1. Introduction
The critical energy scenario in China necessitates the urgent augmentation of unconventional hydrocarbon reserves, including gas hydrates [
1]. In response to the rational and efficient extraction of tight reservoirs, the advent of horizontal well fracturing technology was facilitated by hydraulic fracturing [
2]. In 2010, M. J. Mayerhofer proposed the Stimulated Reservoir Volume (SRV) concept while studying the microseismic technology and fracture changes in Barnett shale [
3]. As the injection of fracturing fluid continues, the net pressure of the primary fracture continually increases. When this pressure surpasses the difference in the principal horizontal stress in the in situ stress and the tensile strength of the rock, secondary fractures will form [
3,
4]. During the segmented fracturing process of horizontal wells, the already-formed hydraulic fractures introduce a stress shadow effect, also known as induced stress. This stress shadow effect increases the subsequent perforation fracturing pressure, altering the original geo-stress state and the fracture expansion direction. The fracture formation process exhibits the “3S” effect of stress accumulation, stress transfer, and stress shadow [
5,
6]. In the 1980s, Warpinski and Teufel began studying the interaction between fractures, providing an analytical solution for the stress field of a hydraulic fracture with constant height and finite length [
7]. In 2004, Fisher confirmed this phenomenon through microseismic mine research and explained the stress shadow effect using microseismic detection methods. He pointed out that when hydraulic fractures interact with natural fractures, various forms such as crossing, termination, and offset occur, with the impact range of the stress shadow effect being 1.5 times the height of the fracture [
8,
9,
10,
11,
12]. Numerous studies have been conducted on the topic commonly referred to as the stress shadow area. A literature review was also undertaken on this subject, detailing the stress disturbances generated in the zone between fractures, confirmed by applying microseismic technology [
12,
13,
14]. Hence, the stress shadow phenomenon is the principal element of forming an extensive, intricate fracture network, thereby increasing complexity within the shale reservoir [
15,
16]. Roussel and Sharma pointed out that a preceding hydraulic fracture would alter the stress field distribution of subsequent fractures within a specific fracture spacing, causing the fracture to turn and become tangent to the primary fracture [
14,
15,
17,
18]. Negal simulated the stress shadow effect of multiple fractures under varying reservoir conditions, spacing, and in situ stress ratios using continuous and discrete elements [
19]. Therefore, deciphering the intrinsic mechanism of the stress shadow effect and utilizing it to enhance the fractures’ complexity and connectivity are paramount.
The primary objective of volumetric hydraulic fracturing in large-scale shale reservoirs is to facilitate the communication between artificial and natural fractures, thereby establishing a three-dimensional fracture network. This has been confirmed by means of micro-seismic measurements and surface tilt-meter measurements [
7,
9,
10,
11,
20]. Numerous studies have scrutinized the interplay between hydraulic fractures and pre-existing fractures. Blanton conducted a series of experiments exploring the impact of the intersection angle between hydraulic and natural fractures and the differential stress on the state of fracture crossing [
21]. Gu and others proposed a simple standard for the intersection extension of hydraulic fractures and frictional natural fractures based on linear elastic fracture mechanics theory, verified through triaxial hydraulic fracturing experiments [
22,
23]. Olson and Bahorich replaced rocks with gypsum blocks for hydraulic fracturing experiments, studying the crossing process of hydraulic fractures and bonded natural fractures [
24,
25]. Should there exist an abundance of natural fractures, they would be prioritized as the principal determinant in altering the trajectory of the fractures in contrast to other influences.
Various models and simulators that consider the coupling of multiple physical fields have been suggested to emulate the process of hydraulic fracturing [
26]. Although some new methods have been developed, such as the phase field method [
27] and the peri-dynamic method [
28], methods including the finite difference (FD) method [
29], finite element (FE) method [
30], the extended finite element method (XFEM) [
31,
32], the displacement discontinuity method (DDM) [
33], and the discrete element method (DEM) [
34], are still standard and effective methods for hydraulic fracturing simulation. Fu [
35] suggested a linked hydro-mechanical model utilizing the finite element and finite volume method to address the geomechanics and hydrodynamics issue. Riahi [
36] proposed a model using DEM to anticipate the progression of intricate fractures on diverse natural fracture patterns. Coupled thermal–hydro–mechanical (THM) FEM techniques were implemented to simulate the flow of fluid and heat reaction in fracture problems [
37,
38]. Li [
39] constructed a thoroughly integrated three-dimensional thermo–hydro–mechanical (THM) hydraulic fracturing (HF) model to optimize multi-well completion design, utilizing a hybridized finite volume/finite element approach, contemplating both HF propagation and natural reactivation. Owen [
40] presented a unified finite element–discrete element (FE-DE) methodology to address large-scale pragmatic issues, encompassing fracture solids and particulate media. This synergized finite/discrete element approach offers a more accurate resolution in the context of hydraulic fracturing challenges. Traditional methods, such as the continuum FEM models, are dependent only on the combination of damage and the tensile model to capture porous flow; the integration of the proppant transport model presents challenges when executing a pure continuum FEM [
41]. The discrete element approach often assumes that the edge of an element pre-determines the direction of fracture propagation, thereby precluding the formation of genuinely complex fractures that are frequently observed in highly heterogeneous reservoirs [
42,
43]. In addressing the difficulties associated with the traditional FE and DE methods, this study employs an adaptive FE-DE method that dynamically updates geometries and meshes to enhance fracture propagation modeling, providing more accurate stress analyses and reliable fracture paths [
44].
The conventional metric utilized for assessing the efficacy of multistage fracturing is typically the volume of the hydraulic fracture [
39]. Indeed, the most credible gauge to appraise the fracturing outcome is the volume of gas generated post-hydro-fracturing, flow-back, and a series of procedures. However, few techniques can accurately portray the dynamics inherent in the exhaustive procedure of multistage hydro-fracturing and gas generation. There remain substantial challenges in evaluating the fracturing impact based on gas production or recovery rate. In this investigation, we employ the adaptive FE-DE technique incorporated in the ELFEN TGR software, utilizing local re-meshing technology to enhance precision. To juxtapose the impact of the stress shadow effect with the natural fracture, we simulated two scenarios employing two fracturing sequences, namely, sequential and simultaneous fracturing, adjusting the perforation cluster space from 12.5 m to 100 m.
3. Numerical Models and Procedure
As demonstrated in
Figure 3a, a 2D model of multistage fracturing was utilized, extending across a 1000 m length and 600 m height and encompassing five clusters. The initial perforation fracture lengths are approximately 2 m near the horizontal well. The inter-cluster spacing ‘s’ was systematically varied from 12.5 m to 100 m to appraise the stress interference derived from neighboring perforation clusters. The target reservoir lies at a perpendicular depth of 3000 m, situated within an environment featuring normal crustal stress and a moderate temperature of 45 °C. The horizontal in situ stress
in the x direction and vertical in situ stress
in the y direction are 40 MPa and 44 MPa, respectively. To authenticate the influence of natural fractures on the propagation of fractures within the reservoir, we incorporated two types of natural fractures near the perforation clusters in the model. This was done to simulate the stress variations typically seen among pre-existing fractures. Such a simulation aims to understand better the behavior of fracture propagation under realistic conditions, where natural fractures within the reservoir can significantly impact the direction and extent of new fractures created during the fracturing process. As evidenced by
Figure 3b, the realm of the pre-existing natural fracture is centrally situated within the reservoir, measuring 500 m in length and 200 m in height. The size and location of this pre-existing natural fracture zone are strategically chosen to align with the refined mesh constraint, which is meant to enhance the precision and efficiency of the computation. Doing so ensures that the computational resources are focused on the most critical areas of the model, providing a higher resolution of results where it matters most. Maintaining consistency in the simulations, the dimensions of the refined mesh area within the reservoir remained constant across models for various scenarios. This allows for a meaningful comparison of results across different scenarios, as the same level of precision is applied in each case. Adaptive refinement at the fracture tips in the original finite element meshes is a strategic approach to improve the accuracy and reliability of the resulting fracture network. This adaptive refinement focuses computational resources on the areas of most significant importance, which are the fracture tips in this case.
Regarding the stress shadow effect, fracture sequence is another significant triggering element. In our model, we hypothesized two types of fracturing sequences: sequential and simultaneous, as illustrated in
Figure 4. For the sequential sequence, fracture propagation is stage-by-stage, where fracturing advances from the cluster on the right (the first) to the one on the left (the fifth). Conversely, the simultaneous sequence sees all clusters commencing fracturing concurrently. Here, the volume of the injected fluid equals that of sequential fracturing. Still, the volume for each cluster is allocated by initial conditions and stress interaction within the fracture zone—the ultimate fracture network results from stress re-orientation and fluid volume rivalry between the perforation clusters. To negate the influence of fluid competition, we introduced a parallel fracturing case, depicted in
Figure 4c, as a comparator within the simultaneous sequence. In this sequence, the total volume of injected fluid remains constant. However, each cluster receives an equal fluid volume allocation. Consequently, fractures at different clusters propagate independently, and propagation deflection is induced solely by the stress shadow effect. By incorporating this scenario into simultaneous fracturing, we can delve deeper into the internal mechanisms of the stress-shadow effect.
5. Discussion
The total fracture lengths under different fracturing sequences and perforation spacings are depicted in
Figure 9a. Sequential fracturing exhibits the maximum total length across various perforation spacings among the three fracturing sequences. This can be attributed to the comparatively weaker stress shadow effect during each stage of fracture expansion in sequential fracturing. In subsequent stages, the stress shadow effect only alters the direction of expansion, with minimal impact on the overall length. For simultaneous fracturing, the total length of hydraulic fractures is the shortest and remains relatively consistent across different perforation spacings. This is due to the intense stress shadow effect between fractures during simultaneous fracturing, which restricts the overall expansion of fractures, resulting in a total length of approximately half that of sequential and parallel fracturing. The fracturing effect of parallel fracturing lies between simultaneous and sequential fracturing. Owing to the individual supply of fracturing fluid at each stage, the fractures expand fully in this stage; the stress shadow effect generated by adjacent fractures alters the expansion direction. The average fracture length per stage in sequential fracturing is 10.8% longer than in parallel fracturing.
The total fracture lengths in the naturally fractured model under sequential and simultaneous fracturing are illustrated in
Figure 9b. It can be observed that when the perforation spacing is less than 25 m, the total length of hydraulic fractures in the naturally fractured model is not significantly different between sequential and simultaneous fracturing, amounting to 127.1 m and 86.91 m, respectively. When the spacing is small, the fracture networks formed by sequential and simultaneous fracturing interconnect after initial expansion to create a common network, resulting in negligible differences between the two. However, when the perforation spacing increases to over 50 m, sequential fracturing, compared to simultaneous fracturing, can connect to natural fractures to the greatest extent during each expansion stage, thereby forming a more complex fracture network. The fracture lengths are 3.1 times, 3.3 times, and 3.8 times that of simultaneous fracturing, respectively. This observation reflects the importance of perforation spacing and fracturing methods in exploiting the presence of natural fractures to maximize hydraulic fracture network complexity and length. Similar results can be derived when comparing the numerical model results with laboratory experiments [
44,
58,
59].
The fracture volume and leak-off volume of three fracture sequences is shown in
Figure 10; we observe a consistent increase in leak-off volume as the process progresses. This reveals that the high leak-off effect significantly influences the dynamics of fractures during hydraulic fracturing. In sequential fracturing, upon completion of fracture extension in the current stage, the in situ stress in the formation exerts considerable compressive force on the formed fracture surfaces, leading to fracture closure and a reduction in fracture volume. The decline phase of the fracture volume occurs between each stage of sequential fracturing. Meanwhile, although minor fluctuations in fracture volume are observed in simultaneous and parallel fracturing, the fracture volume ultimately reaches its maximum value at the end of the fracturing process. Owing to the high leak-off effect, the volume of fluid loss constitutes a substantial proportion of the total fracturing fluid volume across different fracturing sequences, accounting for approximately 40% to 80% of the total fracturing.
Due to the intense stress shadow effect, simultaneous fracturing inhibits rapid growth in fracture length. Under the same leak-off parameters, more fracturing fluid is retained within the fracture. Thus, even though partial closure occurs under in situ stress, the overall fracture volume in simultaneous fracturing still surpasses that in sequential fracturing. On the other hand, parallel fracturing, which involves individual fluid supply at each perforation location, results in the most enormous fracture volume among the three fracturing sequences; consequently, it has the smallest fluid loss volume.
The leak-off volume generally decreases with increasing perforation spacing, as shown in
Figure 11. Parallel fracturing exhibits the smallest fluid leak-off, amounting to just 43% of the total fracturing fluid volume when the spacing is 50 m. Compared to simultaneous fracturing and sequential fracturing, the former records the most significant fluid leak-off volume, peaking at 79% of the total fracturing fluid volume. The fluid leak-off volume of sequential fracturing lies between simultaneous and parallel fracturing. However, as the perforation spacing increases, the total fluid leak-off volume shows a declining trend. This is attributable to the diminishing stress shadow effect, allowing more fracturing fluid to be retained within the hydraulic fractures.