and availability of well log data and laboratory test data.

Estimation of vertical stress

Vertical stress is the pressure applied by the overlaying lithostatic column, at the depth (z), and can be estimated by the following equation (Plumb et al. 1991):

$$S_{v} = \int\limits_{0}^{z} {\rho \left( z \right)~g~dz}$$
(1)

where, \(S_{{\text{v}}}\) is the vertical stress, ρ(\(z\)) is the bulk density, and g is the gravitational acceleration.

The vertical stress is estimated from density logs and the results for well F1A are shown in Fig. 6.

Fig. 6
figure 6

The graph includes a comparison of measured pore pressure \({p}_{mes}\) with the results of the Eaton model \({p}_{p}\), a comparison of measured minimum horizontal stress \(FIT and LOT\) with the results of the Blanton model \({\sigma }_{h}\), hydrostatic pressure \({p}_{hyd}\), maximum horizontal stress \({\sigma }_{H}\), and vertical stress \({\sigma }_{V}\) for well F1A

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Estimation of minimum horizontal stress

Minimum horizontal stress can be estimated from well-log data using different empirical equations such as Eaton (1968), Blanton and Olson (1999), poroelastic equations (Fjar et al. 2008), etc. In this study, the minimum horizontal stress \(S_{h}\) is calculated using the Blanton’s relationship (Blanton and Olson 1999), as follows:

$$S_{h} = k*\left( {S_{{\text{v}}} - p_{{\text{p}}} } \right) + p_{{\text{p}}} + S_{{{\text{tec}}}}$$
(2)

where \(S_{v}\) is the vertical stress, \(p_{p}\) is the pore pressure, and \(k\) is the coefficient of the earth’s stress. The tectonic stress \(S_{tec}\) is represented as (Daines 1982):

$$S_{{{\text{tec}}}} = F - \left[ {\left( {S_{{\text{v}}} - p_{{\text{p}}} } \right)*\left( {\frac{{\upnu }}{{1 - {\upnu }}}} \right) + p_{{\text{p}}} } \right]$$
(3)

where F is the fracture reopening and \(\nu\) is the Poisson’s ratio.

The minimum horizontal stress is estimated from sonic logs using Blanton’s equation The results of the well F1A show that the Blanton model can estimate minimum horizontal stress accurately as illustrated in Fig. 6.

Maximum horizontal stress

To calculate the maximum horizontal stress, the poroelastic theory can be used. The poroelastic relationships are presented as follows (Blanton and Olson 1999):

$$S_{H} = \frac{{\upnu }}{{1 - {\upnu }}}\sigma_{{\text{v}}} - \frac{{\upnu }}{{1 - {\upnu }}}\alpha p_{{\text{p}}} + \alpha p_{{\text{p}}} + \frac{{\text{E}}}{{1 - {\upnu }^{2} }}\varepsilon_{{\text{x}}} + \frac{{{\upnu } {\text{E}}}}{{1 - {\upnu }^{2} }}\varepsilon_{{\text{y}}}$$
(4)

where \(E\) is the Young modulus, \(\nu\) is the Poisson’s ratio, \(\alpha\) is the Biot’s coefficient, \(\varepsilon_{{\text{x}}}\) and \(\varepsilon_{{\text{y}}}\) are the strain in the x and y direction, respectively.

The maximum horizontal stress is estimated from poroelastic theory and the results for well F1A are shown in Fig. 6.

Estimation of pore pressure

Pore pressure can be estimated from well log data using different empirical equations such as Eaton (1975), Gardner et al. (1974), Bowers (1995), etc. In this study, pore pressure is estimated from resistivity and sonic logs using the well-known Eaton’s equation (Eaton 1975). The equation of Eaton is:

$$p_{{\text{p}}} = S_{{\text{v}}} - \left( {S_{{\text{v}}} - p_{{{\text{hyd}}}} } \right) \times \left( {\frac{{DT_{{\text{n}}} }}{DT}} \right)^{a}$$
(5)

where \(S_{v}\) is the overburden stress and DT is the measured sonic transit time available from compressional sonic well logs. The \(DT_{n}\) is the sonic transit time on the normal line which can be estimated using Zhang’s equation (Zhang 2011), as follows:

$$DT_{{\text{n}}} = DT_{{{\text{sh}}}} + \left( {DT_{{{\text{ml}}}} - DT_{{{\text{sh}}}} } \right)e^{ - c z}$$
(6)

where \(DT_{sh}\) is the compressional transit time in the shale matrix, \({DT}_{ml}\) is the mudline transit time, \(c\) is the constant, and \(z\) is the depth. The hydrostatic pressure \(p_{hyd}\) is:

$$p_{{{\text{hyd}}}} = \int\limits_{0}^{z} {\rho _{w} \left( z \right)~g~dz}$$
(7)

where \(\rho_{w}\) is the density of water. In this study, the average gradient of hydrostatic pressure is \(1.03{\text{ g}}/{\text{cm}}^{3}\).

The pore pressure is estimated from sonic logs using Eaton’s equation The results for well F1A indicate that the Eaton model can estimate pore pressure precisely as shown in Fig. 6.

In Fig. 6, laboratory core data are used to evaluate the accuracy of geomechanical properties. It shows a good agreement between measured and numerical results of pore pressure and minimum horizontal stress. The same procedure described for building the 1D model of pore pressure, minimum horizontal stress, maximum horizontal stress, and vertical stress for well F1A as presented in Fig. 6 is performed for other wells.

Elastic properties

The elastic parameters of the rock can be obtained using the density logs and sonic wave velocity (dynamic method). Given the assumption of a homogeneous and isotropic formation, the dynamic Young modulus \({\text{E}}_{{\text{d}}}\) \(\left[ {{\text{Gpa}}} \right]\) and dynamic Poisson ratio \({\upnu }_{{\text{d}}}\) [%] are computed as (Fjær et al. 2008):

$${\text{E}}_{{\text{d}}} = {\rho V}_{{\text{s}}}^{2} \frac{{3{\text{V}}_{{\text{p}}}^{2} - 4{\text{V}}_{{\text{s}}}^{2} }}{{{\text{V}}_{{\text{p}}}^{2} - {\text{V}}_{{\text{s}}}^{2} }}$$
(8)
$${\upnu }_{{\text{d}}} = \frac{{{\text{V}}_{{\text{p}}}^{2} - 2{\text{V}}_{{\text{s}}}^{2} }}{{2\left( {{\text{V}}_{{\text{p}}}^{2} - {\text{V}}_{{\text{s}}}^{2} } \right)}}$$
(9)

where \({\uprho }\) is the bulk density of the formation [g/cm3], \({\text{V}}_{{\text{p}}}\) is the compressional wave velocity \(\left[ {{\text{km}}/{\text{s}}} \right]\), and \({\text{V}}_{{\text{s}}}\) is the shear wave velocity \([{\text{km}}\)\(/\mathrm{s}]\).

Static elastic properties

To estimate the static parameters from dynamic parameters, several relationships have been proposed. In this study, the static Young modulus \({\mathrm{E}}_{\mathrm{s}}\) \([\mathrm{Gpa}]\) is estimated using the John Fuller model (Schlumberger 2018) as expressed by the following equation:

$${\text{E}}_{{\text{s}}} = 0.032 \times \left( {{\text{E}}_{{\text{d}}} } \right)^{1.632}$$
(10)

The static Young modulus is estimated from sonic logs using the John Fuller model. The results to well F1A show that the John Fuller model can estimate the static Young modulus accurately as illustrated in Fig. 7.

Fig. 7
figure 7

The comparison of measured (top-left) static Young modulus \(E\) with the results of John Fuller model, (top-right) Poisson ratio \(\nu\) with the results of the position ratio with the multiplier, \(\mathrm{ml}=1\), and (bottom) unconfined compressive strength \(UCS\) with the results of the Dick Plumb model for well F1A

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The static Poisson ratio \({\upnu }_{{\text{s}}}\) is computed from the dynamic Poisson ratio \({\upnu }_{{\text{d}}}\) by the following relationship:

$${\upnu }_{{\text{s}}} = {\text{ml}} \times {\upnu }_{{\text{d}}}$$
(11)

where \({\text{ml}}\) is the multiplier. Some researchers, including Archer and Rasouli (2012) expressed that the \(\mathrm{ml}=1\) means the static Poisson ratio is equal to the dynamic Poisson ratio. However, Afsari et al. (2009) presented that the \(\mathrm{ml}=0.7\). In this study, the results for estimating the static Poisson ratio show that the best multiplier is \(\mathrm{ml}=1\), as presented in Fig. 7.

Unconfined compressive strength

The unconfined compressive strength (UCS) can be obtained using compressional wave velocity, density, and porosity logs. In this study, the unconfined compressive strength \(\mathrm{UCS}\) \([\mathrm{Gpa}]\) is estimated using the Dick Plumb model (Schlumberger 2018) as expressed by the following equation:

$${\text{UCS}} = 4.242 \times {\text{E}}_{{\text{s}}}$$
(12)

The unconfined compressive strength is estimated from the static Young modulus using the Dick Plumb model. The results to well F1A show that the Dick Plumb model can estimate the unconfined compressive strength accurately as shown in Fig. 7.

In Fig. 7, laboratory core data are used to evaluate the accuracy of geomechanical properties. It shows a good agreement between measured and numerical results of Young modulus, Poisson ratio, and unconfined compressive strength. The same procedure described for building the 1D model of static Young modulus, Poisson ratio, and UCS for well F1A as shown in Fig. 7 is performed for other wells.

The 1D reservoir properties modeling

Well-log data are used to calculate reservoir (petrophysical) properties such as porosity, permeability, and hydrocarbon saturation. The 1D reservoir properties were computed and verified by Equinor (Equinor 2021) and in this study, the same procedure is done in order to be able to build the 3D static reservoir modeling. The process that the Equinor was done to make 1D reservoir properties are as follows:

Porosity

Total porosity \(\phi\) is derived from the density log which is calibrated to overburden corrected core porosity (Tiab and Donaldson 2015). The Neutron log, NPHI, has been used to correct for varying mud filtrate invasion.

$$\phi = \phi_{D} + A \times \left( {{\text{NPHI}} - \phi_{D} } \right) + B$$
(13)

where

$$\phi_{D} = \frac{{\rho_{ma} - \rho_{b} }}{{\rho_{ma} - \rho_{fl} }}$$
(14)

where \(\rho_{ma}\) is the matrix density [g/cm3], \(\rho_{b}\) is the measured bulk density (RHOB) [g/cm3], \(\rho_{fl}\) is the pore fluid density [g/cm3], and \(A\) and \(B\) are regression coefficients.

The porosity for well F1A is obtained from Eq. 13 and the results are illustrated in Fig. 8.

Fig. 8
figure 8

The graph includes: (top- left) porosity \(\phi\), (top-right) permeability \({\text{k}}\), and (bottom) hydrocarbon saturation \({\text{S}}_{h}\)

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Permeability

The permeability \(\mathrm{k}\) is derived from the following equation based on multivariable regression analysis between log porosity and shale volume against overburden corrected core permeability.

$${\text{k}} = 10^{{\left( {2 + 8*\phi - 9 \times {\text{VSH}}} \right)}}$$
(15)

where \({\text{VSH}}\) is the shale volume.

The permeability for well F1A is obtained from Eq. 15 and the results are indicated in Fig. 8.

Hydrocarbon saturation

Hydrocarbon saturation can be calculated from water saturation. The water saturation \({\text{S}}_{w}\) is obtained using Archie equation (Tiab and Donaldson 2015) as follows:

$${\text{S}}_{{\text{w}}} = \left( {\frac{{a \times R_{{\text{w}}} }}{{\phi^{m} *R_{{\text{t}}} }}} \right)^{\frac{1}{n}}$$
(16)

where \(a\) is the Archie factor, \(R_{w}\) is the resistivity of formation water \(\left[ {ohmm} \right]\), \(m\) is the cementation exponent, \(R_{t}\) is the true resistivity \(\left[ {ohmm} \right]\), and \(n\) is the saturation exponent.

The hydrocarbon saturation \({\text{S}}_{h}\) can be obtained as follows (Tiab and Donaldson 2015):

$${\text{S}}_{h} = 1 - {\text{S}}_{w}$$
(17)

The hydrocarbon saturation for well F1A is obtained from Eq. 17 and the results are shown in Fig. 8.

The same procedure described for building the 1D model of porosity, permeability, and hydrocarbon saturation for well F1A as illustrated in Fig. 8 is performed for other wells.

The 3D model generation

The three-dimensional modeling of reservoir geomechanics comprises the 3D geomechanical model and 3D static reservoir model. Each model comprises the reservoir geology, structure, stratigraphic, reservoir sublayers, and faults in 3D structural and properties models.

The 3D geomechanical model

The 3D static geomechanical model is built up using the structural/geological model with high resolution. The 3D geomechanical properties are distributed by geostatistical methods based on the 1D geomechanical models from the log data. The geostatistical methods such as the Kriging and Gaussian (Sequential Gaussian Simulation (SGS)) method are used to populate the 3D geomechanical model.

The 3D reservoir model

The 3D static reservoir model is built up using the structural/geological model with a high resolution. The reservoir volume is divided into a 3D mesh of cells and for each cell, rock properties are assigned. The 3D reservoir properties are distributed using the upscaled 1D (reservoir) petrophysical models from the log data. The geostatistical methods such as the Kriging and Gaussian (Sequential Gaussian Simulation (SGS)) method are used to populate the 3D reservoir model.

Results

The 3D geomechanical properties modeling

The geomechanical properties such as pore pressure, horizontal stresses, vertical stress, Young’s modulus, Poisson’s ratio, and unconfined compressive strength are upscaled and populated by the Kriging and Gaussian (Sequential Gaussian Simulation (SGS)) methods. The geomechanical properties show distinct vertical variations, but only slight lateral changes because of changes in lithology. The spatial distribution of each property is obtained using the upscaled methods and the results of them are presented as follows:

The spatial distribution of pore pressure using both Kriging and SGS methods are presented in Fig. 9. The pore pressure ranges from 19.46 Mpa to 41.62 MPa. The average value of the pore pressure is 33.63 MPa.

Fig. 9
figure 9

Static pore pressure distribution through upscaled computed 1D well log properties: (top) Kriging (Krig) method and (bottom) SGS method

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The spatial distribution of minimum horizontal stress using both Kriging and SGS methods is presented in Fig. 10. The minimum horizontal stress ranges from 40.79 MPa to 57.03 MPa. The average value of the minimum horizontal stress is 53.23 MPa.

Fig. 10
figure 10

Static minimum horizontal stress distribution through upscaled computed 1D well log properties: (top) Kriging (Krig) method and (bottom) SGS method

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The spatial distribution of maximum horizontal stress using both Kriging and SGS methods is presented in Appendix A. The maximum horizontal stress ranges from 44.99 MPa to 61.79 MPa. The average value of the maximum horizontal stress is 56.49 MPa. Moreover, the spatial distribution of vertical stress using both Kriging and SGS methods is presented in Appendix A. The vertical stress ranges from 49.15 MPa to 66.54 MPa. The average value of the vertical stress is 60.51 MPa. In addition, the spatial distribution of the Young modulus using both Kriging and SGS methods is presented in Fig. 11. The Young modulus ranges from 0.55 GPa to 33.58 GPa. The average value of the Young modulus is 7.50 GPa.

Fig. 11
figure 11

Static Young modulus distribution through upscaled computed 1D well log properties: (top) Kriging (Krig) method and (bottom) SGS method

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The spatial distribution of the Poisson ratio using both Kriging and SGS methods is presented in Appendix A. The Poisson ratio ranges from 0.12 to 0.44. The average value of the Poisson ratio is 0.28. Furthermore, the spatial distribution of unconfined compressive strength UCS using both Kriging and SGS methods is presented in Appendix A. The UCS ranges from 2.34 MPa to 142.42 MPa. The average value of the UCS is 31.93 MPa.

The 3D reservoir properties modeling

The reservoir properties such as porosity, permeability, and hydrocarbon saturation are upscaled and populated by the Kriging and SGS methods. The spatial distribution of each property is obtained using the upscaled methods and the results of them are presented as follows:

The spatial distribution of porosity using both Kriging and SGS methods is presented in Fig. 12. The porosity ranges from 0 to 0.28. The average value of the porosity is 0.15.

Fig. 12
figure 12

Static porosity distribution through upscaled computed 1D well log properties: (top) Kriging (Krig) method and (bottom) SGS method

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The spatial distribution of permeability using both Kriging and SGS methods is presented in Appendix B. The permeability ranges from 0 mD to 34,148 mD. The average value of the permeability is 412.99 mD. Moreover, the spatial distribution of hydrocarbon saturation using both Kriging and SGS methods is presented in Appendix B. The hydrocarbon saturation ranges from 0 to 0.99. The average value of the hydrocarbon saturation is 0.38.

Discussions

Future dynamic scenarios

The 3D reservoir geomechanical model gives the opportunity to assess the drilling and production conditions (potential and risk) of a new well at any location within the domain of the reservoir. For instance, in the drilling operation, planning of a well trajectory requires information about the pore pressure and the full stress tensor, i.e., the magnitude and the orientations of all three principal stresses. This fundamental information is essential, especially near faults and other geological complexities. Such a comprehensive prognosis of pore pressure and stress tensor can be provided by the 3D reservoir geomechanical model, as in this section, the pore pressure and stress tensor for a new position well, namely F1 are shown in Fig. 13. This method of stress tensor prediction based on the 3D reservoir geomechanical model can be universally used.

Fig. 13
figure 13

The comprehensive prognosis of vertical stress, maximum horizontal stress, minimum horizontal stress, and pore pressure for a new position well, namely F1: (up) Kriging, and (bottom) SGS method

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Summary and Conclusions

A workflow of reservoir geomechanical numerical modeling has been proposed. It combines the 1D and 3D models of geomechanics and reservoir to construct the 3D static reservoir geomechanical model. The following remarks can be outcomes of this paper:

  • A systematic procedure for the construction of a 1D geomechanical model is developed to calculate the pore pressure, horizontal stresses, vertical stress, Young modulus, Poisson ratio, and unconfined compressive strength. The numerical results of this procedure are in agreement with measured data.

  • The same procedure as the 1D geomechanical model is presented to determine the reservoir characteristics such as porosity, permeability, and hydrocarbon saturation.

  • The 3D reservoir and geomechanical properties are distributed by using geostatistical methods such as Kriging and Gaussian from the 1D reservoir and geomechanical model. The 3D static reservoir geomechanical model is composed of the combination of the 3D reservoir and geomechanical model. This 3D model is an effective tool for a better understanding of the spatial distribution of reservoirs and geomechanical properties.

  • The 3D static reservoir geomechanical model provides the complete state of parameters such as pore pressure, in-situ stresses, elastic properties, and reservoir characterization at any location in the model domain. This information as prior tools can be applied to the planning of future wells and reservoir scenarios.