Cracking mechanism of shale cracks during fracturing

In this paper, we set up a model for calculating the shale fracture pressure on the basis of Huang’s model by the theory of elastic-plastic mechanics, rock mechanics and the application of the maximum tensile stress criterion, which takes into account such factors as the crustal stress field, chemical field, temperature field, tectonic stress field, the porosity of shale and seepage of drilling fluid and so on. Combined with the experimental data of field fracturing and the experimental results of three axis compression of shale core with different water contents, the results show that the error between the present study and the measured value is 3.85%, so the present study can provide technical support for drilling engineering.


Introduction
The shale formation is mainly composed of illite and mixed layer clay minerals, the horizontal bedding, crack and micro crack development of shale are obvious [1]. When the shale meets water, it can easily hydrate [2][3][4], the mechanical strength of it is reduced, the expansion stress is produced, and the pore pressure of the rock is increased [5]. Furthermore, water molecules in drilling fluids flow through the formation pores, then the additional stress field yields around the well wall [6], which leads to the decrease of the effective circumferential stress of the well. Besides, rock wall temperature changes can also cause wellbore instability, especially for the shale with high elastic modulus and low Poisson's ratio [7]. During the process of drilling fluid circulation, the thermal expansion of the surrounding rock of the shaft wall is not allowed to expand freely due to the limitation of wellbore liquid pressure and the surrounding rock of the shaft wall, so that the expansion stress will generate, which is the thermal stress affecting the shale fracture pressure [8].
At present, experts have proposed various methods for predicting formation fracturing pressure. Huang's model [9] is mainly for general reservoirs. It considers the relationship between overburden pressure and depth of rock strata, the influence of stress concentration on wellbore, the effect of tectonic stress of underground nonuniform distribution and the strength of strata and so on. Therefore, it can be applied to different deformations. Yan et al [10] established a fracture pressure model considering hydration based on Huang's model, which improved the prediction accuracy, but did not consider effect of perforation and natural fractures on the cracking of hydraulic cracks. Li et al [7] 1234567890''""  [11] established a model for calculating the fracture pressure considering the influence of high temperature and seepage, the prediction accuracy increased by 7%. However, this model is only fit for sandstone.
However, the fracture pressure model of shale considering the additional stress fields caused by the crustal stress field, chemical field, temperature field, tectonic stress field, the porosity of shale and seepage of drilling fluid is still absent. This paper is based on Hang's model, considering all these factors and established a new model for calculating the fracture pressure of shale.

Influence of seepage and porosity on crustal stress field
On the infinite plane, a circular hole receives uniform internal pressure, at the same time, the infinite distance in the plane is subjected to two horizontal stresses ( 1  and 2  ) and overlying pressure in the vertical direction. Considering that the rock is a small deformation elastic body, the principle of linear superposition is applicable to it. The stress state of the wall rock of the wellbore can be obtained by studying the stress contribution of the stress components to the wall rock of the wellbore, and then superimposing them. It is assumed that the strata are homogeneous isotropic and linear elastic porous materials, the rocks around the borehole are in a state of plane strain [2]. The stress model of the wellbore wall is shown in figure 1. R is the maximum borehole radius, m; r is the radius of borehole, m;  is the polar angle of the reverse clockwise rotation of the maximum radial direction in the radial direction. The wall rock of the borehole wall is porous medium, so the flow of fluid satisfies Darcy's law, the radial flow of drilling fluid filtrate in the stratum will generate additional stress field around wellbore rocks. Combined with drilling fluid column pressure and crustal stress, the distribution of the circumferential stress in the crustal stress field around the shaft wall in a vertical well is [2].
The formation cracks produced by the fluid in the well are mostly vertical, the reason is that the tensile strength of the tangent stress exceed the tensile strength of the rock from compression to tension. When

Effect of chemical field on crustal stress field
When water-based drilling fluid encounters shale, under the action of water pressure difference, chemical potential difference and the pressure difference between the drilling fluid pressure and pore pressure, water molecules flow into the pores between the cracks and particles, they cause the transfer of water molecules and ions, at this time, hydration occurs [2]. Under certain conditions, the shaft wall can be considered as a semi permeable membrane, and the two sides are drilling fluid system and stratigraphic fluid system. On this basis, Chenevert and Pernot [12] put forward the calculation model of hydration stress. The calculation formula is It is assumed that the structure of shale can make the fluid in the pores circulate in them and form the uniform pore pressure to the skeleton of the rock, so the effective stress is equal to the positive stress minus the pore pressure and the hydration stress. Therefore, the effective circumferential stress is as follows:

Influence of temperature on crustal stress field
Boas [13] and Maury [14] put forward the idea that the thermal pressure caused by the temperature change of wellbore rock can cause the instability of the wellbore wall. The effect of temperature should be considered when the vertical depth of the open hole is more than 1 000 m. It is proved that the stress of 0.4 MPa is produced by 1 C increases in the temperature of the medium hardness rock [14]. According to the generalized Hoek's law and thermoelastic theory, the circumferential stress caused by the variation of the wellbore and the formation temperature is as follows: In the upper formula, w T is the temperature of the wellbore, 0 T is the original stratum temperature.

Influence of tectonic stress field
According to the stress superposition criterion of small pore elastic deformation, under the influence of in-situ stress, seepage, temperature field and chemical field, the effective circumferential stress of the stress field around the borehole can be expressed as: According to the maximum tensile stress criterion, the condition of stratum cracking is  If (8) can be written as The main stress in the vertical direction is called the overlying stress 3  , which is produced by the weight of the overlying strata and varies with the depth of the rock strata. The effective overlying pressure ' 3  is: According to the view of Huang [6], if tectonic pressure exists, the effective stresses added to the two horizontal main directions in general are not equal. They are expressed as Where ' 1  is the maximum horizontal effective stress, MPa; ' 2  is the minimum horizontal effective stress, MPa. Submitting formula (12) into the formula (10), the shale stratum fracture pressure model is got.
Where F p is the stratum fracture pressure, MPa;  ,  are stress coefficients of geological structure.

Example calculation of shale fracture pressure
According to the crustal stress and measured fracture pressure parameters of oilfield [6,11,12,14], the basic parameters are selected as shown in tables 1 and 2. According to the formula (13), the calculation results of the fracture pressure of shale are shown in figure 2.  The calculation model is compared with other classic models as shown in figure 2. It can be seen that the fracture pressure of shale gradually rises with the rise of the well depth. The shale fracture pressure calculated by the present study is higher than that of other models at the same well depth. The reason is that the model of this paper not only considers the effect of these factors: the relationship between the overlying stress, the depth of the rock layer, stress concentration in wellbore wall, the effect of tectonic stress on underground nonuniform distribution and rock stratum strength, but also considers the additional stresses caused by percolation of drilling fluid, the hydration stress, the thermal stress caused by the temperature difference in the wellbore. The results show that the prediction error of the model of this paper is 3.85% compared with the measured results. The predicted results of this model are in good agreement with the actual fracture pressure. So the density of the equivalent drilling fluid calculated is more practical.

Summary
• Combined with the theory of elastic mechanics and rock mechanics, the application of the maximum tensile stress criterion, the present study established a model of shale fracture pressure. Based on Huang's model, the model of the present study has considered the additional influence of seepage of drilling fluid, hydration stress, shale porosity，temperature variable stress and tectonic stress and so on. Compared with the measured results, the error of this model is 3.85%, so the calculation accuracy is improved. • Based on the stress superposition principle of small deformations on elastic material with pores, the present study made the linear superposition of crustal stress, tectonic stress, additional stresses caused by seepage and temperature and hydration stress. However, the interaction between them has not been considered yet, which needs to be studied in the subsequent work.