Modeling the Crack Interference in X80 Oil and Gas Pipeline Weld

Based on the numerical simulation method of the virtual crack closure technique (VCCT), an interference model was established to investigate the physical problem of two interacting cracks of different sizes in the welding zone of oil and gas pipelines. The obtained results of the current interference problem were compared with those of single crack case. Various crack configurations, such as different crack spacing and different crack sizes, were analyzed. The characteristic quantity fluid pressure load P during the crack propagation process, the peak value of the von Mises stress distribution field of the crack growth path, as well as the difference ∆Bx between the peak value of the magnetic induction intensity component at the crack and the value of the magnetic induction intensity component at its symmetrical position were calculated. The crack interaction scale factors, including γP, γMises, and γΔBx, were compared and analyzed. The numerical modeling results show that when the unequal-length double cracks interfere with each other, the cracks are more likely to propagate toward each other. The tendency of the double-cracks to propagate toward each other gradually weakens as the distance between the crack tips increases and is finally the same as that of single-crack cases. It was also found that the effect of large-sized cracks on small-sized cracks is greater than that of small-sized ones on large-sized ones. The numerical modeling results could be applied for the prediction and analysis of multicrack damage in oil and gas pipeline welds.


Introduction
Oil and natural gas are dangerous goods under national key supervision due to their flammable, explosive, and toxic characteristics. Large storage tanks and oil and gas pipelines used to store and transport oil and gas resources easily constitute a major hazard. Under the combined action of internal and external loads, this storage and transportation equipment will inevitably produce defects and cracks, especially in stress-concentration areas such as welds. Leakage and explosion accidents of storage and transportation equipment frequently occur owing to weld defects. For example, in 1974, an accident occurred at the Mitsubishi Mizushima Oil Refinery in Japan. The weld between the oil tank wall and the side plate of the tank bottom that was 5 × 10 4 cubic meters fractured, and the oil leaked out instantaneously and destroyed the fire embankment, posing a serious threat to the environment and the safety of people's lives and property. The southwest oil and gas field's "1.20" explosion and fire accident is another example. The direct cause of the accident was that the pipe was torn under internal pressure due to defects in the spiral weld of the pipe, and the ferrous sulfide powder carried by the leaked natural gas oxidized and spontaneously ignited in the air, causing an explosion outside the natural gas pipe. decision of fatigue crack growth thresholds from crack propagation data was investigated by Schonherr et al. The results show that the proposed method decreases the artificial conservatism led by the evaluation method as well as the sensitivity of test data dispersion and the impact of test data density [14]. Zhang and Collette investigated the use of dynamic Bayesian networks to predict the propagation and interaction of multicracks in structural systems. Using the crack lengths measured in the experiment, the ability of predicting future crack size evolution is assessed. For this sample, it must capture a high level of crack-to-crack interaction so that the pair of cracks can realistically track the evolution of the system [15]. Ahmed et al. used the boundary cracklet method to study the fatigue crack propagation of multiple interacting cracks in a porous perforated plate. The boundary cracklet method (BCM) was further used to model the interaction of multiple fatigue cracks in a perforated plate geometry with multiple holes and cracks. They concluded that the BCM could be a reliable tool for simulating the reality of multiple fatigue crack interactions in 2D [16].
Among the finite element methods for studying crack growth, the virtual crack closure technique (VCCT) is one of the most widely used methods [17][18][19][20][21][22][23]. For example, Krueger outlined a few virtual crack closure techniques. The methodology used was discussed, the history was summarized, and insights into its application were provided. Necessary modifications to the use of geometrically nonlinear finite element analysis methods and corrections required for crack-tip elements of different lengths and widths were analyzed [17]. Banks-sills and Farkash studied the specification of the virtual crack closure technique for a crack at a bimaterial interface. The energy release rate's dependence on the virtual crack growth size of the interfacial crack was analyzed and accounted for so that the stress intensity factor can be obtained accurately when using a fine finite element mesh, as well as by more than one element [18]. Yu et al. studied the crack growth behavior of simulated nuclear graphite using the extended finite element method (XFEM), VCCT, and cohesive zone model (CZM) methods. The dependence on the element size that the peak load P c has was analyzed, and the numerical results showed that, using the three methods, P c is sensitive to the mesh size, and most sensitive when using VCCT [19]. Yao et al. investigated a 3D-VCCT-based fracture analysis method for multicracked gas pipelines. If the crack horizontal spacing is greater than six times the major semiaxis of the main crack, the interference between parallel collinear cracks and parallel offset cracks is negligible, and the analysis of multiple cracks can be simplified to a single crack in the process [20]. Based on the previous research work of the research group [24,25], this paper carried out the research on the interference problem of unequal-length cracks in the weld seam of X80 oil and gas pipeline, and the research results can provide theoretical basis for guiding the actual safety assessment of X80 pipeline with multiple cracks.

Unequal-Length Cracks Interference Method
The flow chart of the established unequal length crack interference method is shown in Figure 1. Step 1: Establish the unequal-length crack model of the X80 oil and gas pipeline weld, and set the geometric dimensions, material parameters, load parameters, and other information of the pipeline weld. Geometric dimensions and material parameters are shown in Table 1 and Figures 2 and 3 in the previous study [24] and Tables 1 and 2 in the previous study [25]. The initial fluid pressure in the pipeline is 1 MPa; the maximum fluid pressure is 30 MPa. The model properties are shown in Table 1 of the paper.
Step 2: Set information such as the position of the unequal-length crack, the initial length of the crack, and the initial distance of the crack tip.
Step 3: Based on VCCT technology; discretize the prepropagation path into INTER202 interface units; select TARGE169 as the target unit at the interface unit; select CONTA171 as the contact unit; and create a contact relationship.
Step 4: Apply load and displacement boundary conditions; constrain the degrees of freedom in the x-direction and y-direction at 0 • and 180 • in the circumferential direction of the pipeline; constrain the degree of freedom in the x-direction at 270 • in the circumferential direction of the pipeline.
Step 5: Based on the energy release rate criterion, carry out the unequal-length crack propagation calculation and extract the first characteristic quantity during crack propagation process, the fluid pressure load P in the pipeline, and the second characteristic quantity, the von Mises stress distribution field peak value, of the crack propagation path.
Step 6: Reconstruct the calculation domain around the unequal length crack growth. The specific implementation process is to extract the propagation result of each load step during the crack propagation process, update the node coordinates according to the deformation during the process, and perform mesh reconstruction around cracks with unequal lengths.
Step 7: According to the structural characteristics of the pipeline weld, construct the calculation domain of the pipeline external excitation structure of the permanent magnet, armature, and pole shoe.
Step 8: With the dynamic application of the fluid pressure, the crack initiation pressure is reached, and the crack is initiated. The fluid pressure continues to be dynamically increased, which accelerates the crack propagation process, causes the deformation of the pipeline weld structure, and leads to the change of the leakage magnetic field distribution. In the multifield coupling calculation process, the difference ∆Bx between the peak value of the magnetic induction intensity component at the crack and the value of its symmetrical position of the third characteristic quantity in the process of crack propagation is extracted. In the process of unequal-length crack propagation, the incremental crack propagation of each load step is completed, and the excitation structure model is established based on the reconstructed mesh. The loop iteratively calculates the multifield coupling of crack propagation until the fracture condition is reached.

Simulation Study of Unequal-Length Cracks Interference
Based on the unequal-length crack interference method above, the simulation research of unequal-length crack interference was carried out, and the characteristic quantity fluid pressure load P in the process of crack propagation, the peak value of von Mises stress distribution field in the crack propagation path, the difference value ∆Bx between the peak value of magnetic induction component at the crack, and the value at its symmetric position were compared and analyzed. On this basis, the crack interaction ratio factor was introduced to study the influence of crack spacing on crack interference and crack size on crack interference.

Interference Phenomenon of Unequal Length Cracks
For the problem of ferromagnetic pipeline welds with double cracks, in order to study the interference effect of unequal-length double cracks, a model of unequal-length double cracks in pipeline welds was established, as shown in Figure 2a.

Simulation Study of Unequal-Length Cracks Interference
Based on the unequal-length crack interference method above, the simulation research of unequal-length crack interference was carried out, and the characteristic quantity fluid pressure load P in the process of crack propagation, the peak value of von Mises stress distribution field in the crack propagation path, the difference value ∆Bx between the peak value of magnetic induction component at the crack, and the value at its symmetric position were compared and analyzed. On this basis, the crack interaction ratio factor was introduced to study the influence of crack spacing on crack interference and crack size on crack interference.

Interference Phenomenon of Unequal Length Cracks
For the problem of ferromagnetic pipeline welds with double cracks, in order to study the interference effect of unequal-length double cracks, a model of unequal-length double cracks in pipeline welds was established, as shown in Figure 2a. The double crack consists of a crack in the outer wall of the pipeline weld and a crack in the inside of the pipeline weld. In order to compare the interference effect, a single crack model on the outer wall of the pipeline weld was established at the same position, as shown in Figure 2b, and another single crack model inside the pipeline weld was established, as shown in Figure 2c. Three models of cracks are set in Figure 2, with the yellow lines representing cracks. The two unequal-length cracks in Figure 2a are located at the same position and have the same length as the single cracks in Figure 2b,c, that is, the initial length of the outer wall crack is l o = l o = 2 mm, and the initial length of the internal crack is l i = l i = 4 mm, and the initial distance between the crack tips of two unequal-length cracks is s = 2 mm. In Figure 2a, the crack tips are indicated by T 1 , T 2 , and T 3 ; in Figure 2b, the crack tips are indicated by T 1 ; and in Figure 2c, the crack tips are indicated by T 2 and T 3 . EN represents the finite element, EN T1 represents the number of finite elements extended by the crack tip T 1 , and so on. When meshing, the discrete size of the unit on the crack propagation path was 0.25 mm.
The von Mises stress during the propagation process of the outer wall single crack (p = 20.7534 MPa), the internal single crack (p = 21.3534 MPa), and the unequal-length double cracks (p = 18.3534 MPa) in the loading step before the crack propagation were extracted. Their distribution cloud maps are shown in Figures 3a, 4a and 5a, and the von Mises stress distribution fields of their corresponding crack propagation path are shown in Figures 3b, 4b and 5b. It can be seen from Figure 3 that the peak value of the von Mises stress distribution field of the crack propagation path is located at 2 mm from the outer wall, which is at the crack tip T 1 of the single crack on the outer wall. Additionally, it can be seen from Figure 4 that the two peaks of the von Mises stress distribution field of the crack propagation path are located at 4 mm and 8 mm from the outer wall, which are the crack tips T 2 and T 3 of the internal single crack. In Figure 5, it appears that the three peaks of the von Mises stress distribution field of the crack propagation path are located at 2 mm, 4 mm, and 8 mm from the outer wall, which are the double crack tips T 1 , T 2 , and T 3 . From Figures 3-5, it can be seen that the place where the von Mises stress is the largest is at the crack tip; the distance between the peak value of the von Mises stress distribution field and the internal or outer wall of the pipeline is consistent with the distance from the crack tip to the internal or outer wall of the pipeline. Therefore, the position of the single crack or double crack tips can be judged from the peak position of the von Mises stress distribution field.
In Figure 2a, the crack tips are indicated by T1, T2, and T3; in Figure 2b, the cra indicated by T1′; and in Figure 2c, the crack tips are indicated by T2′ and T3′. EN the finite element, ENT1 represents the number of finite elements extended by th T1, and so on. When meshing, the discrete size of the unit on the crack propag was 0.25 mm. . It can be seen from Figure 3 that the peak value of the von Mises st bution field of the crack propagation path is located at 2 mm from the outer w crack tip; the distance between the peak value of the von Mises stress distributio and the internal or outer wall of the pipeline is consistent with the distance from the tip to the internal or outer wall of the pipeline. Therefore, the position of the single or double crack tips can be judged from the peak position of the von Mises stress bution field.    Figure 6 shows the required pressure load for the crack tips corresponding to the outer wall single crack, internal single crack, and double crack to extend by one finite element (EN), respectively. It can be seen from the figure that due to the interference effect of the internal crack, the pressure load required for the outer wall crack tip T 1 to extend by 1 EN in the double cracks is reduced by 21.0534 − 18.6534 = 2.4 MPa compared to the pressure load required for the outer wall single crack tip T 1 to extend by 1 EN. Similarly, due to the interference effect of the outer crack, the pressure load required for the internal crack tip T 2 to extend by 1 EN is reduced by 21.6534 − 18.8715 = 2.7819 MPa compared to the pressure load required for the internal single crack tip T 2 to extend by 1 EN. Additionally, the pressure load required for the crack tip T 3 to extend by 1 EN in the double cracks is reduced by 21.8775 − 19.0113 = 2.8662 MPa compared to the pressure load required for the internal single crack tip T 3 to extend by 1 EN. Judging from the pressure load P required for crack propagation, the interference effect between the double cracks intensifies the crack propagation process compared with a single-crack case.
11.002mm     Figure 6 shows the required pressure load for the crack tips corresponding to outer wall single crack, internal single crack, and double crack to extend by one fin element (EN), respectively. It can be seen from the figure that due to the interference eff of the internal crack, the pressure load required for the outer wall crack tip T1 to exte by 1 EN in the double cracks is reduced by 21.0534-18.6534 = 2.4 MPa compared to pressure load required for the outer wall single crack tip T1′ to extend by 1 EN. Similar due to the interference effect of the outer crack, the pressure load required for the inter crack tip T2 to extend by 1 EN is reduced by 21.6534-18.8715 = 2.7819 MPa compared the pressure load required for the internal single crack tip T2′ to extend by 1 EN. Additi ally, the pressure load required for the crack tip T3 to extend by 1 EN in the double cra is reduced by 21.8775-19.0113 = 2.8662 MPa compared to the pressure load required the internal single crack tip T3′ to extend by 1 EN. Judging from the pressure load P quired for crack propagation, the interference effect between the double cracks intensif the crack propagation process compared with a single-crack case.  Figure 6 shows the required pressure load for the crack tips corresponding to the outer wall single crack, internal single crack, and double crack to extend by one finite element (EN), respectively. It can be seen from the figure that due to the interference effec of the internal crack, the pressure load required for the outer wall crack tip T1 to extend by 1 EN in the double cracks is reduced by 21.0534-18.6534 = 2.4 MPa compared to the pressure load required for the outer wall single crack tip T1′ to extend by 1 EN. Similarly due to the interference effect of the outer crack, the pressure load required for the interna crack tip T2 to extend by 1 EN is reduced by 21.6534-18.8715 = 2.7819 MPa compared to the pressure load required for the internal single crack tip T2′ to extend by 1 EN. Addition ally, the pressure load required for the crack tip T3 to extend by 1 EN in the double cracks is reduced by 21.8775-19.0113 = 2.8662 MPa compared to the pressure load required for the internal single crack tip T3′ to extend by 1 EN. Judging from the pressure load P re quired for crack propagation, the interference effect between the double cracks intensifies the crack propagation process compared with a single-crack case. The von Mises stress distribution field of the crack propagation paths for the outer wall single crack, internal single crack, and unequal-length double cracks under the same Mises stress at the outer wall single crack tip T1′. Because of the interference effect o outer wall crack on the internal crack, the von Mises stress peak value of the crack t in the double cracks increases by 817.03-691.5 = 125.53 MPa compared to the peak Mises stress at the internal single crack tip T2′. The peak von Mises stress at the crac T3 increases by 782.97-693.4 = 89.57 MPa compared to the peak von Mises stress a internal single crack tip T3′. Judging from the von Mises stress at the crack tip, comp with the single crack, the interference effect between the double cracks intensifies crack propagation process.  Figure 8. S the numerical model only establishes the relevant cracks on the right side, and the p ous research results show that the magnetic induction intensity curve is symmet when there is no crack in the pipeline weld, the characteristic quantity ∆Bx is define the difference between the peak value of the magnetic induction intensity compone the crack and its symmetrical position [24,25]. In this study, there are, in total, three characteristic quantities, with ∆Bxo p representing the outer wall single crack, ∆Bxi p fo internal single crack, and ∆Bxd p for the double cracks. In  The component curves of magnetic induction intensity for the outer wall single crack, internal single crack, and unequal-length double cracks under the same pressure load p = 18.3534 MPa without any propagation were extracted and are shown in Figure 8. Since the numerical model only establishes the relevant cracks on the right side, and the previous research results show that the magnetic induction intensity curve is symmetrical when there is no crack in the pipeline weld, the characteristic quantity ∆Bx is defined as the difference between the peak value of the magnetic induction intensity component at the crack and its symmetrical position [24,25]. In this study, there are, in total, three ∆Bx characteristic quantities, with ∆Bx o p representing the outer wall single crack, ∆Bx i p for the internal single crack, and ∆Bx d p for the double cracks. In Figure 8, ∆Bx o p = 0.0065 T, ∆Bx i p = 0.0025 T, ∆Bx d p = 0.0246 T. The interference effect between the double cracks leads to the superposition of the leakage magnetic field of the outer wall crack with that of the internal crack. Furthermore, it leads to the detected peak value of the magnetic induction intensity component of the double cracks increasing by ∆Bx d p − ∆Bx o p = 0.0181 T compared to the peak value of the magnetic induction intensity component of the single crack in the outer wall. So, the detected peak value of the magnetic induction intensity component of the double cracks increases by ∆Bx d p − ∆Bx i p = 0.0221 T compared to the peak value of the magnetic induction intensity component of the internal single crack. Judging from the peak value of the magnetic induction intensity component, compared with the single crack, the interference effect between the double cracks intensifies the crack propagation process. It can be seen that when the double cracks interfere with each other, the cracks are more likely to propagate toward each other.
Materials 2023, 16, x FOR PEER REVIEW 12 can be seen that when the double cracks interfere with each other, the cracks are m likely to propagate toward each other. Since the peak position of the von Mises stress distribution field is at the crack the position and number of cracks can be judged. Extracting s = 2 mm during the c propagation process of the double crack in the Figure 2a above, the von Mises stress tribution field of the crack tip T1 extending by one, two, three, and four finite elem that is, the von Mises stress distribution fields of ENT1 = 1, ENT1 = 2, ENT1 = 3, and ENT are depicted in Figures 9a, 10a, 11a and 12a. In Figures 9b, 10b, 11b and 12b, the leng the outer wall crack is expressed by lo e , and the length of the internal crack is expresse li e . The distance between the crack tip T1 and the crack tip T2 is denoted by s e . Grid re structions are shown in Figures 9c, 10c, 11c and 12c. It is seen from Figures 9-12 th the crack propagates, the peak position of the von Mises stress distribution field changes, and there is a one-to-one correspondence. For example, in Figure 12b, when = 4 and ENT2 = 3, the distance between the crack tip T1 and the crack tip T2 is a finite elem s e = 0.25 mm, while in Figure 12a Figure 13 crack propagation process and position can be monitored more clearly. Figure 14a-d s the magnetic induction intensity nephograms when extracting crack tip T1 extendin one, two, three, and four finite elements, and Figure 14e shows the magnetic indu intensity component curves when extracting crack tip T1 extending by one, two, three four finite elements. With the increase in the crack propagation process, the greate peak value of the von Mises stress at the crack tip, the greater the deformation at the c position, which affects the magnetic field distribution of the pipeline weld during crack propagation process, so the peak value of the detected magnetic induction inte component shows an increasing trend. Since the peak position of the von Mises stress distribution field is at the crack tip, the position and number of cracks can be judged. Extracting s = 2 mm during the crack propagation process of the double crack in the Figure 2a above, the von Mises stress distribution field of the crack tip T 1 extending by one, two, three, and four finite elements, that is, the von Mises stress distribution fields of EN T1 = 1, EN T1 = 2, EN T1 = 3, and EN T1 = 4, are depicted in Figures 9a, 10a, 11a and 12a. In Figures 9b, 10b, 11b and 12b, the length of the outer wall crack is expressed by l o e , and the length of the internal crack is expressed by l i e . The distance between the crack tip T 1 and the crack tip T 2 is denoted by s e . Grid reconstructions are shown in Figures 9c, 10c, 11c and 12c. It is seen from Figures 9-12 that as the crack propagates, the peak position of the von Mises stress distribution field also changes, and there is a one-to-one correspondence. For example, in Figure 12b, when EN T1 = 4 and EN T2 = 3, the distance between the crack tip T 1 and the crack tip T 2 is a finite element, s e = 0.25 mm, while in Figure 12a Figure 13. This figure shows that when the crack tip begins to propagate, the von Mises stress value at the crack tip increases as the crack propagation process intensifies. From Figure 13, the crack propagation process and position can be monitored more clearly. Figure 14a-d show the magnetic induction intensity nephograms when extracting crack tip T 1 extending by one, two, three, and four finite elements, and Figure 14e shows the magnetic induction intensity component curves when extracting crack tip T 1 extending by one, two, three, and four finite elements. With the increase in the crack propagation process, the greater the peak value of the von Mises stress at the crack tip, the greater the deformation at the crack position, which affects the magnetic field distribution of the pipeline weld during the crack propagation process, so the peak value of the detected magnetic induction intensity component shows an increasing trend.

Effect of Crack Spacing on Crack Interference
In order to analyze the influence of crack spacing on crack propagation, finite element models with initial crack spacings s = 2 mm, 4 mm, 6 mm, and 8 mm were established, respectively. Figure 15a-c, respectively, show the model diagrams of the crack tip T 1 extending by four finite elements when s = 4 mm, s = 6 mm, and s = 8 mm. When s = 2 mm, the model of crack tip T 1 extending by four finite elements is depicted in Figure 12b.

Effect of Crack Spacing on Crack Interference
In order to analyze the influence of crack spacing on crack propagation, finite element models with initial crack spacings s = 2 mm, 4 mm, 6 mm, and 8 mm were established, respectively. Figure 15a-c, respectively, show the model diagrams of the crack tip T1 extending by four finite elements when s = 4 mm, s = 6 mm, and s = 8 mm. When s = 2 mm, the model of crack tip T1 extending by four finite elements is depicted in Figure 12b. For different distances, the fluid pressure required is calculated when the double crack tip T1 extends by one, two, three, and four finite elements. That is, at distances s = 2 mm, 4 mm, 6 mm, and 8 mm, respectively, the required fluid pressures are given in Table  2 and Figure 16 when ENT1 = 1, ENT1 = 2, ENT1 = 3, and ENT1 = 4. In Table 2 and Figure 16, to compare the influence of crack spacing on crack interference, the fluid pressures required For different distances, the fluid pressure required is calculated when the double crack tip T 1 extends by one, two, three, and four finite elements. That is, at distances s = 2 mm, 4 mm, 6 mm, and 8 mm, respectively, the required fluid pressures are given in Table 2 and Figure 16 when EN T1 = 1, EN T1 = 2, EN T1 = 3, and EN T1 = 4. In Table 2 and Figure 16, to compare the influence of crack spacing on crack interference, the fluid pressures required for the expansion of a single outer wall crack T 1 extending to one, two, three, and four finite elements are also listed. The interference effects at different spacing from the required pressures can also be found in Table 2 and Figure 16.  When the double-crack distances are s = 2 mm, 4 mm, 6 mm, and 8 mm, respectively, under the same pressure load without any propagation (taking p = 18.3534 MPa as an example), the von Mises stress distribution field is extracted, which is shown in Figure 17. To compare the interference effect with various crack spacing, the von Mises stress distribution field of the outer wall single crack is also listed. Figure 17a     Based on Table 2 and Figure 16, the required fluid pressure increases with the initial spacing s of the double cracks. From Figures 7 and 17, under the same pressure load, with the increase in the different initial distances between the double cracks, the peak value of the von Mises stress at the crack decreases. It can be seen from Figure 18 that with the increase in the different initial spacing s of the double cracks, the difference between the peak values of the detected magnetic induction intensity components of the double cracks and the values of their symmetrical positions decreases. In order to describe the effect of crack spacing on crack interference more conveniently, the crack interaction scale factors γ P , γ Mises , and γ ∆Bx are introduced. γ P is the ratio of the pressure required for the outer wall single crack to the pressure required for the double cracks when extending by the same number of finite elements; γ Mises is defined as the ratio of the peak value of von Mises stress at the crack tip T 1 of the double cracks to the peak value of the von Mises stress at the crack tip T 1 of the outer wall single crack under the same pressure load; γ ∆Bx is defined as the ratio of the peak value difference of the magnetic induction intensity component of the double crack to the peak value difference of the magnetic induction intensity component of the single crack in the outer wall under the same pressure load. The expressions are: where P o is the pressure required for the outer wall single crack, in units of MPa; P d is the pressure required for the double cracks, in units of MPa.
where Mises T1 is the peak von Mises stress of the crack tip T 1 of double cracks, in units of MPa; Mises T1 is the peak von Mises stress of the crack tip T 1 of the outer wall single crack, in units of MPa.
where ∆Bx d p represents the peak value difference of the magnetic induction intensity component of the double cracks, in units of T; ∆Bx o p is the peak value difference of the magnetic induction intensity component of the outer wall single crack, in units of T. Table 3 is obtained from Table 2; Table 4 is obtained from Figures 7 and 17. According to the extraction method of the characteristic quantity ∆Bx value in Figure 8, the value of ∆Bx d p when s = 2 mm, the value of ∆Bx o p of the outer wall single crack, and the value of ∆Bx d p when s = 4 mm, 6 mm, and 8 mm are extracted from Figure 18 and summarized in Table 5. It can be seen from Table 3 that when T 1 and T 1 are extended by the same number of finite elements, the value of γ o P decreases as the crack spacing increases. That is, the gap between the pressure required for the outer wall single crack and the pressure required for the double cracks decreases. Taking EN T1 = 3 and EN T1 = 3 for example, when s changes from 2 mm to 4 mm to 6 mm to 8 mm, γ o P changes from 1.1623 to 1.0918 to 1.0283 to 1.0036. It can be seen from Table 4 that under the same pressure load, as the crack spacing increases, the value of γ Mises decreases. That is, the difference between the peak von Mises stress of the crack tip T 1 of the double crack and the peak value of the von Mises stress of the crack tip T 1 of the outer wall single crack decreases. It can be seen from Table 5 that under the same pressure load, as the crack spacing increases, the value of γ ∆Bx decreases. That is, the peak difference between the magnetic induction intensity component of the double crack and the magnetic induction intensity component peak difference of the outer wall single crack decreases. As the crack spacing increases, γ o P , γ Mises , and γ ∆Bx decrease, and the interference effect between the double cracks becomes smaller and smaller. When the crack spacing s = 8 mm (double cracks model: the initial length of the outer wall crack l o = 2 mm, the initial length of the internal crack l i = 4 mm), the values of γ o P , γ Mises , and γ ∆Bx are close to 1, and the interference effect of the double cracks is negligible. The interference trend of the double cracks gradually weakens with the increasing tip distance and finally tends to the situation of the single crack. Thus, when conducting a safety assessment for multicrack oil and gas pipelines, it is possible to directly simplify the multicrack treatment to the single crack treatment without considering the interaction between cracks.

Effect of Crack Size on Crack Interference
Through the analysis in Section 3.1, compared with the single-crack case, the interference effect between double cracks intensifies the crack propagation process. Three characteristic values are defined as the pressure P required for crack propagation, the peak value of von Mises stress, and the difference ∆Bx between the peak value of the magnetic induction intensity component at the crack and the value of its symmetrical position. These characteristic values can be used to measure the progress of crack propagation. According to the analysis in Section 3.2, the double cracks' interference effect weakens as the crack tip distance increases. To further study the interference effect of the unequal-length double cracks, two sets of the above three characteristic quantities are established and extracted. For the first set, the initial crack length in the outer wall is l o = l o = 2 mm, the initial length of the internal crack is l i = l i = 6 mm, and the initial distance between the crack tips of two unequal-length cracks is s = 2 mm. For the second numerical example, the initial length of the outer wall crack is l o = l o = 2 mm, the initial length of the internal crack is l i = l i = 8 mm, and the initial distance between the crack tips of two unequal-length cracks is s = 2 mm. The model diagrams of the first calculation example are shown in Figure 19a (1), that is, when extending by the same number of finite elements, the pressure ratio required for the internal single cracks and the double cracks is expressed as: where P i is the pressure required for the internal single crack, in units of MPa, and P d is the pressure required for the double cracks, in units of MPa. The pressure P required to extract the characteristic quantity crack propagation process is listed in Tables 6 and 7. Table 8 is obtained from Formulas (1) and (4) and Tables 6 and 7.
where Pi is the pressure required for the internal single crack, in units of MPa, and Pd is the pressure required for the double cracks, in units of MPa. The pressure P required to extract the characteristic quantity crack propagation process is listed in Tables 6 and 7. Table 8 is obtained from Formulas (1) and (4) and Tables 6 and 7.   By comparing the interference scale factors γ o P and γ i P in Table 8, it can be seen from the first set of calculation examples (l o = l o = 2 mm, s = 2 mm, l i = l i = 6 mm) that when the crack tip extends by one finite element (EN T1 = 1, EN T1 = 1 or EN T2 = 1, EN T2 = 1 or EN T3 = 1, EN T3 = 1), due to the interference of the 6 mm crack on the 2 mm crack, γ o P = P o P d = 1.3197 (EN T1 = 1, EN T1 = 1), namely, the extension pressure P o required for the 2 mm single-crack case is 1.3197 times as large as the expansion pressure P d required for the 2 mm-and-6 mm double-crack case. Owing to the interference of the 2 mm crack on the 6 mm crack, γ i P = P i P d = 1.1742 (EN T2 = 1, EN T2 = 1) and γ i P = P i P d = 1.1805 (EN T3 = 1, EN T3 = 1), and the required propagation pressure P i for the 6 mm single-crack case is 1.1742 times (EN T2 = 1, EN T2 = 1) and 1.1805 times (EN T3 = 1, EN T3 = 1) as large as P d required for the 2 mm-and-6 mm double-crack case. Therefore, the change range of the crack tip propagation pressure of the 2 mm crack under the interference of the 6 mm crack is larger than that of the crack tip propagation pressure in the opposite case. In other words, the proportional factor γ o P > γ i P ; the interference effect of 6 mm crack on 2 mm crack is more severe than that of the 2 mm crack on the 6 mm crack. This phenomenon can still be observed when the crack tip extends by two finite elements.    1.1805 = 1.0281 (EN T3 = 1, EN T3 = 1) times as large as the interference factor γ i P = 1.1742 (EN T2 = 1, EN T2 = 1) and γ i P = 1.1805 (EN T3 = 1, EN T3 = 1) of the 2 mm crack on the 6 mm crack. The interference effect of the 2 mm crack on the 6 mm crack and 8 mm crack is not much different. The interference effect of large-size cracks such as 8 mm and 6 mm on small-size cracks such as 2 mm is significantly greater than the influence of the small-size crack such as 2 mm on large-size cracks such as 8 mm and 6 mm. The influence of the 8 mm crack on the 2 mm crack is much greater than that of the 6 mm crack on the 2 mm crack.
In Formula (5), the von Mises stress ratio is defined as, under the same pressure load, the ratio of the von Mises stress peak value at crack tip T 2 of the double crack to that at crack tip T 2 of the internal single crack. In Formula (6), the von Mises stress ratio is defined as, under the same pressure load, the ratio of the von Mises stress peak value at crack tip T 3 of the double crack to that at crack tip T 3 of the internal single crack.
Under the same pressure load, p = 15.6534 MPa is taken as the first example and p = 13.2534 MPa is taken as the second example. The von Mises stress distribution field is shown in Figures 20 and 21        In Tables 9 and 10, for the first example with l o = l o = 2 mm, s = 2 mm, and l i = l i = 6 mm, under the same pressure load p = 15.6534 MPa, the interference scale factors including γ 1 Mises , γ 2 Mises , and γ 3 Mises are compared. Owing to the interference of the 6 mm crack on the 2 mm crack, γ 1 Mises =1.3897, i.e., the von Mises stress peak value at crack tip T 1 for the double-crack case (2 mm and 6 mm, respectively) is 1.3897 times as large as the peak value at the crack tip T 1 when only the single 2 mm crack exists. Owing to the interference of the 2 mm crack on the 6 mm crack, γ 2 Mises =1.2028 and γ 3 Mises =1.1254, i.e., the von Mises stress peak value at the crack tip T 2 for the double crack case is 1.2028 times as large as the one at crack tip T 2 when only the single 6 mm crack exists. The von Mises stress peak value at the crack tip T 3 for the double crack case is 1.1254 times as large as the one at crack tip T 3 ' when only the single 6 mm crack exists. Therefore, the change amplitude of the von Mises stress peak at the crack tip under the interference of the 6 mm crack to the 2 mm crack is larger than that of the other way around. The scaling factor γ 1 Mises > γ 2 Mises and γ 1 Mises > γ 3 Mises , and the interference effect of 6 mm crack to 2 mm crack is stronger than that of the 2 mm crack to 6 mm crack. This phenomenon is more obviously observed in the second example, where l o = l o = 2 mm, s = 2 mm, and l i = l i = 8 mm. Under the same pressure load p = 13.2534 MPa, γ 1 Mises =1.4138, γ 2 Mises =1.2130, and γ 3 Mises =1.1262, that is, the proportional factor γ 1 Mises > γ 2 Mises and γ 1 Mises > γ 3 Mises , and the interference effect of the 8 mm crack to 2 mm crack is more severe than the other way around.

Conclusions
In the current investigation, a new numerical simulation methodology was developed to study the failure procedure of two interacting cracks in the welding zone of a pipeline. The influence of the space between the two unequal-length cracks as well as the crack sizes were analyzed. Some useful conclusions are drawn as follows: (1) In pipeline weld joints, cracks with different sizes exist and interfere with each other.
It is found that when the unequal length cracks interfere with each other, the cracks are more likely to propagate toward each other, and this trend gradually weakens with the increase in the tip distance, and finally tends to the condition that each crack exists separately. (2) The crack interaction scale factors (γ P , γ Mises , γ ∆Bx ) are introduced to gauge the interference intensity. Those scale factors are very user-friendly for practical applications in failure analysis and prevention. (3) By comparing the degree of mutual interference of two unequal length cracks, the influence of the larger crack on the smaller cracks is greater than that of the smaller one on the larger one. Funding: This research was funded by the Natural Science Foundation of Heilongjiang province, grant number LH2021E020.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.