Shale oil and gas development commonly encounters challenges such as low initial production, rapid production decline rates, and low recovery degree of single well ^{[1⇓⇓⇓⇓-6]}. In the context of horizontal well hydraulic fracturing with smaller cluster spacing as a predominant reservoir stimulation technique, the development of shale beddings and natural fractures (NFs), coupled with reservoir heterogeneity, presents challenges in achieving balanced initiation and propagation of fractures of densely multi- cluster perforation ^{[7⇓⇓⇓-11]}. Within this framework, the com-bination of intra-fracture temporary plugging (IFTP) and intra-segment temporary plugging (ISTP) emerges as a pivotal strategy for enhancing hydraulic fracture (HF) complexity and achieving more uniform distribution ^[12-13]. Therefore, elucidating the mechanisms governing the propagation and diversion of multi-cluster fractures from temporary plugging in horizontal well holds significant implications for optimizing the applicability of temporary plugging and diversion fracturing (TPDF).

In order to explore the propagation law of multi-fractures after TPDF, numerous scholars have conducted laboratory experiments and theoretical research. They have found that the effectiveness of temporary plugging is influenced by reservoir conditions (such as in-situ stress state, beddings/natural fractures, etc.) and operational parameters (such as temporary plugging agent (TPA) type, TPA particle size, TPA concentration, displacement, and fracturing fluid viscosity, etc.) ^{[14⇓⇓⇓⇓⇓-20]}. Zou et al. ^[21], Zhou et al. ^[22], and Chen et al. ^[23] suggested that the horizontal stress difference and inter-cluster stress difference are the primary controlling factors determining the diversion effects of HF by temporary plugging within fractures and within segments. The larger the horizontal stress difference or inter-cluster stress difference, the more difficult it is for fractures to divert, requiring more TPAs, earlier plugging timing and more plugging cycles. Tang et al. ^[24] indicated that the higher the density or the azimuth angle of NFs, the later the optimal timing for temporary plugging. Lu et al. ^[25] proposed that the efficiency of NF opening is influenced by the plugging strength and the approaching angle of NFs. Forming an effective plugging zone is crucial for inducing HF diversion. Guo et al. ^[26] proposed that the primary controlling factors affecting the formation mode of the plugging zone are the ratio of particle size to fracture width, and the displacement and viscosity of carrying fluid. Li et al. ^[27] pointed out that under the influence of TPA particle concentration, the plugging zone exhibits different bridging modes, with high concentration conditions tending towards dual-particle bridging and low concentration conditions tending towards single-particle bridging. Different diverted fractures are formed in different plugging positions. Li et al. ^[28] found that the larger the horizontal stress difference, the closer the initiation position to the fracture tip. Yuan et al. ^[29] proposed a method based on the normal width of the fracture to describe the characteristics of the plugging position, suggesting that the narrow region inside the fracture is the characteristic plugging position. Wang et al. ^[30] discovered that when the plugging occurs at the fracture tip, the diverted fracture is perpendicular to the main fracture; and when the plugging occurs in the middle of the fracture, the fracture morphologically exhibits a "S" shape. Hu et al. ^[31] found that the closer the plugging position is to the fracture mouth, the more balanced the overall propagation of multi-cluster fractures. Zhang et al. ^[32] suggested that the primary controlling factor influencing the plugging position and the propagation of diverted fractures is TPA concentration. Regarding the mechanism of temporary plugging for pressure buildup, Wang et al. ^[33] and Shi et al. ^[34] qualitatively pointed out that plugging at the fracture mouth changes the circumferential stress around the wellbore, plugging inside the fracture changes the closure stress of the fracture, and plugging at the fracture tip hinders the conduction of fluid stress and achieves fracture arrest.

The above-mentioned studies on TPDF simulations mainly focus on open-hole completions for vertical/horizontal wells ^{[16,32 -33,35⇓ -37]} or polyvinyl chloride casing perforation completions ^[38], which differ significantly from the on-site perforation completion methods. In addition, there is currently limited experimental research on the influence of the number of single-cluster perforations in horizontal wells on the propagation of multi-fractures, and there is still controversy over the primary controlling factors affecting the diversion effect of HFs in shale formation. This paper adopted the shale outcrop of Silurian Longmaxi Formation in the Sichuan Basin, SW China, to carry out true triaxial TPDF simulation experiment of horizontal well with multi-cluster sand jetting perforation, to study the effects of TPA particle size, TPA concentration, number of single-cluster perforations, and cluster numbers on the initiation, propagation, and diversion of multi-fractures. The aim is to provide a basis for the design of key parameters in TPDF processes for shale reservoirs with developed beddings.

The experimental rock samples were obtained from the outcrop of Silurian Longmaxi Formation shale in the Sichuan Basin (underground mine in Changning County). The mineral compositions of the shale are predominantly silica and clay minerals, with quartz content of 51.8%, clay mineral content of 32.8%, and carbonate and pyrite contents of 9.5% and 5.9%, respectively. The samples exhibit widespread development of beddings and NFs filled with pyrite and carbonate mineral (Fig. 1). The matrix porosity of the rock samples ranges from 4.4% to 5.1%, with strong anisotropy in permeability and mechanical parameters. The formation permeability parallel and perpendicular to the bedding direction are 0.000 236× 10⁻³ μm² and 0.000 014×10⁻³ μm², respectively. The elastic moduli parallel and perpendicular to the bedding direction are 49.1 GPa and 40.3 GPa, respectively, and the tensile strengths parallel and perpendicular to the bedding direction are 6.8 MPa and 9.2 MPa, respectively. The permeability of mineral-filled NFs is (0.001 22-0.010 63)× 10⁻³ μm², with tensile strengths ranging from 4.7 MPa to 7.2 MPa. Overall, bedding planes (BPs) and NFs serve as high-permeability mechanical weak surfaces.

The large shale outcrop was cut into 30 cm×30 cm×30 cm cubic rock samples along parallel and perpendicular bedding directions. To simulate horizontal well fracturing, holes with a diameter of 30 mm and a length of 26 cm were drilled at the center of the sample surface along the parallel bedding direction. Steel pipes with inner diameter of 22 mm and outer diameter of 27 mm were selected to simulate casing, and threads were added to the outer wall to enhance bonding strength. High- strength epoxy resin was then used to bond the steel pipes to the hole walls. Subsequently, the perforation gun of the hydraulic sand jetting perforation equipment was inserted into the wellbore to the predetermined depth. The perforation gun was rotated to align the nozzle with the desired perforation direction, fixed in place, and the high-pressure sand-carrying pump was turned on to complete the perforation (Fig. 2). After the sand-carrying pump was turned off, the perforation gun was rotated again to complete the remaining perforations in the cluster. Starting from the cluster closest to the toe, the perforation gun was gradually lifted step by step to complete the perforation of all clusters. The duration of sand-carrying for each individual perforation process (the perforation equipment allows for manual control of whether it carries sand, in which the jet cannot penetrate the wellbore or the rock matrix without carrying sand; The perforation is effective only during the sand-carrying period) was kept constant at 1 min to ensure uniformity of the perforation hole sizes (diameter of 4 mm, depth of 10 mm). Multiple clusters with different numbers of perforations or different phases of perforation were completed using the above method (Fig. 3).

The experiment utilized an integrated true triaxial sand fracturing simulation device ^[39-40]. Key experimental parameters were determined based on the performance of the experimental equipment and the similarity criteria, including three-axis stress (maximum horizontal principal stress σ_H, minimum horizontal principal stress σ_h and vertical stress σ_v), displacement, and fracturing fluid viscosity, etc. To make the experimental results closer to reality, the actual in-situ stress conditions of a certain block in the Longmaxi Formation shale reservoir were included in the experiment ^{[39,41 -42]}, which follows a stress mechanism of strike-slip reverse faulting (σ_H>σ_v>σ_h). Additionally, the relative values of 3D in-situ stress in the reservoir were taken into account, i.e., a vertical stress difference of 10 MPa and horizontal principal stress difference of 15 MPa, to set the experimental vertical stress at 20 MPa, maximum horizontal principal stress at 25 MPa, and minimum horizontal principal stress at 10 MPa. To simulate the on-site process of viscous-dominated fracture propagation, the main injection parameters were calculated using similarity criteria ^{[43⇓⇓-46]}, as shown in Eqs. (1) and (2). Considering on-site displacements ranging from 14 m³/min to 18 m³/min, fracturing fluid viscosity from 5 mPa·s to 10 mPa·s, and fracture propagation characteristic radius from 15 m to 25 m (i.e., assessing half-fracture height), while the experimental fracture propagation characteristic radius was approximately 0.15 m (i.e., half the length of the rock sample), the experimental displacement was calculated to be approximately 200 mL/min.

According to the geometric similarity criteria ^[47-48], considering an on-site cluster spacing of 5 m to 15 m and a fracture half-length of 100 m to 150 m, with the experimental fracture half-length approximately 0.15 m (half the length of the rock sample), the experimental cluster spacing can be calculated to be approximately 2 cm to 6 cm using Eq. (3). In this study, the fixed-face perforation method was adopted ^[49-50], and based on an experimental horizontal well section length of 26 cm, the number of perforation clusters per segment was set in the range of 3-7.

The main experimental steps are as follows: (1) The rock sample is placed in a triaxial core chamber, and a variable frequency loading method is employed. The minimum horizontal principal stress, maximum horizontal principal stress, and vertical stress are successively applied along the X, Y and Z axes using a triaxial hydraulic servo system until reaching predetermined values, which are then maintained stable. (2) The fracturing process is divided into three phases: conventional fracturing (CF) phase (pre-plugging phase), IFTP phase, and ISTP phase. Based on previous experimental experience, fine-grained TPAs tend to migrate into the fracture to form plugs, while coarse-grained TPAs tend to plug at the fracture mouth or perforation orifice. Therefore, fine-grained TPAs are mainly used in the test of the IFTP phase, while coarse-grained TPAs are mainly used in the test of the ISTP phase. Fracture generation during each fracturing phase is identified based on the distribution of different colored tracers combined with acoustic emission localization. (3) Micro-CT scanner is used to scan the grayscale images of the rock sample. The rock core is reconstructed using high-precision CT data. Combined with the tracer distribution and rock sample dissection results, the HFs, opened NFs and beddings, and proppant/TPA distribution inside the rock sample were comprehensively identified and analyzed.

In the CF phase, the proppants sized between 75 μm and 106 μm (140-200 mesh) were adopted to maintain a certain width of HFs. The experiment disregarded the process of TPAs migrating from the perforation orifice to the point with critical fracture width of the bridge. It focused on the possibility of TPAs forming a plugging zone at the critical width of the fracture. To mitigate potential experimental biases arising from differences in TPA particle size and performance, on-site TPAs were utilized in the experiment, primarily including small- sized TPAs ranging from 125 μm to 180 μm (80-120 mesh) and 180 μm to 850 μm (20-80 mesh), as well as large-sized TPAs ranging from 1 mm to 2 mm and 1 mm to 3 mm.

A total of 13 sets of experiments were designed. Among them, samples 1# to 5#, 6# to 8#, and 9# to 11# were respectively utilized to analyze the effects of TPA particle size, TPA mass concentration, and the number of single-cluster perforations on the initiation, propagation and diversion of multi-cluster fractures. Samples 7#, 12#, and 13# were used to analyze the impact of cluster numbers. The specific plans are detailed in Table 1.

The influence of TPA particle size, TPA mass concentration, single-cluster perforation number, and cluster number on the TPDF effect of multi-cluster sand jetting perforation was evaluated with the pressure increase amplitude, HF quantity and HF complexity after TPDF as indicators. Based on the post-fracturing HF occurrence of rock samples, three representative HF orientations can be summarized: transverse fractures extending perpendicular to the wellbore, longitudinal fractures extending along the wellbore axis, and horizontal fractures extending along beddings (Fig. 4). When encountering beddings or NFs, HFs exhibit various behaviors, including penetration, deviation and termination. During CF phase, HFs tend to initiate in low-strength beddings near the wellbore, resulting in difficulties in subsequent fracture penetration and forming simple fracture morphology overall (as seen in samples 5# and 10#). However, when weak BPs are distributed away from the wellbore or NFs are present, HFs can propagate through beddings or NFs, forming complex branched fractures (as observed in samples 1#, 3# and 6#).

To further investigate the HF formation patterns of TPDF under multi-cluster perforation, typical rock samples 7#, 8#, and 12# were selected. Cylindrical rock samples with a diameter of 12 cm and a height of 26 cm were drilled with the wellbore as the center. Three-dimensional CT scans of near-wellbore fracture morphology and TPA distribution were conducted, as shown in Fig. 14. The laboratory experiments were conducted at a much smaller scale compared with field conditions. Due to constraints imposed by the wellbore and fracture dimensions, the experiments demonstrated superior plugging effects compared with field observations, which showed that the TPAs for IFTP were easily distributed widely within the fracture, and the TPAs for ISTP tended to accumulate in the wellbore near the perforation hole. Specifically, during the IFTP phase, fine-grained TPAs primarily entered the fractures through the initiated perforation holes and were concentrated at the intersections of fractures. During the ISTP phase, coarse-grained TPAs exhibited a preference for distribution near the perforation cluster holes corresponding to complex fracture regions near the wellbore, and additionally tended to accumulate at the toe of the wellbore.

The fracture plugging situations were analyzed based on fracture morphology and TPAs distribution. It was found that in sample 7#, the volume of distributed TPAs reached 10.1 cm³, forming a complex network of intersecting HFs near the concentrated distribution area; in sample 8#, the volume of distributed TPAs reached 10.3 cm³, forming a complex network of HFs near the S1 surface of the wellbore; while in sample 12#, the volume of distributed TPAs was only 1.3 cm³, with 3 longitudinal fractures propagating in parallel, exhibiting a single morphology. Overall, the larger distributed volume of TPAs (samples 7# and 8#) indicates effective migration of the TPAs into the fractures, thereby achieving a higher probability of plugging; whereas the smaller distributed volume (sample 12#) suggests ineffective migration of the TPAs into the fractures, resulting in a lower probability of effective plugging. On the other hand, the intricate intersecting pre-existing HFs (samples 7# and 8#) provide more favorable plugging positions for the TPAs, leading to fracture diversion to form a more complex HF network; whereas in samples with simpler pre-existing fractures and smoother fracture surfaces (sample 12#), the TPA tends to migrate towards the rock surface, making it difficult to form a stable plugging zone to promote fracture diversion.

Through observation of the accumulation trends of TPAs inside the wellbore, the plugging conditions within the segment were analyzed as follows: In sample 7#, TPAs were sparsely distributed near the toe, plugging all opened perforations of the second cluster and the opened perforations in the right side of the first cluster. In sample 8#, TPAs were heavily accumulated inside the wellbore, plugging all perforations. In sample 12#, TPAs were widely distributed at the toe of the wellbore, with a smaller amount distributed in the middle of the wellbore, plugging the opened perforations of the first cluster and the third cluster. Overall, the distribution of TPAs near initiated perforation clusters indicates effective plugging, while heavy accumulation near unopened perforation clusters indicates ineffective plugging. Additionally, in the case with fewer clusters (sample 12#), even if all perforation clusters are opened, the amount of TPAs entering the fractures is limited. Under laboratory conditions, simply increasing the concentration of TPAs (medium concentration in sample 7#, high concentration in sample 8#) did not significantly increase the amount of TPAs entering the fractures; excess TPAs only accumulated inside the wellbore, leading to ineffective plugging.

The peak pressures of both IFTP and ISTP phases are higher than the breaking pressure of CF phase, and their variations show consistency. The frequent and significant fluctuations in pressure during the plugging phases are the response of complex fractures, accompanied by dense acoustic emission events.

Using a combination of small particle size TPAs for IFTP and large particle size TPAs for ISTP is more conducive to increasing the plugging pressure, promoting the multi-stage diversion of HFs, and achieving better plugging effects. Adding fibers to granular TPAs can rapidly increase the pressure to ultra-high levels, but the effect of plugging the HFs is poor. When the concentration of TPAs increases to a certain value, the maximum plugging peak pressure can be reached, but further increasing the concentration of TPAs does not significantly increase the plugging peak pressure. Under conditions where the particle size of TPAs, the number of perforations per cluster, and the number of clusters are constant, there exists an optimal TPAs concentration that matches these parameters. The breaking pressure and plugging peak pressure decrease with an increase in the number of perforations per cluster. A lower number of perforations per cluster are more conducive to increasing the breaking pressure and plugging peak pressure, especially in controlling the propagation of multi-cluster fractures. A lower number of clusters hinder the increase in the total number and complexity of artificial fractures, while a higher number of clusters make effective plugging difficult to achieve.

The transport of TPAs within fractures is a complex process that significantly influences the effectiveness of plugging. TPAs within fractures are primarily distributed in the complex fracture areas where fractures intersect, while those within the segment are preferentially distributed near the perforation with multiple clusters corresponding to the complex fracture areas near the wellbore. To more accurately describe the distribution patterns of TPAs, it is necessary to further explore visualization methods.

E°—elastic modulus of plane strain, Pa;

K°—corrected fracture toughness, Pa·m^1/2;

L—half-length of the fracture, m;

Q—pumping displacement, m³/s;

R_max—characteristic radius of fracture propagation, m;

S—cluster spacing, m;

t_max—characteristic time of fracture propagation, s;

X, Y, Z—Cartesian coordinate system, m;

α, β—similarity coefficient;

μ—viscosity of fracturing fluid, mPa·s;

σ_h—minimum horizontal principal stress, MPa;

σ_H—maximum horizontal principal stress, MPa;

σ_v—vertical stress, MPa.

l—laboratory parameters;

f—field parameters.