Volume stimulation in horizontal well has become a key technology for effective development of unconventional oil and gas reservoirs^[1,2,3]. Better understanding on the theory of volume stimulation technology has important significance for optimizing design and application of fracturing in the field. In this study, we explain the core connotation, summarize the research results and key technologies and look into the future development direction of volume stimulation based on the theoretical study and field application of this technology. The ultimate goal is to promote the application of this technology in a wider range.

In 2002, through micro seismic monitoring, Maxwell^[4] found that the fractures generated in the staged fracturing of horizontal well showed a complex network propagation geometry in the plane and longitudinal direction, rather than the two-wing symmetric fractures stated by the traditional fracturing theory. This discovery was the starting point for the development of volume stimulation technology. In 2006, Mayerhofer^[5] first proposed the concept of “stimulation of reservoir volume (SRV)”, followed by an in-depth analysis of SRV in 2010 but did not include technical aspects^[6]. In 2008, our research team first proposed the concept of “fracture network” fracturing technology in the industry. After one year of research and practice, an article^[7] was published in 2009 to elucidate the technology and its application effect. In January 2009, the author officially proposed the concept of “volume stimulation” technology in PetroChina. After more than one year of research and summary, the first article was published in 2011^[1], which clearly stated that volume stimulation technology is a major technological breakthrough. In 2013, the theory and design method of volume stimulation technology were initially established.

Generally speaking, the volume stimulation technology is the general fracturing technology under modern theory. The “fracture network” is the fracture geometry pursued by volume stimulation, and the “fracture network” fracturing technology is an expression of volume stimulation technology. The volume stimulation technology invented by the author’s research team is an interpretation of the classic Darcy’s law from different perspectives in the field of unconventional oil and gas exploration and development. Its core theory mainly includes the following points: (1) one method: “breaking up” the reservoir to form network fractures and artificial permeability; (2) three connotations: the maximum contact area between the fracture wall and the reservoir matrix, the shortest flow distance of fluid from the matrix to the fracture, and the minimum pressure difference for the fluid flow to the fracture; and (3) three functions: to increase single well production, improve oil recovery, and maximize the producing degree of reserves. Studies by researchers at home and abroad have shown that^[8,9,10], for seepage of oil and gas in unconventional reservoir matrix, whether considering non-Darcy flow, starting pressure flow or multi-scale flow, its model of non-Darcy flow is still in the expression form of Darcy’s law, and the difference is that the model is modified by using different parameters, but the seepage characteristics are still controlled by the seepage area, flow distance, and driving pressure difference. These studies have confirmed theoretically rationality of the core idea of volume stimulation technology, that is “the maximum contact area, the shortest flow distance, the smallest pressure difference”. Therefore, it can be argued that Darcy’s law is the theoretical basis for establishing volume stimulation technology, and “the maximum contact area, the shortest flow distance, the smallest pressure difference” is a new expression of Darcy’s law in the field of reservoir stimulation. Volume stimulation technology can be applied not only to unconventional reservoirs, but also to the development of low-saturation reservoirs, heavy oil reservoirs, and even conventional reservoirs. Volume stimulation technology is also applicable to deep unconventional reservoirs. The main technical bottlenecks are the technical capability and level of deep well operation, the pressure resistance of wellheads and equipment, and greater investment. Therefore, how to reduce cost and increase efficiency is an important research direction for effective development of deep shale gas reservoirs.

Since the emergence of volume stimulation technology, it has not only been widely used in PetroChina, but foreign researchers^[11] also agree that it is the future development direction of fracturing technology to “break up” the rock near the wellbore through multi-cluster and large-scale fracturing. Pearson et al.^[12] compared the fracturing effects obtained by using different tools and techniques in the Bakken Basin and found that volume stimulation using bridge plug, clustering perforation and slick water in large-scale and large displacement has the best result. Romanson et al.^[13] verified the general applicability of volume stimulation technique by monitoring the complex fractures formed by the volume stimulation in the Bakken Basin through microseismic monitoring (Fig. 1).

In recent years, the new progress of volume stimulation technology is mainly reflected in the continuous reduction of well spacing and cluster spacing, the use of three-dimensional volume stimulation in multi-layer or thick-layer reservoirs, and the significant cost reduction and improvement of Estimated Ultimate Recovery (EUR) of ultra-long horizontal wells.

Microseismic monitoring and well test analysis show that the actual length of the hydraulic fracture is much smaller than the range of the microseismic event^[14,15]. Under the condition of large well spacing (such as 400 m), the length of the artificial fracture is not enough to cover the interwell reservoir fully, so the reserve utilization rate is limited and stimulation efficiency is not maximized. So, reduction of well spacing is taken in North America^[16,17,18], with well spacing shortened from 400 m in 2009 to less than 200 m, and the minimum well spacing being just 76 m^[19]. Reducing the well spacing can reduce the fracture length designed and make field operation easier. By strengthening the support ability of proppant to the fractures in distal end of formation, the fractures can fully control the reserves between two adjacent wells, thus greatly reducing the volume of unstimulated reservoir far from the wells. In addition, while reducing the well spacing, it is often necessary to reduce the cluster spacing, they are closely related to each other. At present, the cluster spacing of horizontal well stimulation in unconventional reservoirs in North America has been reduced from 20-30 m^[20] to 5-10 m^[21]. Under the condition of small cluster spacing, it is not necessary to form network fractures between clusters. The seepage distance between oil and gas and fractures in the matrix within two fractures is only a few meters. For micro- and nano-scale permeability reservoirs, the driving pressure difference required for fluid to flow into the fracture in the matrix has been greatly reduced, which means that the oil and gas recovery in the matrix is basically no problem. “All” recovery is the core and ultimate goal of the volume stimulation technology.

Increasing the producing degree of longitudinal sections of layered or thick unconventional reservoirs has always been an important research topic in the field of reservoir stimulation. Multi-layer fracturing with packer sliding sleeves and coiled tubing hydraulic sand blasting in vertical well, and multi-lateral horizontal well fracturing are effective technical methods. With the development of drilling technology, the drilling speed has been greatly improved. For example, in the Eagle Ford block, for typical wells 4 853 m deep and 2 198 m long in horizontal length, it only takes 6.02 d to finish drilling from spud-in. Drilling costs have fallen from 60%-80% of the cost of completion to 21%-34%. The speed of drilling has prompted the development of spatial-mode volume stimulation technology (Fig. 2)^[22], that is, a horizontal well is drilled separately for each target layer, and multi-lateral horizontal wells are no longer used. Compared with multi-lateral wells, multi-layer horizontal wells are simple in operation, high in efficiency, low in risk, and low in overall cost. This technology applies the concept of “breaking up” reservoirs from plane to vertical direction by volume stimulation. With reference to the horizontal well development model in the wellpad mode, the horizontal sections are stacked in the longitudinal direction, the fractures are placed in a staggered pattern, so network fractures can be created by the interference of effective stress generated in the longitudinal direction by the fractures height propagation to greatly increase the reserve utilization rate of the longitudinal reservoirs. Farhan et al.^[22] found that when the Wolfcamp block was developed using three stacked multi-layer horizontal wells, 78% of the wells with production were of multi-layer completion. Field practice in the blocks of Eagle Ford and Niobrara shows that poor reservoirs with no economic value of development interpreted by the logging data, can reach development result similar to good reservoirs after using reservoir simulation technology^[23].

Utra-long horizontal well fracturing technology has been attempted in North America since 2013, with the horizontal section increasing from 1 900 m at first to 5 700 m in 2016. For example, the company of Eclipse successfully implemented ultra-long horizontal well fracturing in Well Purple Hayes1H in the Utica block^[24]. The well has a depth of 8 244 m, horizontal section length of 5 562 m, and vertical depth of 2307 m. It took 17.5 d to finish its drilling, and 23.5 d to finish its fracturing. The fracturing had 124 stages, with each stage being 45.72 m. 5.3 stages with 5 clusters per each stage were fractured per day. After the fracturing, the well had a natural gas production of 14.16×10⁴ m³/d and condensate production of 190.8 m³/d. According to the production data, the EUR for 3 years would reach 2.76×10⁸m³. The ultra-long horizontal well can reduce the drilling and completion cost per meter, improve the economic indicators of the well, reduce the cost of fracturing and drilling by 20%-30%, and increase the profit by 35%-70%, compared with the short horizontal well. By increasing the number of fractures cutting matrix and the contact area between fractures and matrix, the ultra-long horizontal well can maximize the recovery rate of reserves in the horizontal section, so as to delay the decline of the production, greatly increase the EUR, reduce cost and increase profit.

How to understand and apply volume stimulation technology, North America gives its own answer, that is, “tight spacing, multi-dimension, super long horizontal well”, the core of that is to further shorten the seepage distance of fluid in the matrix to the fracture, greatly reduce the driving pressure difference, and increase the contact area between the matrix and the fracture. Volume stimulation has evolved from a two-dimensional plane to a multi-dimensional space, surpassing the idea of “breaking up” the reservoir on the plane only. According to the theory of volume stimulation technology, Tuha Oilfield has carried out tests in other types of reservoirs, and achieved good results, which is also a practice and proof for the “maximum contact area, shortest distance, and smallest pressure difference” of core theory of volume stimulation.

Multi-cluster perforation is the key to the application of volume stimulation. When multiple clusters (3 clusters or more) are used for each stage, the key to ensure the initiation and growth of each cluster under constant injection rate is to limit the perforation numbers in each stage. If the total perforation number can ensure sufficient entry resistance, all the perforation clusters can be initiated^[3]. In this case, it is not necessary to use the temporary blocking technique to activate the unopened clusters. Due to the influence of reservoir heterogeneity and the phase of perforation hole, how to realize the uniform stimulation of all clusters needs to be studied and analyzed from the aspect of multi-fracture propagation (Fig. 3).

When the fracture expands, the energy from the injected liquid is converted into the strain energy and fracturing energy of the rock, the energy that overcomes the far-field stress and interactive stress, the perforation friction and the energy consumed by the fluid flow in the fracture^[25]. Considering the case of N clusters of fractures in each stage, the energy balance equation for each fracture can be expressed as:

Assume that the interactive stress of ith fracture caused by the jth fracture is σ_I,i,j, meanwhile as the hydraulic fractures are largely viscosity-dominated fractures^[26], W_c can be negligible, so the equation (1) can be simplified as:

According to equation (2), for the propagation of multi- cluster hydraulic fractures, two kinds of cases could occur, the cases dominated by stress interference and cases dominated by perforation friction. When the stress interference effect is much larger than the perforation friction, the stress interference would dominate the energy consumption. Now, the equation (2) can be converted into:

Under the influence of the interaction stress between the fractures, the fractures in a cluster need to overcome more stress work from other fractures in the cluster. Since the hydraulic fractures would propagate in the least energy-consuming manner, less amount of fluid would get into interior fractures and more injected fluid would go into exterior fractures. Consequently, the amount of fluid entering the fractures won’t be even, and the propagation of the fractures inner the cluster will be insufficient than the fractures in the outer side of the cluster.

When the perforating friction is much greater than the interaction stress, the perforating friction would dominate the energy consumption. In this case, the equation (2) can be converted into:

According to equation (4), there is no difference in energy dissipation between different fractures in each cluster. At this time, the amount of liquid injected can be evenly distributed into the fractures of each cluster, thereby achieving uniform propagation of fractures in each cluster.

The above analysis shows that in the volume stimulation of horizontal well, uniform stimulation of each cluster can be achieved by multi-cluster limited entry technique. This conclusion has been proved by Lecampion and Wu et al.^[27,28] through numerical simulation. The extreme limited entry (XLE) perforating technique proposed by Somanchi et al.^[29] can achieve multi-cluster simultaneous opening and uniform expansion by a greater degree of multi-cluster limited entry. The technology was tested in the Montany block, where with 3 clusters per stage and 2-3 perforation holes per cluster, the fracturing was done at the injection rate of 5 m³/min and perforation entry resistance of 8.3 MPa. Optical fiber diagnosis shows that the technology results in more balanced amount of sand per cluster and the efficiency of perforation clusters 33% higher than conventional limited entry perforating. Weddle et al.^[30] reported the effect of XLE in the Bakken block: the horizontal section of the wells were 4 313 m long, and were fractured in 40-50 stage, with 12-15 perforation clusters in each stage and 2 holes in each cluster; the holes in the same cluster were perforated at 180° phase angle; the fracturing was done at the sand adding rate of 0.98-1.51 t/m, injection rate of 12.7 m³/min, and the perforating entry resistance of 10-14 MPa. Post-fracturing gamma logging shows that the efficiency of XLE is 80% to 90%, while the efficiency of limited entry perforating is often only 30% to 80%. However, the injection rate of XLE is lower, mainly because the number of holes is too few, which leads to excessive entry resistance and great increase of the injection pressure at the wellhead. But, lower injection rate often results in lower net pressure in the fractures, which is not conducive to the formation of complex fractures and increase of SRV.

Usually, perforation tunnels are subject to erosion by the sand carrying fluid during fracturing, which would lead to increase in the radius and reduction in the friction of perforated holes. Therefore, in the optimized design, the number of tunnels should be appropriately reduced according to the calculated entry resistance to ensure the effective opening of each cluster. In addition, it is necessary to quickly increase the injection rate during fracturing, to establish the bottom-hole pressure to ensure sufficient entry resistance for opening of each cluster, the even entry of fracturing fluid into fractures in each cluster, and effective propagation of multi-cluster fractures.

The main purpose of volume stimulation is to increase the complexity of fractures far from well^[31]. The near-well fractures should be as simple as possible in shape to avoid distortion or multiple fractures which may cause smaller fracture width and subsequently sand plugging at the perforation tunnel or near the well. Therefore, directional perforation, equal hole size perforation and other techniques can be taken to make the perforation phase as consistent as possible with the direction of the maximum principal stress, to effectively avoid the distortion of near-well fractures^[32,33]. In staged fracturing from horizontal wells, the perforation spacing of the same perforation cluster should generally be less than 4 times the diameter of the wellbore^[34] to ensure that the fractures at different tunnels are single.

Beugelsdijk et al.^[35] studied the fracture geometry in naturally fractured formations, and pointed out that the product Qμ was the key factor affecting the natural fracture propagation and fracture propagation geometry. When Qμ was 8.3×10^-8 N·m, the liquid would flow along the natural fracture and no main artificial fracture would be generated; and when Qμ was 8.3×10^-6 N·m, main artificial fractures would be created and the natural fracture would not open. The study also showed that the change rate of injection rate had a significant effect on the fracture initiation, if the injection rate was increased slowly, the pressure curve would have no signs of breakdown, the injected liquid may lose along the natural fracture, causing multiple fractures near well to open (Fig. 4a); when the injection rate was increased quickly, the pressure curve would show obvious breakdown, in this case the natural fracture would not open, only a single hydraulic fracture would be created (Fig. 4b). The experiments of Fu Haifeng et al.^[36] verified the function of product Qμ; Lecampion et al.^[37] also found that for reservoirs with natural fractures near the wellbore, rapid increase of injection rate to establish bottom-hole pressure could avoid initiation of multiple fractures, thereby reducing the complexity of fractures near wellbore.

Beugelsdijk et al.^[35] studied the difficult degree of main fracture diverting and opening of natural fractures and concluded that the diverting ability of artificial fractures should be determined by dimensionless net pressure (ratio of net pressure to horizontal stress difference). The greater the dimensionless net pressure, the easier the fracture will deviate from the direction of the main fracture and form complex fractures. He also introduced a dimensionless horizontal stress difference coefficient to characterize the open ability of natural fracture:

Under the condition of the same stress difference, the larger the minimum horizontal principal stress, the smaller the k_h, and the smaller the opening degree of the natural fracture, the more difficult it is for the fluid to lose in the natural fracture. Therefore, the dimensionless horizontal stress difference coefficient reflects the ability of the fluid to enter the natural fracture and has nothing to do with the diverting ability of the hydraulic fracture.

The extended shape of the fracture away from wellbore is mainly affected by the far field stress, the stress interference

between fractures and the natural fractures^[38]. During the on-site fracturing, the clustering perforation can be used to improve the complexity of fractures in far from wellbore by making use of the superposition of inter-cluster stress interference and the diverting technique in fractures^[7]. In addition, reducing the cluster spacing can make multi-cluster fractures deflect away from each other, thus increasing the complexity of fractures and the lateral stimulation range^[39]. For reservoirs with natural fractures susceptible to sand plugging near wellbore, a combination of “breaking up rock by gel + carrying proppant by slickwater” can be considered to reduce the risk of sand plugging and improve the complexity of fractures far from wellbore.

From the perspective of proppant migration, the smaller the particle size of the proppant, the slower the setting rate, the farther the proppant will move in the fractures, and the better the placement effect of proppant will be. The equation for the proppant settlement within the fracture can be expressed as:

According to the equation (6), when the particle diameter of the proppant is reduced to 1/2 of the conventional particle diameter, the setting speed of the proppant will be reduced to 1/4 of the setting speed of the conventional particle size proppant. At present, the clusters in a fracturing stage of horizontal well volume stimulation are going up from 3 to more than 10^[30], so the injection rate in each cluster is greatly reduced. Since the injection rate cannot be increased indefinitely to increase the fracture width, and the limit dynamic fracture width to avoid sand screenout should ideally be 2 to 3 times the proppant particle size^[40], so it is sensible to select small particle size proppant to reduce the risk of sand screenout and increase the migration distance of proppant in the fracture.

Since slickwater volume stimulation is likely to create complex fractures, the small particle size proppant would not only settle in the main fractures, but also enter the branch fractures and micro fractures in the forms of corner support and single particle support. This type of placement manner conforms to the early research consensus, that is: “Proppant setting in single layer has the best conductivity”^[41], which is an important reason why the complex fracture network has better conductivity. Ely et al.^[42] compared the production of the Eagle Ford and Bakken blocks, and found that the small-size quartz sand worked better than the large-size quartz sand in fracturing. At present, the field test of smaller particle size proppant was carried out in Grassland block of Texas, North America^[43,44]. In the test, 11 horizontal wells were stage fractured with 3 clusters in each stage at an average injection rate of 8.6 m³/min. The concentration of proppant of 0.150 mm (100 mesh) and 0.425/0.212 mm (40/70 mesh) were both 300 kg/m³, and that of 0.045 mm (325 mesh) micro-size silt was 12 kg/m³. Fracturing with the above method, the cumulative gas production per well over 210 days increased by 20%-30%. Through experiment and numerical simulation, Dahl et al.^[43] pointed out that injecting small particle size proppants could increase the micro-fracture permeability of shale and enhance production.

In the application of volume stimulation technology, slick water is generally used to expand the sweeping volume because of its low viscosity. In this method, large fracturing fluid volume is made use to supplement the energy, and large injection rate is used to carry sand to the distal end of the fracture. The follow-up proppant migration and placement by slickwater in the fracture is shown in Fig. 5. The proppant continuously settles with the liquid migration, and gradually spreads from the bottom of the fracture in the longitudinal direction to form a support for dynamic fractures. After the completion of the fracturing, it is not recommended to do rapid flowback or shut-in, as at a lower flowback rate, fracturing fluid retained will support and stop the fracture from closing. The proppant that continues to settle only increases the height of the sandbank under the condition of the fracture width created by the stacking, which makes the dynamic fracture width generated by the large injection rate basically the support fracture width, that is the reason why the slick water fracturing doesn’t need to pursue the higher sand concentration. If a small particle size proppant or a small particle size proppant with low density is selected, the proppant can be transported further in the fracture, which is more conducive for improving the stimulation effect.

The design of hydraulic fracturing relies on theoretical models and calculation methods. The early classic models included mainly PKN, GDK models etc.^[45], and later Settari and Palmer et al.^[46,47] developed them into a pseudo-three-dimensional model. The model considers the influence of the inter-layer stress difference, but the fracture height equation does not consider the vertical flow and the interlayer mechanical properties. Therefore, when the thin interbed and perforation segments are high-stress layers, the fracture height predicted by this model would have larger error. A full three-dimensional model established by Lee and Carter et al.^[48,49] doesn’t use the fracture height equation but obtains the fracture width and the fluid pressure in the fracture by coupling the two-dimensional flow equation of the fluid in the fracture with the three-dimensional linear elastic mechanics model. In addition, the fracture boundary is determined by the stress intensity factor of the discrete grid element to work out the fracture length and height.

In recent years, the application of volume stimulation technology has driven the development of unconventional fracturing models. Although the unconventional fracturing model is still based on the theory of linear elastic fracture mechanics, the problem solved is more complicated, such as: the propagation of multi-cluster fracture, shear fracture and bedding weak interface, as well as interaction between artificial fracture and natural fracture. The unconventional fracturing model mainly means to solve two key problems: (1) the description of natural fractures, including its geometric characteristics, petrophysical and mechanical parameters, etc.; (2) the propagation simulation of complex fractures, which includes the propagation form of artificial fractures after encountering natural fractures, as well as the propagation of tensile fractures and shear slip fractures. At present, there are four main types of propagation simulation solutions for complex fractures: finite element, extended finite element, boundary element and discrete element.

Hossain et al.^[50] first proposed a method of modeling natural fracture networks inversely based on fractal theory. The model considers the propagation of shear fractures and can be used for the analysis and design of slick water fracturing in the formation with natural fractures. The accuracy of the model has been verified by micro-seismic monitoring results.

Robert et al.^[51] proposed a method based on discrete fracture network model (DFN) and productivity data to model the natural fracture network of the formation inversely. The model is mainly used for the morphological interpretation of fractures after fracturing and is not suitable for design analysis before fracturing.

Olson^[52] established a boundary element model for multi- cluster fracture and natural fracture propagation. The model considers the shear and tensile breaking of natural fractures and can simulate the propagation of complex fracture under the condition of stress interference. Later, Wu et al.^[53] extended the model to consider the effect of perforation and wellbore friction. Dahi-Taleghani et al.^[54] studied the interaction between hydraulic fractures and a single natural fracture based on the extended finite element method and established a method to evaluate the difficulty of natural fracture and rock matrix failure using energy release rate. Moreover, they also studied the extended morphology of fractures created by the stimulation in formations where natural fractures with different length and angle are evenly distributed^[55]. The model can be used to simulate the fracturing of reservoirs with natural fractures.

Meyer et al.^[56] used the orthogonal fracture model to describe the natural fracture network. The model assumes the shape of the fracture to be an orthogonal fracture ellipsoid only, which is different from the actual situation. In the same year, Maxwell et al.^[57] proposed a method to model the geometry of natural fractures based on micro-seismic modified model. This model combines micro-seismic events obtained from monitoring in horizontal well fracturing with geological modeling and 3D seismic data and pioneered the research of geological-engineering integration.

Kresse et al.^[58] proposed an unconventional fracture model (UFM). Based on the pseudo-three-dimensional model, the model can be used to predict the intersecting pattern of the hydraulic fracture with natural fracture based on the hydraulic fracture tip stress field established by Gu et al.^[59]. Later a semi-analytical intersection criterion^[60] considering the injection rate and viscosity was introduced into the model. At present, the UFM model considers the stress interference of multi-cluster fracture propagation and the transport of proppant in the fractures. It can be combined with micro-seismic monitoring to correct the results of fracture propagation, thus realizing numerical simulation of fracture propagation in staged multi-cluster fracturing of multi-layer reservoirs^[61].

Xu et al.^[62,63] proposed an extended model of multi-cluster fracturing in horizontal wells. The model considers the effects of stress interference and friction between the fractures and can be used for fluid-solid coupling simulation of three-dimensional fracture propagation in multi-layer reservoirs. However, without considering natural fractures, it cannot be used to simulate fracture propagation in strata with natural fractures.

Nassir et al.^[64] advanced a fluid-solid coupling model for hydraulic fracturing in reservoirs with natural fractures. The model uses the elastoplastic constitutive equation to describe the shear and tensile deformation characteristics of natural fractures and predicts the stimulated reservoir volume based on the tensile failure and the molar Coulomb shear failure criterion in three-dimensional space. Considering factors such as stratigraphic heterogeneity and elastoplastic deformation of natural fractures, the model can be used for numerical simulation of fracture propagation in the formation with natural fractures.

Profit et al.^[65] developed a staged multi-cluster fracturing simulator for tight gas reservoirs. The simulator achieves full coupling of geomechanics and fluid flow through adaptive mesh simulation technology. Considering fracture tensile, shear or mixed failure modes, the model can simulate multiphase flow of fluid.

Alfataierge^[66] put forward a 4D/9C (four-dimensional/nine- component) microseismic interpretation method for analyzing the effectiveness of hydraulic fractures, and proposed a geological engineering integration simulation method by combining 4D delayed multi-component seismic technology, microseismic interpretation and 3D hydraulic fracture simulation technology in 2018. This method combines 3D geomechanical model with 3D fracture propagation simulator and can be used to correct the simulation results of fracture propagation by seismic technology and microseismic interpretation^[67]. Based on the study of the Niborara strata in the Wattenberg gas field in Colorado, the effective length of the fracture is 60-180 m, which deviates from the locations of the micro-seismic events to some extent. Therefore, the authors believe that the initial stimulation of the Niborara formation is insufficient, which means that the fracture spacing and well spacing need to be further optimized by using the geological engineering integration model.

Chinese researchers have also carried out research on fracture propagation models. Chen Mian^[68] deduced the controlling equations for hydraulic fracture activation and diverting in three-dimensional space and established an extended model of two-dimensional natural fractures. Zeng Qinglei et al.^[69] worked out a fluid-solid coupled finite element algorithm for multi-cluster fracture propagation. His model considers perforating friction and wellbore friction. Wang Lixiang et al.^[70] proposed a finite element and discrete element hybrid method for two-dimensional hydraulic fracture propagation and simulated the fracture propagation pattern of multi-cluster fracturing in the formation with natural fractures. Guo et al.^[71] introduced a pore elastic bonding unit and established a two-dimensional fluid-solid coupling finite element fracture propagation model. He also studied the effect of natural fractures on the hydraulic fracture propagation morphology. Zou et al.^[72] developed a finite element/discrete element model based on the random distribution of natural fractures in the formation and studied the fracture propagation pattern of shale formations. Xu Yun et al.^[39] established a boundary element model for multi-cluster fracture propagation and examined the deflection law of multi-cluster fracture propagation. Chen Ming et al.^[73,74] further discussed the factors controlling the deflection of fractures and how to create multi-fractures. Zhao Jinzhou et al.^[75] proposed a pseudo-three-dimensional multi- cluster hydraulic fracture extension boundary element model and a perforation optimization method based on PKN model. Wu Qi et al.^[76] expounded the application of geological-engineering integration technology in shale gas development and demonstrated the promising application prospect of this technology. However, the current fracture propagation simulation in China is mainly based on two-dimensional model, and there is no three-dimensional model. The research on geological-engineering integration mainly relies on foreign software. Our research team is carrying out relevant research with the help of national research projects and is expected to strengthen the construction of the volume stimulation optimization design platform for the geology-engineering integration of unconventional reservoirs in China, so as to promote the advancement of China’s unconventional reservoir simulation technology.

The research progress at home and abroad tells us that: on the one hand, the volume stimulation design model can accurately describe the strata with natural fractures, characterize the relationship between artificial fractures and natural fractures to ensure the matching between main fractures, branch fractures and network fractures; on the other hand, it is also able to efficiently and accurately calculate the tensile-shear propagation of complex fractures. The integration of geomechanical modeling, microseismic monitoring and hydraulic fracture simulation is an important development direction for future model research, while fractal damage mechanics is the basis for studying fracture initiation and constructing new fracturing mechanics theory.

4.1.1. Tight spacing mode and optimization of fracturing volume

Early studies^[77] believed that the best cluster spacing should be 20-30 m in the case of multiple-cluster perforation in horizontal wells. If three clusters are used, the length of each fracturing stage is generally 60 to 90 m. However, Mayerhofer et al.^[5] argued that when the reservoir permeability was as low as 0.000 1×10^-3 μm², if the fracture spacing was 8 m, the production could be greatly increased and the oil recovery improved. Zhu et al.^[21] found through study that reducing the cluster spacing could greatly improve the ultimate recovery of the reservoir. Years of field practices have proved the tight cluster spacing can greatly shorten the distance of fluid seepage in the matrix, thus realizing volume stimulation of the reservoir with strong plasticity and large stress difference that is difficulty to form complex fractures. At present, the cluster spacing has been gradually reduced from 20 m to 4.6 m^[78] in North America, and the small cluster spacing has been widely used in horizontal well staged fracturing of all unconventional reservoirs, rather than reservoirs hard to form fracture network only.

The reasonable matching of tight cluster spacing and well spacing is the key to the volume stimulation of well groups from wellpad. In 2017, Pioneer^[78] reduced the fracture cluster spacing and stage length of its horizontal wells; for horizontal wells with the same length, if the well spacing is constant, tight spacing will lead to an increase in single well injection volume and sand addition (Fig. 6). Although the amount of injected fracturing fluid volume and sand addition of a well increased, the ratio of proppant to injection volume did not change much and was between 95.4 and 110.9 kg/m³. Taking fracturing with two clusters in one stage as an example, if the injected fluid volume is 2 000 m³, the amount of fluid injected per fracture is 1 000 m³; if the number of clusters is increased to 4, the injected fluid volume of each fracture is 500 m³, which will lead to insufficient length of the created fractures, large area not fractured between the two wells, and reduction of the utilization rate of reserves, this goes against the original intention of fully fracturing the reserves. Therefore, the tight spacing mode requires shortening the well spacing, deploying infill wells, or increasing the injected fluid volume.

Similarly, increasing the number of clusters will result in an increase in proppant of single stage, assuming that the length of a stage is 60 m, each stage has three clusters with the cluster spacing of 20 m, and the injected proppant is 120 t, then the proppant per fracture is 40 t. If tight spacing is used, the clusters of each stage would increase to 6 with a cluster spacing of 10 m. According to the arithmetic comparison, then this stage requires 240 t of proppant, which seems to be the major reason behind the increase of proppant usage in each stage in foreign countries. However, when the cluster spacing is reduced from 20 m to 10 m, the amount of oil and gas controlled by each fracture will be reduced by half, and the re-quired fracture conductivity should be changed. Whether the amount of proppant per cluster should be 40 t or 50 t needs to be optimized by simulation research and field practice. Although the use of quartz sand instead of ceramic beads in North America has achieved significant cost reduction, excessive increase of sand volume would still increase material and transportation costs, and even increase equipment wear and tear. We believe that learning the North American model cannot simply calculate the times of increment sand added per meter, but the factors such as the increase in the number of clusters and the reduction of well spacing should be considered comprehensively. It is more scientific to use the amount of sand added per cluster to characterize the fracturing scale than the amount of sand added per meter.

The characteristics of tight spacing can be summarized as follows: (1) The well spacing is constant, the number of clusters is increased, and the required fracture length is constant, which means that the injected fluid volume and sand volume must increase. (2) If the well spacing is reduced, the number of clusters is constant, the required fracture length will be shorter, which means the amount of injected fluid and sand will reduce. (3) If the well spacing is reduced, the number of clusters is increased, the required fracture length will be shorter, which means the injected fluid volume will reduce (or unchanged), and the amount of injected sand will increase. The change of well spacing and cluster number is the basic factor determining the increase or decrease of injected fluid volume and sand volume. Good understanding of the essence of "less fluid and more sand" in North America is the key to the application of tight spacing technology.

4.1.2. Fracture height control and spatial-mode stimulation

Field practice and research have shown that the hydraulic fractures can intersect with bedding surfaces (Fig. 7) in several ways, including passing, terminating, slipping, and communicating high-angle fractures. Slipping is the main mechanism controlling fracture height propagation. When the fracturing fluid is lost along the bedding surface, the pressure in the fracture would reduce and the bedding slippage causes the flow friction of fluid to increase, both of which would make artificial fracture fail to pass through the bedding, limiting the fracture propagation in height. Similarly, in slick water fracturing, the proppant continuously settles along the direction of fracture height and rapidly deposits at the bottom of the fracture to prevent the fracture from expanding downward. Monitoring shale gas horizontal wells with inclinometer, Xiu Nailing et al.^[14] found that vertical fracture volume makes up 90% of the total fracture volume at most, but for some sections, this is only 50% to 60%. The artificial fracture system consists of vertical and horizontal fractures, indicating that the artificial fractures have the characteristics of extending in the horizontal bedding. The shale gas field data of PetroChina (Table 1) shows that the position of the horizontal well section in the high-quality reservoir is closely related to the stimulation effect, and the propagation of the fracture in the height

direction is limited, which overthrew the traditional fracturing theory that the fracture height isn’t limited and prompted people to study the effects of horizontal bedding and weak surface on fracture propagation^[79,80]. Finding new ways to improve the effective utilization rate of vertical reserves has become an important research direction at present and in the future.

The faster drilling in North America has made the drilling cost drop significantly. Many companies have begun to explore spatial stimulation development model, seeking to break the limitations of well trajectory and fracture height, thus improving the stimulation effect on reservoir profile. Carrizo has implemented a spatial stimulation project in the Niobrara formation^[23]. The projected included 47 wells with horizontal section of 1 426 m and horizontal section space of 90-100 m. The wells are arranged in 3 layers, Fig. 8 shows a side view of the horizontal wells stacked in three layers (i.e. toe end position of horizontal section of each horizontal well). The wells of layers A and C are in the same façade in the vertical direction, and the wells in layer B are misplaced, we called this as three-dimensional staggered zipper fracturing technology. The order of fracturing is from left to right: C1—C2—B1—A1— C3—B2—A2—A3—…… The method can make the first fracturing well generate an additional stress field at the bottom. Due to the dislocation fracturing of different layers, the additional stress generated by the fracturing can increase the complexity of the adjacent layer fractures, thereby improving the stimulation effect of the longitudinal small layers.

Energen^[81] applied the same volume stimulation technology to the Wolfcamp formations in Delaware and Midland Basin. In the second quarter of 2017, they placed 8 and 10 wells on two vertical layers respectively in their own block. These wells with horizontal section of 2 281-3 210 m were fractured at 45 m a stage and the cluster spacing of 9 m, with the amount of injected proppant of 2.5 to 3.0 t/m and injected fluid volume of 6.4-7.1 m³/m. The maximum production of the 18 wells 30 days after fracturing reached 300 t/d.

Although the current drilling in China speed is low in speed and high in cost, making it hard to implement spatial stimulation, this technology is an effective means for the development of tight reservoirs of multi-layer systems in Sichuan, Changqing and Xinjiang and is an important development direction in the future. Especially in the context of limited of mineral right area and the shortage of undiscovered high- quality reserves, this technology is the best choice for tapping potential and improving utilization rate of existing proven reserves. Xinjiang Oilfield has designed a two-layer spatial- mode development well pattern and carried out pilot test, which will provide useful experience for China to further explore and promote the application of this technology.

4.2.1. Shear fractures can greatly improve the conductivity of fractures

Many researchers at home and abroad have studied the relationship between shear fracture and conductivity. A study by Ramurthy et al.^[82] showed that the number of fractures per meter was logarithmically proportional to the increase in permeability, and the high-hardness rock can maintain high self-supporting fracture conductivity; on the basis of the formation of self-supporting fractures, hydraulic fracturing should pursue high complexity of fractures rather than high conductivity of the main fractures. Sharma et al.^[83] analyzed the fluid loss characteristics, interwell pressure and productivity, and concluded that fracturing of unconventional reservoir would form a large number of self-supporting shear fractures. Shear fractures allow the nano-darcy level reservoir to have sufficient conductivity, which will help increase production. Through numerical simulation, Weng et al.^[84] studied the permeability of shear fracture induced by hydraulic fracturing. They found that shearing could significantly improve the conductivity of natural fractures, and the conductivity of shear fractures could reach 600×10^-3 μm²•cm. Wu et al.^[85] performed a shear fracture conductivity experiment on rock samples from the Eagle Ford Basin. The experimental results show that when the closure stress is 28 MPa, the conductivity of the shear fracture is 3×10^-3 μm²·cm. Through the fracture diagnosis test, Wang et al.^[86] found the conductivity of the self-supporting fracture under the closed stress of 15 MPa was (10-70)×10^-3 μm²·cm. In addition, Sierra et al.^[87] concluded that increasing the seepage area was the key to increasing the productivity of shale gas reservoirs with a matrix permeability of less than 500×10^-9 μm².

In recent years, the conductivity experiments conducted by PetroChina Research Institute Petroleum Exploration and Development show that (Fig. 9): under the same conditions, the conductivity of tensile fracture is the lowest and most affected by the closure stress; the shear fracture has a high conductivity due to the support of the rough surface of the fracture; under the 20 MPa closing stress, the conductivity of the shear fracture is about two orders of magnitude higher than that of the tensile fracture. Characterized by permeability, the non-proppant shear fracture permeability is 25.18×10^-3 μm² at a closure stress of 50 MPa, which is significantly higher than the shale reservoir with nano-darcy level permeability. Therefore, no proppant fractures are still effective fractures. Similarly, in the conductivity experiment of fractures supported with proppant, we reached a similar conclusion: the conductivity of shear fracture is the highest when the same concentration of proppant is added.

If the three stresses of the reservoir meet the conditions for the formation of shear fracture (σ_H>σ_v>σ_h), or the shear slip can be induced by large injection rate of slick water, and considering the support effect of the fracturing fluid on the fractures under the low flowback rate, the proper reduction of the amount of proppant in the optimization design can reduce costs and increase profit.

4.2.2. No need to pursue high conductivity of the main fractures during volume stimulation

The difficulty in effectively developing unconventional reservoirs is the extremely low matrix permeability, and the main purpose of reservoir stimulation is to reduce the seepage resistance of the matrix. Cipolla et al.^[88] found through study that when the reservoir permeability was lower than 0.01×10^-3 μm², the contribution rate of secondary fracture network to production was about 40%; and when the reservoir permeability was less than 0.000 1×10^-3 μm², the contribution of secondary fracture network to production was about 80%. It can be seen that the productivity of micro-Nano-darcy-level reservoirs are controlled by the fracture morphology rather than by the conductivity of main-fracture. It is generally considered that the critical dimensionless conductivity of tight reservoirs is generally 10 to 50^[89], and that of shale gas reservoir is around 30^[90]. Gu et al.^[90] pointed out that when fractures had branches, the dimensionless conductivity of shale or tight gas reservoirs would reduce by 15 to 25, which further demonstrates that complex fracture patterns can reduce the requirement on conductivity.

The classical study has shown^[91] that when the dimensionless conductivity of a hydraulic fracturing well is higher than that of critical dimensionless conductivity, further increase in dimensionless conductivity would not increase production. The dimensionless conductivity can be expressed as:

According to the equation for calculating permeability of the supported fracture, it can be known that:

According to the equations (7) and (8), the relationship between the fracture width and the dimensionless conductivity can be obtained as follows:

Assuming that the half length of the fracture is 200 m, the porosity of the supporting fracture is 5%, and the tortuosity is 2, the fracture width required to achieve different critical dimensionless conductivity is calculated (Fig. 10). Studies show that for reservoirs with a matrix permeability of (100 to 1 000) × 10^-9 μm², the fracture width required for the dimensionless conductivity to reach 30 is 0.13 mm, and a width of 0.16 mm is required to reach 50. It can be seen that the volume stimulation does not require too high sand volume added to meet the effective development of unconventional reservoirs.

In view of the current concept of “less fluid and more sand” in the fracturing circle of China, this paper focuses on two key factors in the optimization design of stimulation, to make it clear that the ultimate goal of the unconventional reservoir volume stimulation is to maximize the SRV and greatly re-duce the seepage distance of the fluid in the matrix by forming complex fractures or tight spacing, so as to achieve maximum control and “full” recovery of reserves. Studies have shown that shear fractures and complex fractures can reach the conductivity required for development under moderate sand adding conditions, while the “high sand adding” mode that pursues the high conductivity of the main fractures (such as the traditional gel fracturing) does not conform to the idea of volume stimulation.

Unconventional oil and gas development in North America is usually limited by the area of leased blocks. Infill wells are usually used to maintain or increase the production of blocks. At present, the infill wells of several basins such as Eagle Ford, Bakken and Haynesville have exceeded the wells deployed in the new blocks^[92]. According to the timing of the production of adjacent wells in the platform, in the North America, the child-well is defined as a new well put into production one year later than the adjacent well, and the parent-well is an old well with a production time of no less than one year^[92]. Usually, the child-well appears in two forms: one is the infill well, which is drilled in between existent wells when the well spacing was too large. This mode would form a pressure drop zone on both sides of the infill well; the second is well drilled from a new wellpad, the well in the outer side of new wellpad is a child-well for the old well in the outer side of the adjacent parent-wellpad, in this case, the pressure drop is a single-sided mode, for the child-well, the asymmetry of fractures may occur in the fracturing, and even “attracted” by the parent-well, the fracturing could be insufficient in the other side^[93].

Gakhar et al.^[94] showed that only when the distance between the child-well and the parent-well in the Eagle ford basin was over 243 m, could the child-well have the same cumulative production as the parent-well (Fig. 11). The results were normalized by the length of the horizontal stage and the amount of proppant, and did not consider the completion

method, number of fracturing stages, type and size of the liquid, injection rate, and updip or downtilt of the wellbore trajectory. Gakhar thought that the poor fracturing effect of the child-well was caused by the fracturing size of the child-well and the insufficient fracture length.

When the two wells are fractured in staggering fracture arrangement, the penetration ratio would be 0.85 and the fracture length 207 m. If the fractures are arranged symmetrically, the penetration ratio would be 0.45 and the fracture length 109 m. At this time, the production of the child-well and the parent-well are comparable, which indicates that the length of the fractures generated during the fracturing of the parent-well does not reach the resolution of the microseismic, and the reserves between the two wells are not fully utilized. The well spacing (243 m) only reflects the fracturing state of the parent-well and the extent of the pressure drop, but is not necessarily the optimal well spacing. If the well spacing is 122 m, the child-well production is only 70% of the parent-well. Assume that the fractures propagate symmetrically along both sides of the wellbore, and the reserve utilization rate contributed by the artificial fracture of the parent-well to the child-well is only 30%, which indicates that under the control of such short fractures (the length of the fractures is 61 m), the matrix seepage is still insufficient, there is still a large amount of reserves not produced. Therefore, the optimization of the well spacing should be dominated by the reduction of the well spacing, and then by assessing the fracturing scale of the parent-well and the range of pressure drop caused by the cumulative production, and the differences between the parent and child-wells in terms of fracturing mode and scale, etc, and comparing the development results of the child-well and parent-well, the optimal well spacing can be selected at last.

We hold that reducing the well spacing is an inevitable choice to greatly increase the utilization rate of reserves, and also the technical direction to achieve efficient development for unconventional reservoirs. Since the current well spacing of domestic shale gas and tight oil and gas reservoirs is generally large, in order to avoid the deployment of infill wells, the shortcomings in North American well spacing optimization research can be set aside for now, their experience and the concept of reducing well spacing can be learned, which combines with tight spacing, can realize the full use of reserves with one-time development, to greatly reduce development costs.

Usually, after the mother-well is produced for a period of time, a pressure drop zone would gradually form in the range of the fracture. The energy deficit would change the geostress field, and even cause stress reversal, which could make the artificial fractures created by the repeatedly fracturing of the parent-well approach the pressure drop zone and deflect^[95,96]. The fracturing of child-well could also be affected by the “attraction” of the reversal stress, approaching the fracturing zone of the parent-well or even ending in fracture hit (Fig. 12a), and thus unsatisfactory fracturing effect. In order to prevent the fractures of the child-well from getting close to the fluid deficit zone (the stimulated zone of the parent-well), North America proposed a protective fracturing measure for the parent-well^[97]. The specific implementation method is that before the child-well fracturing, the parent-well is refractured and doesn’t immediately flow back (Fig. 12b). This can supercharge the energy storage of the initial fractures and increase the in site stress in the fluid deficit area to achieve reconstruction of the stress field. In this way, the fractures of the child-well can be prevented from deflecting to the stimulated area, effectively improving production of the new wellpad group and the infill well^[84].

Since the flow follows the principle of minimum resistance, when the pressure drop zone makes the stress reverse, the main flow direction of the fluid would also change accordingly, that is the oil and gas in the matrix between the two wells flow to the low stress zone. Especially after the formation of the mainstream channel, the change of seepage field is particularly obvious. For example, in the Wangyao Oilfield of Changqing, after 20 years of water flooding, a detection well was drilled 50 m away from an oil well, and the analysis of cores taken from the well showed that the reservoirs still had the original oil saturation. The conclusion was that the proportion of unflooded reservoirs reached 48%. Therefore, to tap the potential of unutilized reserves by repeated re-fracturing of old wells, it should be ensured that the injected fracturing fluid of the child-well uniformly spread on both sides of the horizontal section, to this end, reconstructing the seepage field is crucial. In the fracturing of the mother well, large injection rate and large liquid storage energy repeated fracturing, multiple rounds of water injection to supplement the energy of the formation, or temporary plugging diverting technology can be adopted. In the fracturing of child-well, water injection and volume stimulation can be selected to change the main seepage direction and improve the stimulation effect of the not stimulated area. The Well Group 56-101H in the Santanghu Block of the Tuha Oilfield adopted the well group model, in which the synergy effect of the well group has rendered high production of the fracturing wells in the old area, benefit to the adjacent wells, and good development effect. The oil production after fracturing was 63 t/d, which was 3.5 times higher than that of the adjacent well. At the same time, four old wells in the well group were benefited, and their daily production nearly doubled (from 13.8 t to 25.8 t). Therefore, according to the cumulative production of the parent-well, calculating the deficit volume and pressure drop range of the reserve, determining the scale of the storage liquid, and reconstructing the stress field and the seepage field with the corresponding technology, can reduce the “traction” effect of the parent-well on the fractures of the child-well, so as to protect the parent-well and improve the stimulation effect of the child-well.

How to achieve effective segmentation is the biggest problem in the current horizontal well re-fracturing. Double- packer fracturing tool and coiled tubing fixed-point fracturing, difficult to increase injection rate, are limited in working range, making it unable to meet the needs of volume stimulation. Whereas the re-fracturing by ball dropping or temporary plugging agent can’t achieve effective sealing and thus realize the reconstruction of the stimulation object, so they all belong to the general fracturing category. At present, the main method for re-fracturing the wellbore is the expansion tube technology^[98], in which the horizontal section is completely sealed by the expansion tube, and then re-fractured. This method has certain advancement. However, this technology still has the problem of new fractures propagating towards old fractures. Therefore, the authors proposed an idea of re-fracturing: First, the seepage field and the stress field were changed by means of energy storage, so that the direction of the new fractures would not be affected by the old fractures; the second was to inject degradable high-intensity temporary plug agent into the formation, the agent would completely seal the old fractures, realizing the re-fracturing of the wellbore. When the re-opened expansion fracture encounters the old fracture, since the old fracture is blocked by high-strength and high-plastic material, the fracture would find a new opening and extending path, thereby realizing the stimulation of the un-fractured area. Moreover, domestic high water cut oilfields can be re-fractured by similar technique.

Re-fracturing in the form of volume stimulation is to rebuild the seepage field, stress field, and stimulated object to tap the potential of the old well and improve the benefit of the well group.

Volume stimulation technology has experienced nearly 10 years of development and practice since it was proposed formally. Combining the latest theoretical research progress and application status, as well as the authors’ research and field practice experience in the past years, this article explains the essence of horizontal well volume stimulation, and makes it clear the “maximum, shortest, smallest” stimulation mechanism is the key to volume stimulation technology and the basis for the application of this technology in a wider range of fields. Meanwhile, the tight spacing and spatial-mode stimulation is the expansion and sublimation of the application of the volume stimulation technology. In the future, through the technical methods of tight spacing and decreasing well spacing, the development model of “fracture-controlled” recoverable reserves will be established, overthrowing the fixed ideas of traditional well-controlled reserves calculation and development, and guiding the effective application of volumetric re-fracturing technology to achieve the recovery of unutilized reserves. This will be an important direction for the future development and application of volume stimulation technology, and it is also an important technical guarantee for sustainable development.

Nomenclature

C— correction factor, ranging from 0.56 to 0.89;

D_i— perforation diameter corresponding to the ith fracture, mm;

d_p— particle diameter of proppant, m;

F_CD— dimensionless conductivity;

g— gravity acceleration, m/s²;

K_f, K_m— permeability of fractures and matrix, mm²;

k_h— dimensionless horizontal stress difference coefficient;

L_f — half length of the fracture, m;

N— number of fractures;

n_i — number of perforations for the ith fracture;

p_w,i — the bottom hole pressure of the ith fracture, Pa;

Q₀— injection rate in fracturing, m³/min;

Q_i — injection rate of the ith fracture, m³/min;

$\dot{U}$— rock strain energy per unit injection time, J/min;

v— settling rate of proppant, m/s;

${{\dot{W}}_{\text{c}}}$— rock fracture energy per unit time, J/min;

${{\dot{W}}_{\text{f}}}$— energy consumed by the flow friction in the fracture per unit time, J/min;

${{\dot{W}}_{\text{I}}}$— overcoming the energy of additional stress generated by other fractures per unit time, J/min;

${{\dot{W}}_{\text{p}}}$— energy consumed by perforating friction per unit time, J/min;

${{\dot{W}}_{0}}$— energy to overcome the far field stress per unit time, J/min;

w_f — fracture width, mm;

x,y—rectangular coordinate system, m;

μ— fluid viscosity, Pa·s;

ρ— density of liquid or sand carrying liquid, kg/m³;

ρ_f — fluid density, kg/m³;

ρ_p — proppant density, kg/m³;

σ_H, σ_h — maximum and minimum horizontal principal stress, Pa;

σ_v — overlying rock pressure, Pa;

σ_I,i,j — the ith fracture interfered by the stress caused by the jth fracture, Pa;

τ— the tortuosity of the proppant filling fracture, dimensionless;

ϕ_f — fracture porosity, %.