## Abstract

The estimation of the in situ stresses is very crucial in oil and gas industry applications. Prior knowledge of the in situ stresses is essential in the design of hydraulic fracturing operations in conventional and unconventional reservoirs. The fracture propagation and fracture mapping are strong functions of the values and directions of the in situ stresses. Other applications such as drilling require the knowledge of the in situ stresses to avoid the wellbore instability problems. The estimation of the in situ stresses requires the knowledge of the Static Young’s modulus of the rock. Young’s modulus can be determined using expensive techniques by measuring the Young’s modulus on actual cores in the laboratory.

#### Author:

Salaheldin Elkatatny^{1}, Mohamed Mahmoud^{1}, Ibrahim Mohamed^{2}, Abdulazeez Abdulraheem^{1}

^{1}King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. ^{2}Advantek Waste Management Services, Houston, TX, USA

Received: 23 September 2016 / Accepted: 5 January 2017 / Published online: 30 January 2017

©The Author(s) 2017.

The laboratory values are then used to correlate the dynamic values derived from the logs. Several correlations were introduced in the literature, but those correlations were very specific and when applied to different cases they gave very high errors and were limited to relating the dynamic Young’ modulus with the log data. The objective of this paper is to develop an accurate and robust correlation for static Young’s modulus to be estimated directly from log data without the need for core measurements. Multiple regression analysis was performed on actual core and log data using 600 data points to develop the new correlations.

The static Young’s modulus was found to be a strong function on three log parameters, namely compressional transit time, shear transit time, and bulk density. The new correlation was tested for different cases with different lithology such as calcite, dolomite, and sandstone. It gave good match to the measured data in the laboratory which indicates the accuracy and robustness of this correlation. In addition, it outperformed all correlations from the literature in predicting the static Young’s modulus. It will also help in saving time as well as cost because only the available log data are used in the prediction.

#### List of symbols

- E Young’s modulus
- Δt Transit time
- ε Strain
- ν Poisson’s ratio
- ρ
_{b}Bulk density of the rock - σ Stress

#### Subscripts

- s Shear
- p Compressional
- dynamic Dynamic value
- static Static value

## Introduction

The terms Young’s modulus, tensile modulus, elastic modulus, modulus of elasticity, and stiffness are refereeing to the mechanical property that measures the stiffness of a certain material. Young’s modulus is the ratio of the stress applied on the material to strain associated with the applied stress (Chen 2011). Young’s Modulus can be calculated using Hook’s Law (Nguyen et al. 2009) as follows:

where E = Young’s modulus (GPa), σ = stress (GPa), and ε = strain.

Young’s modulus is different for different rock type; the value depends on the rock properties including porosity, lithology, temperature, pore pressure, fluid saturation, and the rock consolidation (William 1969). Soft formations like shale have a low Young’s modulus value (100,000–1,000,000 psi) comparing to medium formations like sandstone (2,000,000–10,000,000) and hard formation limestone (8,000,000–12,000,000 psi) as presented by Nur and Wang (1989).

The ranges presented above show that there is no typical value of Young’s modulus for certain rock, and measuring the Young’s modulus is a must in order to conduct the geomechanical analysis for the formation in interest. Building a geomechanical model is essential for several applications related to mechanical rock failure during well drilling, completion, and stimulation, which include estimation of the formation breakdown pressure, fracture simulation, wellbore stability, and formation compaction (Ciccotti and Mulargia 2004).

Young’s Modulus can be either calculated from the sonic and density logs (dynamic Young’s modulus), or measured directly in the laboratory (static Young’s modulus). The dynamic modulus is usually significantly higher than the static moduli as originally noted by Zisman (1933) and Idle (1936). The difference between the dynamic and static modulus is more pronounced for soft rocks (sandstone) than hard rock (granite) (King 1966).

Idel (1936), Brace (1965), and Walsh (1966) stated that the difference between the static and dynamic modulus is strongly affected by the rock microstructure (natural fractures and pores) and the confining stress. High stresses might close the microcracks and result in increase in the velocity waves as the rock is compacted. The faster waves will be interpreted as a higher elastic modulus (dynamic modulus).

Zisman (1933) explained the difference between the static and dynamic modulus by the loss of energy that the wave pulse might suffer when passing through the rock pores (intergranular pores or natural fractures) due to reflection and refraction at the fluid/rock interfaces (Howarth 1984). Static Young’s modulus is often used in the wellbore stability and in situ stress applications (Ledbetter 1993; Hammam and Eliwa 2013). However, collecting cores from the well to measure the moduli in the laboratory is expensive and usually not feasible.

Several equations have been developed to establish a relationship between the static and dynamic Young’s modulus. However, the applicability of each of those correlations is limited to specific rock type under certain conditions. Belikov et al. (1970) has developed a correlation to estimate the static Young’s modulus from the dynamic one. However, his equation is only applicable for microcline and granite rock. King (1983) established a correlation for igneous and metamorphic rocks. McCann and Entwisle (1992) equation is valid for Jurassic granites. Morales and Marcinew (1993) correlation can be used for rocks with high permeability values.

Wang (2000) has developed two different correlations for hard and soft rocks. Gorjainov (1979) introduced two relationships for clays and for wet soils. Also, there are some other equations that can be used for a wide range of rocks (Eissa and Kazi 1988; Canady 2010). As the Young’s modulus value are strongly dependent on the rock microstructure, mineralogy, and confining stresses, neither of the correlation mentioned above give an acceptable match to the laboratory measurement for carbonate formation in Saudi Arabia as well be shown later in “Triaxial testing” section, and development of a new correlation to calculate static modulus is essential.

The in situ stresses can be determined from the leak-off test, and the fracture pressure of the formation behind the casing can be determined as well as the minimum horizontal principal stress. The nonlinear behavior of the leak-off test could be due to drilling fluid loss to the formation, fractures and cracks in the cement behind the casing (test run before cement setting), and plastic fracturing around the wellbore. The linear behavior during the leak-off test resulted from the drilling fluid compression and wellbore expansion around the well. The leak-off test should be repeated several times to distinguish between the different mechanisms to identify the rock fracture pressure and minimum horizontal stress (Zhou and Wojtanowicz 2002).

The determination of in situ stresses are important during the drilling operations to maintain the hole integrity and wellbore stability to avoid drilling problems. Integrated rock mechanical properties analysis will enhance the drilling process because rock mechanical properties are one of the groups that should be integrated with petrophysical parameters and other parameter to enhance the drilling and avoid wellbore stability problems. Nes et al. (2012) have done a complete analysis by integrating several rock and fluid properties among them is the rock mechanical parameters to reduce the wellbore stability issues during drilling using different types of drilling fluids. They developed a model that can be used to identify the drilling problems and to design the drilling operations. They used their model in high-pressure high-temperature drilling and compared the data to the field observations.

The static Young’s modulus is used to identify the down hole stresses profiles which are important for fracture mapping and fracture design in several rocks such as carbonates and unconventional shales. Li et al. (2014) introduced analytical solution to the stresses induced during the fracturing of unconventional shale and how they can be used in shale gas exploration. Their model can be used to predict the induced stresses due to the fracturing operations and also to predict the minimum spacing between the fractures to prevent the communication between these fractures which will cause loss of gas production due to the interference between these fractures. Zhou et al. (2015) studied the interaction between the hydraulic fractures in shale formations numerically. They concluded that a stress shadow area around the fracture will be generated due to the induced stresses, three areas of compressive stress will be formed, and this will affect the fracture orientation from the normal trends or directions.

Chan and Board (2009) used finite elements calculations to determine the induced stresses to heating and thermal effects. They found out that the in situ induced stresses due to thermal heating and the rock displacement are primarily affected by the temperature relation with the rock thermal expansion coefficient. They considered this coefficient to be the main factor that controls the prediction of the rock stresses.

Based on the literature survey, it can be said that till now there is no such general equation to calculate the static Young’s modulus from the well log data. All the correlations reported in the literature are based on the laboratory measurements to develop the relation between dynamic and static Young’ modulus which is time-consuming and expensive. The relationship should be picked carefully to satisfy the validation conditions of each equation. Otherwise, generalization will always give the wrong answer. The objective of this study is to develop a new correlation to estimate the static Young’s modulus from the well log data for carbonate rocks (limestone and dolomite). This correlation can be used directly to estimate the static Young’s modulus from log data (density, compressional, and shear transit times).

## Triaxial testing

In the laboratory, Young’s modulus can be calculated from the stress–strain curve, and the experiment can be conducted under uniaxial or triaxial stress conditions. It is always preferred to run the test under triaxial conditions because uniaxial test might overestimate the static Young’s modulus value due to closing the fractures parallel to the stress direction (Thill and Peng 1974). The triaxial test is used to measure the mechanical properties of a cylindrical rock sample. The fluid pore pressure, drainage conditions, axial load, and confining stresses can be controlled to simulate the actual formation conditions.

A triaxial test is conducted by loading the sample axially while applying a constant all-around (cell or confining) pressure equivalent to the effective reservoir pressure. The prepared samples were 3.81 cm in diameter and 7.62 cm long. The end faces of all samples were cut and ground parallel. The samples were then cleaned using toluene and were vacuum-dried in the oven at 60 °C. All samples were tested under dry condition. The testing was conducted under room temperature, and confining pressure was kept constant during the loading phase. The confining pressure was calculated using Eq. (2).

where σ′_{3} is the effective confining pressure; ν is the Poisson’s ratio (assumed 0.3); σ_{v} is the vertical stress; α is Biot’s constant (assumed 1); and Pp is the pore pressure. Tectonic component is estimated using leak-off or microfrac tests conducted in the field.

The confining pressure was increased gradually from 0 to the required level with an increment of 0.02068 MPa/s. For the determination of E and ν of the rock sample, the sample was jacketed using heat shrink tubing. The jacketed sample was then placed between the hardened steel plates, and the sample was tightly secured with the platens using steel wires. The sample was then instrumented with LVDTs (Linear Variable Differential Transformer). Two LVDTs were used for recording axial displacement. These two LVDTs were mounted on the steel platens opposite to each other using LVDT holders. The radial displacement was recorded using an LVDT mounted directly on the sample. An LVDT—instrumented rock sample is shown in Fig. 1.

**Fig. 1** A rock sample instrumented with LVDTs for measuring deformation in the sample

The stress–strain response was plotted for all tested samples, and the elastic constants (Young’s modulus and Poisson’s ratio) were computed at 50% of the peak stress (for example, it will be at 80 MPa for Fig. 2). A tangent will be done at stress = 80 MPa, and the static Young’s modulus will be calculated from the slope of the stress–strain line tangent at 80 MPa. The left curve (radial one) the slope of the tangent at 80 MPa will yield the static Young’s modulus divided by the static Poisson’s ratio.

**Fig. 2** Determination of static Young’s modulus from stress–strain curves. E_{static} = static Young’s modulus and ν_{static} = static Poisson’s ratio

## Correlation development

### Formation characterization

The selected formation comprising of carbonates and anhydrite. The overall petrographic characteristics of seven samples showed that the studied section is divided into three units: dolomitized grainstone, lime mudstone, and peloidal bioclastic–intraclastic dolograinstone.

Dolomitized grainstone facies were observed in two samples that consist of medium-to-coarse-grained, moderately sorted grainstone. All the matrix and debris were dolomitized showing dolomitized rhombs, Fig. 3. The grainstone was partly leached and shows both intercrystalline and intracrystalline porosity, which ranges from 9 to 10%.

**Fig. 3** Thin section photomicrograph, X25 under cross-polarized light, of carbonate sample (Depth: XX43.8 m)

The lime mudstone facies were observed in three samples consisting of gray colored compact limestone with scattered dolomite rhombs and anhydrite, Fig. 4. The rock is very tight, and the porosity ranges from 0.5 to 1.0%.

**Fig. 4** Thin section photomicrograph, X25 under cross-polarized light, of carbonate sample (Depth: XX33.2 m)

Peloidal bioclastic–intraclastic dolograinstone facies were observed in two samples consisting of medium-to-fine-grained peloidal grainstone with some skeletal debris and intraclast, Fig. 5. At some places, anhydrite is found replacing some of the leached peloidal and skeletal grains. Leaching of the grains has resulted in good porosity development, which ranges from 9 to 10%.

**Fig. 5** Thin section photomicrograph, X25 under cross-polarized light, of carbonate sample (Depth: XX44.4 m)

### Log data analysis

Figure 6 shows the available log data for the selected formation. The data contain the neutron porosity, bulk density, and sonic time (compressional and shear times). The available data were used to estimate the dynamic parameters; Poisson’s ratio, and Young’s modulus.

**Fig. 6** Log data for the base case. The top depth of the formation is represented by the zero value here, and the bottom depth is the 91.44 m

Figure 6 shows the results of the dynamic geomechanical parameters Young’s Modulus (E_{dynamic}) and Poisson’s Ratio (ν_{dynamic}).

where ∆t_{s} = Shear transit time, μsec/ft, ∆t_{p} = compressional transit time, μsec/ft, ρ_{b} = bulk density, g/cc, ν_{dynamic} = dynamic Poisson’s ratio, dimensionless, E_{dynamic} = dynamic Young’s Modulus, GPa.

### Environmental corrections

The environmental corrections for bulk density, neutron porosity, and sonic time were performed. Corrections such mud cake correction and lithology correction were applied for all logs used. The lithology correction was done using the cross-plot.

### Depth shifting

The log parameters for each static Young’s modulus measured in the laboratory were obtained at the corresponding core depth after adjusting the depth between the log and core data (depth shifting). This correction of depth shift is necessary, as the depth measurement for log values using cables is not the same as that measured from the number of drill strings and core lengths for core data. Core porosity values were also used to estimate the depth shift by comparing them with the corresponding log porosity and density values. From the core and log porosity profiles, it is clarified that the coring depth shift was 1.46 m for the base case, and the core depth was increased by this value to match the log depth. Hence, no depth shift was applied to convert laboratory-based sample depth to corresponding log depth. This was done for all cases tested using the new correlation.

### Static Young’s modulus

Table 1 lists the static Young’s modulus for five core samples that were measured in the laboratory. To obtain the static Young’s modulus for all the depth range, a relation between dynamic Young’s modulus and static Young’s modulus for the five samples should be developed, and it will be used to determine the static Young’s modulus profile from the log data.

**Table 1** Core depth and static Young’s modulus from laboratory measurements

Table 2 lists the log value of the five samples and Young’s modulus values (static and dynamic).

**Table 2** Log data and Young’s modulus (static and dynamic) values for the five core samples