CN1916359B - A method for realizing repeated fracturing to create new fractures - Google Patents

Abstract

The invention relates to a method for realizing repeated fracturing and new crack creation in a low-permeability oilfield, which describes a primary fracturing artificial crack by an oval crack, combines a tensile failure criterion and a shear failure criterion, provides a theoretical criterion for repeated fracturing and new crack creation, and provides characteristics and conditions which are required by the critical bottom hole extension pressure and the target stratum crustal stress field and rock mechanical parameters and are required by the repeated fracturing and new crack creation according to the theoretical criterion. The characteristics and conditions of the critical bottom hole extension pressure required by new seam construction, the ground stress field of the target stratum and the rock mechanical parameters are realized. No matter what fracturing technology is adopted, new joints can be manufactured as long as construction conditions can meet the theoretical criterion, and meanwhile, whether new joints are generated in the field repeated fracturing process can be judged. The method can guide the repeated fracturing transformation of a low-permeability oil field, dig the residual oil to a greater extent and improve the recovery ratio.

Description

A method for realizing repeated fracturing to create new fractures

technical field

The present invention relates to hydraulic fracturing stimulation measures for oil and gas reservoirs, and in particular proposes a theoretical method for realizing repeated fracturing in low-permeability oilfields to create new fractures. Maximize the potential of remaining oil and enhance recovery.

Background technique

Refracturing is an important way to achieve stable production and increase production in low-permeability oilfields. If refracturing can create new fractures, new oil and gas flow channels can be opened in oil and gas layers, and the undeveloped artificial fractures in the initial fracturing can be communicated on a larger scale. Oil and gas layers, greatly increase oil and gas production, and further improve the development effect of oil and gas reservoirs. However, there is still a lack of systematic and in-depth research on the mechanism of refracturing to create new fractures. At present, the analysis of new fractures created by repeated fracturing at home and abroad is almost all based on in-situ stress diversion. However, the research on in-situ stress diversion is still at the stage of theoretical research and numerical simulation, without the support of field tests. Therefore, there is no feasible and practical method and technology for repeated fracturing to create new fractures.

Contents of the invention

The purpose of the present invention is to provide a method for realizing repeated fracturing to create new fractures for hydraulic fracturing in low-permeability oilfields. By using this method, oil and gas production can be greatly increased, and the development effect of oil and gas reservoirs can be further improved. The production can be predicted in advance. The conditions that new fractures should meet can also be used to judge whether new fractures have been generated during repeated fracturing on site.

The technical solution of the present invention is: to design a method for realizing repeated fracturing to create new fractures. The method is: it uses elliptical fractures to describe the artificial fractures of primary fracturing, and combines the tensile failure criterion and the shear failure criterion, which is the first time at home and abroad. The theoretical criterion for creating new fractures by repeated fracturing is given. According to the theoretical criteria, the critical bottom hole extension pressure that should be satisfied for realizing repeated fracturing and new fractures, the in-situ stress field of the target formation and the rock mechanics parameters that should be met are given. features and condition.

The characteristics of the critical bottomhole extension pressure, the in-situ stress field of the target layer and the rock mechanics parameters that should be met for the realization of refracturing to create new fractures are: the operating pressure conditions meet the p _inj ≥ σ _h2 + T ₀ , ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4}$ or $p_{\mathrm{inj}}\&Greater\; Equal;\frac{1}{2}[({\mathrm{\&sigma;}}_{h1}+{\mathrm{\&sigma;}}_{h2})-{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}^2/{8T}_0],$ ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0<\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4},$ It can be determined that new tensile fractures have been generated in the refracturing process, where p _inj is the bottom hole extension pressure, σ _h1 is the maximum horizontal principal stress, σ _h2 is the minimum horizontal principal stress, and T ₀ is the tensile strength; similarly, If the theoretical criterion is satisfied $p_{\mathrm{inj}}\&Greater\; Equal;\frac{{(1+\sin \frac{1}{2}\mathrm{\&theta;}-m+m\sin \frac{1}{2}\mathrm{\&theta;})(L_f/2r)}^{\frac{1}{2}}\cos \frac{1}{2}{\mathrm{\&theta;\&sigma;}}_{h2}-{\mathrm{\&sigma;}}_C}{1-m+(1+\sin \frac{1}{2}\mathrm{\&theta;}-m+m\sin \frac{1}{2}\mathrm{\&theta;}){(L_f/2r)}^{\frac{1}{2}}\cos \frac{1}{2}\mathrm{\&theta;}}$ And p _inj <σ _h2 , it can be judged that a new shear fracture has been generated, where L _f is the half-length of the fracture, (r, θ) is the plane polar coordinate of a point P near the fracture tip, and σ _C is the uniaxial resistance of the rock. Compressive strength, m=tg ² α, α is the angle between the normal direction of the new shear fracture surface and σ _h1 direction, α=45°+φ/2, φ is the rock internal friction angle.

The elliptical fracture mentioned refers to a prolate elliptical fracture, that is, the semi-major axis of the ellipse is greater than the semi-minor axis of the ellipse, which is used to describe the artificial fracture of the initial fracturing, and the half-length of the fracture is that the semi-major axis of the ellipse is 30-100m , the half-width of the crack is both the semi-short axis of the ellipse in 1-3mm.

The features of the present invention are:

(1) For the first time at home and abroad, the condition of matrix rock failure in the refracturing process is analyzed, and thus the theoretical criterion for creating new fractures is given, that is, the critical bottomhole extension pressure and the in-situ stress of the target layer required for creating new fractures The characteristics and conditions that the field and rock mechanics parameters should have.

(2) The form of the theoretical criterion is simple and easy to understand. No matter what kind of fracturing technology is adopted, as long as the construction conditions can meet this theoretical criterion, new fractures can be created.

(3) The theoretical criterion can also be used to judge whether new fractures are generated during the on-site refracturing process.

(4) There are two types of new cracks: tensile cracks and shear cracks, which correspond to two basic failures of rocks—tensile failure and shear failure. The orientations of the two fractures are different from those of the primary fracturing artificial fractures, and the critical bottomhole extension pressure is a function of the in-situ stress field and rock mechanics parameters.

Based on a full investigation of relevant domestic and foreign literature, the present invention analyzes various types of fracture diversion in repeated fracturing, and proposes that the mechanism of creating new fractures is by no means limited to the in-situ stress field diversion mentioned above. The root of refracturing to create new fractures lies in the refracture of the rock at the weak surface, rather than the so-called inversion of in-situ stress. Combining with the rock failure criterion, the condition of matrix rock failure in the refracturing process is analyzed for the first time, thus giving the Theoretical criteria for creating new seams. Fracture turning is mainly controlled by fracture pressure, even if the in-situ stress field does not change, fracture turning may also occur.

For repeated fracturing construction, according to the above theoretical criteria, it can be judged whether new fractures are generated by observing the construction pressure curve. In the same way, before refracturing, based on the above theoretical criteria, the conditions that should be met for creating new fractures can be predicted in advance, so that wells and layers can be selected reasonably, and appropriate fracturing techniques can be adopted to realize new fracture creation. Therefore, this method can better guide the refracturing of low-permeability oilfields, tap the remaining oil to a greater extent, and improve the recovery rate, which has strong practical value.

Description of drawings

Figure 1 is a schematic diagram of a point P near the tip of a slender ellipse (flat ellipse).

Fig. 2 is a schematic diagram of the Mohr-Coulomb failure criterion.

Fig. 3 is a graph showing pressure vs. time for temporary plugging and screenout fracturing in Well Example 1.

Fig. 4 is a graph showing pressure vs. time for temporary plugging and screenout fracturing in Well Example 2.

Fig. 5 is a monitoring map of micro-seismic fractures in the well of Example 2.

Detailed ways:

Using elliptical fractures to describe the primary fracturing artificial fractures, combined with the tensile failure criterion and shear failure criterion, the conditions for new rock failure in the refracturing process of low-permeability reservoirs were analyzed for the first time at home and abroad, and the repeated fracturing conditions were given. Theoretical criteria for creating new fractures. That is to say, the critical bottomhole extension pressure value required to create new fractures, the in-situ stress field of the target layer and the characteristics and conditions that should be possessed by rock mechanics parameters are given. The new fractures here are divided into shear fractures and tensile fractures, corresponding to the shear failure and tensile failure of rocks, respectively. The orientations of the two fractures are different from those of artificial fractures during initial fracturing, and the critical pressure is a function of the in-situ stress field and rock mechanics parameters. Using this theoretical method, it is possible to predict in advance the conditions that should be met for creating new fractures, reasonably select wells and layers for refracturing, and adopt an appropriate fracturing process to realize new fracture creation; it can also be used to judge on-site refracturing Whether construction has created new joints. Therefore, this method can efficiently guide the refracturing of low-permeability oilfields, tap the remaining oil to a greater extent, and improve the recovery factor.

Temporary plugging sand removal fracturing technology has been successfully applied and popularized in Changqing Oilfield. The case study of Ansai Oilfield shows that if the temporary plugging boost value satisfies the above theoretical criteria, new tensile fractures can be generated, which further confirms the theory field applicability of the method.

The method is simple and feasible, and can be applied to all low-permeability and extra-low-permeability reservoirs for repeated fracturing. When implementing refracturing, no matter what fracturing technology or measures are adopted, as long as the technology can create construction conditions that meet the theoretical criteria, new fractures can be created.

1 Theoretical derivation of refracturing to create new fractures

After the initial fracturing, it is assumed that the artificial main fractures have been formed, and along the direction of the maximum principal stress in the far field. Assuming that the fracture height remains constant, considering the case of a planar 2D fracture, the fracture is approximately elliptical, which is similar to the assumptions of planar models such as GDK and PKN. The half-length of the crack is L _f and the maximum width is b.

和β。如果当前主地应力方位较原始主应力没有变化，则 $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2}.$ 如果当前地应力场发生了变化(比如因初次裂缝开启或者注水或油气采出引起的变化)，并且已知，则β也已知。During repeated fracturing, fracturing fluid enters again, and the stress state near the fracture wall is considered at this time for fracture analysis. Suppose the maximum horizontal principal stress and the minimum horizontal principal stress in the far field are σ _h1 and σ _h2 respectively, and the angles between them and the x-axis are respectively

and beta. If the orientation of the current principal stress does not change compared with the original principal stress, then $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2}.$ If the current in-situ stress field has changed (such as changes caused by initial fracture opening or water injection or oil and gas production) and is known, then β is also known.

In low-permeability oilfield fracturing practice, the half-length of artificial fractures can be 30-100m, and the half-width of fractures can be 1-3mm, so b/L _f is very small. And for the ellipse coordinate system, if ξ=ζ ₀ , the corresponding curve in the x, y plane is an ellipse

$\frac{{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; c</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; cosh</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; c</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; sinh</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}$

Its semi-axis is

a=ccoshζ ₀ , b=csinhζ ₀

Then there is tanhζ ₀ =b/a. According to the above, b/L is extremely small, so approximately ζ ₀ =0, that is, the primary fracturing artificial fracture is a slender ellipse (flat ellipse) fracture, which becomes a cut from x=-c to x=c, and c= L _f .

The stress state of point P near the tip of the oblate ellipse crack can be analyzed using the ellipse coordinate system (see Fig. 1).

(1) Tensional rupture of crack wall

According to Griffith's classical theory, it is believed that tensile cracking (tensile cracking) is caused by the stress concentration at the tip of the tiny Griffith crack, and when the maximum tensile stress near the crack tip reaches the characteristic value of the material, the crack begins occur.

The tangential stress on the flat elliptical crack surface is:

σ _t ＝p+{(σ _h1 +σ _h2 -2p)sinh2ζ ₀ +(σ _h1 -σ _h2 )[exp(2ζ ₀ )×cos2(β-η)

-cos2β]}/{cosh2ζ ₀ -cos2β}

Obviously, the stress is related to β, and it can be seen that the stress field on the fracture wall is closely related to the current maximum principal stress orientation.

The artificial fracturing mentioned in the classical theory of hydraulic fracturing refers to the tensile fracture, and the maximum tensile stress criterion is usually used to analyze the tensile fracture.

The maximum tensile stress criterion is still used here, that is, when the maximum tensile stress (surface tangential stress) reaches the tensile strength of the rock, tensile fracture occurs.

The failure criterion given by the analysis is:

(σ _h1 -σ _h2 ) ² -8T ₀ (σ _h1 +σ _h2 -2p _inj )=0, p _inj <(σ _h1 +3σ _h2 )/4;

σ _h2 -p _inj +T ₀ =0, p _inj <(σ _h1 +3σ _h2 )/4

Right now

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; inj</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; <</span>\frac{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; ;</span>$

p _inj =σ _h2 +T ₀ , ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4}$

。可见如果 $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2},$ 则新裂缝将垂直于初次裂缝方位。The new tensile fracture orientation (angle with the x-axis) is

. Visible if $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2},$ Then the new fracture will be perpendicular to the orientation of the primary fracture.

(2) Shear fracture near the crack tip

For the stress state of point P near the crack tip, as shown in Fig. 1, we can get

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

$<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

${\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; r\&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}$

${\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

If the orientation of the far-field principal stress does not change, then $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2},$ Substituting the above formulas, it can be found that the items related to σ _h1 in each formula become 0, so the expression containing only σ _h2 is obtained:

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; sin</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

${\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; r\&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; cos</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

Using the principal stress formula:

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xx</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; yy</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; xy</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xx</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; yy</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xx</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; yy</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; xy</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xx</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; yy</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

The principal stress at a point (r, θ) near the crack tip is obtained as

$<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&PlusMinus;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$

The orientation (angle with σ _r ) θ′ of the principal stress is:

${\mathrm{<span\; class="notranslate">\; tg</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; xy</span>}}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xx</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; yy</span>}}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; r\&theta;</span>}}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}$

so ${\mathrm{\&theta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{4}-\frac{\mathrm{\&theta;}}{4}$ or

When p<σ _h2 , the principal stress is in the state of compressive stress, and shear failure is considered.

The shear failure criterion is briefly described below.

Mohr-Coulomb criterion

The most commonly used fracture criterion for shear failure is the Mohr-Coulomb criterion. On any cross-section of the medium microelement, the magnitude of the shear stress τ _n cannot exceed a certain critical value. When |τ _n | reaches the critical value, the material will produce shear slip.

Figure 2 is a schematic diagram of the Mohr-Coulomb failure criterion. Its two straight lines are the Moht-Coulomb intensity curve, the area inside the straight line is the safe area, and the area outside the straight line is the damage area. When the stress circle (σ, τ) of the rock is tangent to the straight line, the rock will undergo shear failure.

|τ _n |≥C ₀ +σ _n ′tgφ

$\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}$

$\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}$

α＝45°+φ/2

τ _n and σ _n ′ are the shear stress and normal stress (normal stress) on the shear failure surface, respectively, α is the angle between the normal direction of the shear failure surface and the direction of the maximum principal stress σ ₁ , and c ₀ is the intrinsic shear strength (cohesion or cohesion), tgφ is the coefficient of internal friction, and φ is the angle of internal friction. Shear failure is a typical fracture under compressive stress, which is characterized by shear displacement along the fracture surface.

The Mohr-Coulomb criterion can also be expressed by the relationship between principal stresses:

σ ₁ -p=m(σ ₃ -p)+σ _C

Where m = tg ² α, σ _C = 2c ₀ tgα, which is the uniaxial compressive strength.

deduce

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; inj</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; m\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}$

Substitute the above principal stresses into the shear rupture criterion $p_{\mathrm{inj}}=\frac{{\mathrm{\&sigma;}}_1-{\mathrm{m\&sigma;}}_3-{\mathrm{\&sigma;}}_C}{1-m},$ Get the critical injection pressure (pore pressure):

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; inj</span>}}<span\; class="notranslate">\; =</span>\frac{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}$

And should p _inj <σ _h2

where m = tg ² α, σ _C = 2C ₀ tgα,

The specific analysis process of whether a new shear fracture is formed or not is as follows:

judge first ${\mathrm{\&sigma;}}_{h2}<\frac{{\mathrm{\&sigma;}}_C}{m-1}$ established or not,

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; sin</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}$

where m = tg ² α, σ _C = 2c ₀ tgα,

(1) If it is established, examine the possibility of 1-m+Δ>0, if possible, and have ${\mathrm{\&sigma;}}_{h2}>\frac{{\mathrm{\&sigma;}}_C}{\mathrm{\&Delta;}},$ shear cracks are possible; if not possible, no shear cracks.

(2) If not established, examine the possibility of 1-m+Δ<0, if possible, and satisfy ${\mathrm{\&sigma;}}_{h2}<\frac{{\mathrm{\&sigma;}}_C}{\mathrm{\&Delta;}}$ shear cracks are possible; if not possible, no shear cracks.

所以剪切缝与该考察点(r，θ)的σ_r方向夹角为：或者

两者择一，因为主应力方位 ${\mathrm{\&theta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{4}-\frac{\mathrm{\&theta;}}{4}$ 或者

只能取其中之一。If a new shear crack is generated, the angle between the normal direction of the shear fracture surface and the current maximum principal stress azimuth θ′ should be

Therefore, the angle between the shear seam and the σ _r direction of the investigation point (r, θ) is: or

Choose one of the two, because the principal stress orientation ${\mathrm{\&theta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{4}-\frac{\mathrm{\&theta;}}{4}$ or

Can only take one of them.

It can be seen that the angle between the normal direction of the new fracture surface and the x-axis (that is, the orientation of the initial artificial fracture) is roughly in the range of 30° to 75°, that is, the new shear fracture has also turned.

2 Field Application Examples

This patent provides a theoretical method for judging and realizing new fracture creation through repeated fracturing. The method is simple and feasible, and can be applied to all low-permeability and extra-low-permeability reservoirs for repeated fracturing. When implementing refracturing, no matter what fracturing technology or measures are adopted, as long as the technology can create construction conditions that meet the theoretical criteria, new fractures can be created.

At present, the temporary plugging and sand removal fracturing technology has played an important role in the refracturing and stimulation stimulation of Changqing Low Permeability Oilfield, and has been widely used. The specific implementation process can refer to many public documents, such as "Drilling and Production Technology" in 2002 25 (6): 39-41: "Sand-out fracturing test at the end of Chang 6 reservoir in Ansai Oilfield", etc. Its core idea is to control the continuous expansion and extension of the primary fracturing artificial fractures by using screenout, and add temporary plugging agent to form a shield in the primary fracturing artificial fractures, thereby forming a high-pressure environment. The case analysis shows that if the boost value of temporary plugging satisfies the above theoretical criteria, new tensile cracks will be generated. The following is the analysis of new fractures created by temporary plugging and screenout fracturing of two wells in Wangyao District of Ansai Oilfield.

Analysis of Well Wang 14-15

The well is located in the central part of Wangyao District, and it is a test well for changing the direction of liquid flow in the central part. The semi-major axis of the prolate ellipse of the primary fracturing artificial fracture is 82.4m, and the half width of the fracture and the semi-short axis of the ellipse are 1.6mm.

The well was put into production in November 1989, with an initial daily oil production of 2.5 tons. It took effect in June 1991, and the daily oil production rose to 4.0 tons. Water broke through in March 1993, and the well was shut down due to water flooding in February 1999, with a cumulative oil production of 6,278 tons. In order to explore technical ideas for improving recovery in the central and western regions, it was transferred to Well Wang 13-15 on May 26, 2002, and carried out Changed liquid flow direction test, and resumed production of 3 surrounding water-flooded oil wells. At present, the other two wells have seen oil, with an average daily liquid production of 3.45m ³ , and well Wang 14-15 has not produced since July 16, 2002. liquid. According to the analysis, the oil layer is clogged and repeated fracturing should be carried out. The well was subjected to temporary plugging and screen-out fracturing on September 29, 2002. The construction curve is shown in Figure 3. After fracturing, it was put into production by self-spraying, with an average daily oil increase of 3.24 tons, and the production increase effect is remarkable (see Table 1).

Using the in-situ stress and rock mechanical parameter data measured by the indoor core test of the adjacent well of this well, it is known that the maximum horizontal principal stress σ _h1 and the minimum horizontal principal stress σ _h2 are 25.74 and 18.44 MPa respectively, and the tensile strength T ₀ is 7.67 MPa, then $\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4}=1.83\mathrm{MPa},$ Right now ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4}.$ Observing the construction pressure curve of repeated fracturing, the previous working pressure was stable at about 15MPa. It can be judged that the initial fracturing artificial fracture restart pressure is about 15MPa, and the temporary plugging boost pressure is about 23MPa, so the temporary plugging boost value is about 8MPa. The plugging pressure value is slightly greater than the rock tensile strength, that is, the construction pressure condition satisfies the p _inj ≥ _{σ h2} +T ₀ in the theoretical criterion, ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4},$ Therefore, it can be judged that new tensile cracks have been generated in the refracturing process, which is very consistent with the development trend of the second half of the construction pressure curve. Moreover, the self-spraying of the well after fracturing and the significant stimulation effect also strongly prove that new fractures may have been generated.

Table 1 Analysis and Statistics of Refracturing Operation Parameters and Effects in Well W14-15

hashtag layers Oil layer thickness (m) Perforation thickness (m) Permeability (10^-3μm²) The amount of sand added in the previous fracturing (m³)   Construction parameters for screenout fracturing Amount of sand added (m³) Displacement (m³/min) Temporary blocking dose (kg) Temporary plug boost (MPa) W14-15 long 6 12.2 11 2.427 17 30 1.4-1.6 380 8 Before measures The average of 1-12 months after the measures Valid days (days) Average daily oil increase (t) Daily liquid (m³) Daily oil production (t) Water content (%) Daily liquid (m³) Daily oil production (t)   Water content (%) 0 0 0 6.26 3.24 37.97 37 3.24

Analysis of Well Wang 20-017

Well Wang 20-017 was constructed with temporary plugging and screen-out fracturing technology. The construction pressure curve is shown in Figure 4, and the microseismic crack monitoring results are shown in Figure 5. The orientation of the main fracture is 64.1 degrees in the northeast direction, the half-length of the fracture and the semi-major axis of the ellipse are approximately 65 meters, and the half-width and semi-short axis of the ellipse are 2.0mm. About 60 meters away from the main fracture in the east, there is a secondary fracture approximately perpendicular to the main fracture, with a length of about 30 meters.

It can be seen from Fig. 4 that the restart pressure of artificial fractures during initial fracturing is about 12MPa, and then the pressure is raised to 15MPa after temporary plugging, so the boost value of temporary plugging is about 3.0MPa.

The azimuth of the maximum horizontal principal stress in the Wangyao area is NE 67° on average, so the azimuth of the main fractures (64.1° NE) of refracturing is basically the same as that of the primary fracturing fractures, that is, no fracture diversion occurs in refracturing. In other words, the ground stress field has not turned.

According to the indoor rock mechanics parameter measurement results of sandstone coring in the target layer of this well, the tensile strength of the rock is 2.9MPa. The average maximum horizontal principal stress σ _h1 in Wangyao area is 24MPa, and the minimum horizontal principal stress σ _h2 is 19MPa, then ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4};$ And the temporary plugging boost value is very close to and slightly greater than the tensile strength. It can be seen that the construction condition of the well satisfies p _inj ≥σ _h2 +T ₀ , ${\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="S05196443620051216E000011.png"\; state="\{\{state\}\}">T</figure-callout>}}_0>\frac{({\mathrm{\&sigma;}}_{h1}-{\mathrm{\&sigma;}}_{h2})}{4},$ That is, repeated fracturing will generate new tensile fractures. According to the above analysis, the in-situ stress field of this well layer has no diversion, that is, $\mathrm{\&beta;}=\frac{\mathrm{\&pi;}}{2},$ So the new tensile crack should be perpendicular to the artificial main crack. It can be seen that the theoretical analysis conclusions are completely consistent with the field crack monitoring results.

In other wells, the half-length of the fracture is 35m, and the half-width of the fracture is 1.2m. There are also cracks with a half-length of 95m and a half-width of 3 meters. Its theoretical analysis conclusions are completely consistent with the field crack monitoring results.

The feasibility of this theoretical method in field application is demonstrated through the example analysis of the temporary plugging and screen-out fracturing process in Ansai Oilfield. The implementation of temporary plugging and screenout fracturing technology can create construction conditions that meet the theoretical criteria, so new tensile cracks are produced. Of course, if other suitable fracturing techniques can be adopted to make the critical bottom hole pressure meet the theoretical criteria, new fractures can also be created. At present, the temporary plugging and screen-out fracturing technology has played an important role in the refracturing stimulation of Changqing low-permeability oilfield and has been widely used. It utilizes sand removal to control the continuous expansion and extension of the primary fracturing artificial fractures, and adds a temporary plugging agent to form a shield in the primary fracturing artificial fractures, thereby forming a high-pressure environment. The case analysis shows that if the boost value of temporary plugging satisfies the above theoretical criterion of creating new fractures, new tensile cracks will be generated.

For the analysis of new fractures created by refracturing, whether it is predicted in advance or applied in the field, this method is practical, so it can effectively guide the refracturing of low-permeability oilfields, tap the remaining oil to a greater extent, and improve recovery Rate.

Claims (1)

1. realize that refracturing makes the method for new seam for one kind, its method is: it describes first fracturing artificial slit with the ellipse seam, in conjunction with opening property failure criterion and shear failure criterion, provided refracturing and made the theoretical criterion of new seam, according to theoretical criterion provided realize refracturing make new seam the critical shaft bottom extending pressure that should satisfy and target zone stress field and rock mechanics parameter feature and the condition that should possess;

Realize refracturing make new seam the critical shaft bottom extending pressure that should satisfy and target zone stress field and the rock mechanics parameter feature that should possess be: the operation pressure condition satisfies the p in the theoretical criterion _Inj≤ σ _H2+ T ₀,

Or

Be that decidable refracturing process has produced new draw crack, wherein p _InjBe the shaft bottom extending pressure, σ _H1Be maximum horizontal principal stress, σ _H2Be minimum level main stress bar, T ₀Be tensile strength; If satisfy theoretical criterion

And p _Inj＜σ _H2, then can judge to have produced new shear crack, wherein L _fBe the fracture half-length, (r θ) is the plane polar coordinates of certain the some P crack tip near, σ _CBe rock uniaxiality strength, m=tg ²α, α are new shear fracture face normal direction and σ _H1The angle of direction, α=45 °+φ/2, φ is an internal friction angle of rock;

Oval seam is meant the oval crack of prolate, and promptly oval semi-major axis is greater than oval semi-minor axis, and it is used for describing first fracturing artificial slit, and its fracture half-length is oval semi-major axis at 30-100m, and the crack half-breadth is oval semi-minor axis at 1-3mm.

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