Tight oil in China has rich reserves and great exploitation potential. The Ordos Basin is an important production base of tight oil in China, which contributes more than 50% of China’s total output of tight oil ^[1,2]. However, tight oil reservoirs are generally characterized by poor physical properties, strong heterogeneity, and poor pore connectivity, leading to rapid decline of production rate of wells and the difficulty to replenish the reservoir energy by waterflooding. The key and difficult point of tight oil reservoir development is to change development methods, enlarge oil drainage area, and reduce oil flow distance. Nowadays, injection-production well pattern adjustment and well re-fracturing are effective measures to keep tight oil production stable ^[3,4,5,6,7].

During waterflooding, fluid injection and production in a tight sandstone reservoir will cause changes in the reservoir pore pressure, leading to dynamic variations of in-situ stress field over time, i.e., four-dimensional (4D) stress evolution (3D stress changes over time). Through fracture monitoring and seismic test data, Wright ^{[8, 9]}, Kristiansen ^[10], Dons ^[11], Zhang ^[12], and Minner ^[13] et al. proved that injection and production in reservoir would make the in-situ stress change in magnitude and direction, and then changes in in-situ stress would cause fractures in parent wells and infill wells to change in shape. In addition, changes in in-situ stress field will lead to formation deformation, and thus causing fault reactivation ^{[14,15,16,17]} and casing deformation ^[18], and bringing about safety risks to production. Based on field test data and numerical simulation methods, Wang ^[19], Zhao ^{[20, 21]}, Fan ^[22], and Li et al. ^[23] found that long term water injection in reservoir would activate natural fractures or cause initiation of new cracks and propagation of existent cracks. Therefore, studying the stress evolution during waterflooding is vital for the efficient development of tight sandstone reservoirs.

Coupling simulation of multi-physical fields considering stress evolution during reservoir development has long been a hot topic for researchers in China and broad. In 1980, Hagoort ^[24] and Settari ^[25] first established a numerical model coupling reservoir seepage and fracture propagation in consideration of the reservoir’s pore pressure and stress evolution during production and injection. Yue et al. ^[26] and Dong et al. ^[27] studied the stress re-orientation in homogeneous formation during production and its impact on propagation of fractures in re-fractured well by using a theoretical model. By using the displacement discontinuity method (DDM) and finite element method (FEM), Roussel et al. ^[28] investigated the reservoir stress changes induced by injection and production of fracturing wells and their effects on fracture propagation of infill wells. Abou-Sayed examined local stress field changes around the wellbore in a homogeneous formation during injection and production considering the thermo-elastic effect ^[29]. Hwang et al. ^[30] systematically studied in-situ stress changes of homogeneous reservoir and interwell interaction during waterflooding considering rock thermal and pore elasticity. Previous studies mainly focus on the rules of stress field changes around wellbore and between injectors and producers in homogeneous reservoir; while researches on the stress evolution of heterogeneous tight sandstone reservoir are rare.

In this study, Yuan 284 Block of Huaqing Oilfield was taken as an example to investigate the 4D stress evolution during production and injection of tight sandstone reservoir by multi-physical modeling method considering reservoir heterogeneity, hydraulic fracture, and complex production-injection system. The flow and geomechanics coupled model of the tight sandstone reservoir in Yuan 284 Block was built based on 3D geological and geomechanical models of the reservoir, hydraulic fracture simulation and 4D stress evolution of the reservoir. The accuracy of model was verified by well test data, microseismic (MS) monitoring results, and re-fracturing data. The model can be used to study the evolution of reservoir stress field during long term waterflooding. Results of this work can provide theoretical guidance for reservoir well pattern adjustment and re-fracturing design.

The numerical simulation method of stress evolution of tight sandstone reservoir during long term waterflooding in this study considers reservoir heterogeneity and actual production-injection regime of the well network comprehensively. The formation properties, seismic data, logging data, and experimental data, etc are used to build the reservoir geological and geomechanical models. The hydraulic fracture propagation model is embedded to find out the conductivity increase caused by hydraulic fracture. The hydraulic fracture parameters are calculated based on fracturing data and calibrated with MS data. An iterative coupling approach is adopted in this method to ensure practicability and computational timeliness. On the basis of the geological model, a reservoir fluid flow model based on the finite difference method (FDM) is established. The equivalent algorithm is used to convert the hydraulic fracture conductivity into equivalent permeability for fluid flow calculation. Meanwhile, the initial pore pressure and stress data are taken as the initial conditions of the geomechanical model based on the finite element method (FEM), and the pore pressure obtained at each time step from the flow model is used as the boundary condition to simulate the fluid flow and geomechanics of the reservoir during waterflooding. In addition, the stress sensitivity calculation model is included in this method to simulate the effect of stress changes on porosity and permeability. As the flow and geomechanical models have differences in grid data structure, a grid mapping program independently developed is used in this method to convert the data of different grids. This method makes full use of the computational advantages of reservoir flow, geomechanical, and hydraulic fracture propagation models, combines all the models seamlessly, and has good compatibility and flexibility.

1.2.1. Flow equations

According to properties of the tight sandstone reservoir, the two-phase flow model with single porosity and permeability medium is selected. It is assumed that the oil and water in the porous medium are weakly compressible and the porous medium is hydrophilic. The flow model is isothermal and follows Darcy’s law. The two-phase flow equations based on mass balance are expressed as follows:

The relationship between oil and water saturation is expressed as follows:

The pore pressure can be considered as the average oil-water phase pressure:

The relationship between the oil and water phase pressure and capillary pressure is expressed as:

The external boundary (Г) is assumed to be sealed, and the external boundary condition is as follows:

The oil and water phases at the internal boundary (wellbore) satisfy:

1.2.2. Deformation equations of rock

For linear elastic solid media, the geomechanical equilibrium equation is expressed as follows:

The stress-strain evolution of porous medium (reservoir) is calculated based on Biot’s effective stress theory^[31]:

Supposing that the reservoir rock saturated with oil and water has small elastic deformation at certain speed. In this case, the reservoir rock is in dynamic equilibrium state, so Eq. 9 can be re-written as:

The relationship between rock displacement and strain is as follows:

Generally, the mechanical deformation of formation rock is anisotropy, so the relationship between stress and strain of the formation rock can be expressed as ^[32]:

The boundary conditions include displacementboundary condition and stress boundary condition, which are respectively:

1.2.3. Stress sensitivity model

For poro-elastic medium like formation rock, changes in pore pressure and stress field will lead to deformation of formation rock and change in porosity. According to the Carman-Kozeny formula, permeability is closely related to porosity, so prediction formulas of porosity, permeability considering the stress sensitivity can be expressed as follows ^[33]:

1.2.4. Coupling method of flow and geomechanics

Based on coupling solution approaches, flow and geomechanics coupling methods can be classified into fully coupling, iteratively coupling, or one-way coupling. Generally, the fully coupling approach has high calculation accuracy but low efficiency. The coupling calculation is not easy to converge and can’t reflect the heterogeneous characteristic of the reservoir. In contrast, the one-way coupling approach has higher calculation efficiency but lower calculation accuracy ^[34,35]. The iterative coupling can meet the requirements of formation heterogeneity and has higher computational efficiency and accuracy ^[36]. Therefore, iterative coupling was adopted in the coupling simulation of tight sandstone reservoir flow and geomechanics.

In classic iterative coupling, the pore pressure is calculated by the flow model at one time step. The pore pressure is passed to the geomechanical model as the boundary condition, and the stress and deformation are updated. Porosity and permeability changes are obtained from the stress sensitivity equations and then passed to the flow model in the next time step to conduct the coupling calculation. This means that the iteration must be done at least once in each time step before entering the next time step of calculation. This is laborious and time-consuming. Different from the classic iteration method, the iteration method used in this study can complete update of pore pressure, stress, strain, porosity, and permeability at all time steps in one iterative cycle. If the pore pressures at all time steps in one iteration converge compared with those in the last iteration, the coupling calculation is complete. If not, one more iteration should be performed, as shown in Fig. 1. In the coupling simulation of flow and geomechanics, the change rate of the pore volume can be reduced by shortening the time step and adjusting the grid size to improve the iteration convergence. This method can reach the required accuracy through multiple iterations, allow independent operations and calculations of the flow, geomechanical, and stress sensitivity models and smooth data conversion between these models ^[37].

In the process of iterative coupling simulation of flow and geomechanics, the flow model built by FDM and the geomechanical model built by FEM have different grid structures. Therefore, an independently developed grid mapping program is used in this method to enable conversion between different grids. Based on the spherical adaptive search algorithm, this program can transmit data from high-density nodes to target nodes by a proximal point algorithm, and transmit data from low-density nodes to target nodes by linear interpolation algorithm. See details in our previous articles ^[38,39].

According to the reservoir properties and fracturing process, proper models (e.g., analytical model and discrete fracture model) can be selected to simulate fractures and calculate the geometric, conductivity, and permeability of fracture. In Oda method, the permeability of each fracture is projected into the flow model grid in different directions as second order equivalent fracture permeability tensor ^[40,41]. The permeability tensor can be used as fracture permeability in the reservoir model or added to the matrix permeability in reservoir flow simulation.

The re-fracturing pilot, Yuan 284 Block located in the southwest of the Yi-Shan Slope of Ordos Basin, is taken as an example in this study. The Chang 6₃ layer of the Yanchang Formation is the main production layer in this block. This layer has nine sublayers. Chang 6₃¹, the major pay, is at burial depths of 2092 to 2134 m (altitude of -640 to -682 m). The Chang 6₃ is 55 to 90 m thick, and about 70 m thick on average. Reservoirs in this area are generally poor in physical properties, representing tight sandstone oil reservoirs, with porosity from 10.0% to 15.0%, on average 12.4%, and permeability from 0.20 × 10^-3 to 0.50 × 10^-3 μm², on average 0.37 × 10^-3 μm². But due to natural fractures, reservoirs have permeability greater than 1.00 × 10^-3 μm² in local parts, showing strong intra- and interlayer heterogeneity. The maximum horizontal principal stress of formation Chang 6₃ is at the direction of NE75°. The principal vertical stress is the largest among the three principal stresses, followed by the maximum horizontal principal stress and minimum horizontal principal stress. Hence it is regarded as a normal faulting regime. In view of the reservoir properties, this block has been developing by well-pattern with horizontal wells as producers and directional wells as injectors, with satisfactory results achieved ^{[3, 42]}.

According to the injection-production system and calculation requirements, the stress evolution of Chang 6₃ in the horizontal well development zone was simulated in this study. This zone is 3800 m × 1740 m on the plane, and has 16 wells drilled, including five horizontal wells as producers and 11 directional wells as injectors. The horizontal wells are about 720 m apart, and the directional wells are about 650 m apart on average. The horizontal wells were put into production after hydraulic fracturing with guar gel fluid. The directional wells began water injection at high pressure close to the minimum horizontal principal stress of the reservoir after high-energy gas fracturing.

The flow and geomechanical models were built based on the reservoir characteristics and fracturing process, and development scheme. Table 1 shows the main parameters of the models. The models had assumptions: (i) the reservoir was tight oil one without edge and bottom aquifer; (ii) the external boundary displacement of the geomechanical model was zero; (iii) water injection could improve the reservoir permeability around the wellbore, and thus, the permeability around the injection wellbore would be adjusted based on history matching; (iv) the reservoir was a continuous medium.

To balance the calculation efficiency and accuracy, the flow model had grids of 20 m × 15 m on the plane, and 2 m high on average based on the layer thickness. The geomechanical model had grids of 10 m × 10 m × 5 m. The 4D stress evolution in the horizontal well-pattern during long-term production and injection was studied according to the reservoir properties, hydraulic fracturing treatment parameters, and production data.

There are five production wells in the target zone. They are all horizontal ones, with depths ranging from about 2770 to 3120 m and horizontal sections about 450 to 840 m. They were completed by multi-stage hydraulic jet fracturing with guar gel. They were fractured at 7 to 10 stages, with two clusters in each stage. Each stage used about 160 m³ fracturing fluid and about 27.5 m³ proppant on average. The hydraulic fractures generated in the initial fracturing of these wells were simple two-wing ones based on natural fracture characteristics of the reservoir and MS monitoring results. In this work, geometric parameters and conductivities of hydraulic fractures in all the wells were obtained through net pressure matching based on petrophysical and geomechanical parameters, stress field data, fracturing data, and MS monitoring data of the reservoir. The fracturing parameters of each stage and simulated fracture parameters in Well PH2 are shown in Table 2. From the simulation, fractures in this well are 200 to 220 m in length, 35 to 38 m in height, and 180 × 10^-3 to 220 × 10^-3 μm²·m in conductivity. The distribution of fractures in the horizontal wells is shown in Fig. 2. To lower the risk of water fingering, fractures in each well are arranged in spindle shape along the horizontal section, namely, fractures near the injection well are shorter, while fractures further from the injection well are longer. The fractures in the wells range from 170 to 250 m in length and 130 × 10^-3 to 280 × 10^-3 μm²·^.m in conductivity from simulation. Oda’s method was used to convert the fracture conductivity into equivalent permeability, and the permeability of each fracture was projected into the grid of flow model to form the matrix-fracture flow model of the tight sandstone reservoir.

A single-porosity, single-permeability, and two-phase flow model of the formation Chang 6₃ was built on the Eclipse reservoir simulation software, and the geomechanical model was built on the Abaqus platform, to study stress evolution in horizontal and directional well-pattern during waterflooding and the effect of stress on fracture propagation during re-fracturing. The updated pore pressure from the flow model was taken as the boundary condition of the geomechanical model. By using stress sensitivity equations, the flow and geomechanics of the reservoir were simulated jointly based on production and injection data of these wells from the first fracturing (December 2011) to the time before the re-fracturing (December 2016).

2.4.1. Results of flow history matching

The flow simulation was done with observed oil rate and bottom hole pressure (BHP). Fig. 3 shows the history matching results of injector J2-1 and producer PH2 by the flow model. The history matching results show the injection well had a stable flow rate, the producer has kept stable in production (about 3.0 t/d in the later period) and a BHP of 5 MPa. Compared with the observed BHP, the simulation results have errors of less than 8%, showing high credibility.

2.4.2. Coupling simulation results of flow and geomechanics

The wells in the target zone have been put into production or injection successively since December 2011, and several wells were re-fractured after December 2016. In this period, the injectors had an average cumulative water injection volume of about 21 880 m³ and an average flow rate of about 13.98 m³/d. The producers had an average cumulative fluid production of about 7580 m³, an average daily fluid production of about 5.08 m³, average cumulative oil production of 5172 t, and an average daily oil production of about 3.46 t. Fig. 4 shows the simulated pore pressure of the reservoir before and after waterflooding. Due to water injection and production, the pore pressure increased near the injectors but decreased near the producers. The initial pore pressure of the reservoir ranged from 15.5 to 16.2 MPa; the pore pressure rose to 30-38 MPa near the injectors but dropped to 7-10 MPa near the producers by December 2016 (after 60 months of injection and production).

The reservoir in the target zone had an initial vertical principal stress of 55 to 58 MPa, initial maximum horizontal principal stress of 41 to 45 MPa, and initial minimum horizontal principal stress of 33 to 36 MPa. Affected by the changes in pore pressure, the three principal stresses increased near the injectors but decreased near the producers; but they differ in variation pattern on the plane (Fig. 5). By December 2016, the principal vertical stress had risen to 60-68 MPa near the injectors but dropped to 51-54 MPa near the producers. The maximum horizontal principal stress was 48-52 MPa near the injectors but dropped down to 39-45 MPa near the producers. The minimum horizontal principal stress rose to 40-43 MPa near the injectors but dropped down to 29-32 MPa near the producers.

2.4.3. Validation of simulated minimum horizontal principal stress

To a certain extent, the closure pressure of fractures in the re-fractured well can reflect variation in minimum horizontal principal stress. Fig. 6 shows the simulated minimum horizontal principal stresses and actual closure pressures of some stages in the re-fractured wells. Affected by water injection and different production contribution of different fracturing stages, the minimum horizontal principal stress is lower near the toe of PH2 (stages 1-4) and PH3 (stages 2-4) but varies little at all stages of PH4 and PH5. Compared with the closure pressure of fractures in the re-fractured wells, the simulated results of minimum horizontal principal stress have errors of 0.2%-13.2% (7.6% on average), proving the accuracy of the model.

Affected by water injection and oil production, the pore pressure increased near the injectors but decreased in the fractured zones near the producers (Fig. 4). But there are large unaffected zones between the wells because of the large well spacing of injectors and low permeability of the reservoir. Moreover, with different petro-physical properties, and positions and extension of hydraulic fractures, different stages in a horizontal producer differ in flow conditions and production contribution, so different stages along the horizontal section differ in the variation pattern of pore pressure. The variation pattern of pore pressure in this study is similar to the conventional understanding, so it is only briefly explained.

3.2.1. Vertical principal stress

The vertical stress changes largely near the wellbores and in the stimulated zones, and is proportional to the pore pressure, but has slight differences in variation magnitude (Fig. 5a). The vertical stress is mainly affected by compaction and is hardly affected by changes in other stresses, so it has consistent variation zone and pattern with pore pressure. The variation pattern of vertical stress in this study had little difference from classic understanding, and the variation of vertical stress had little guidance to the development plan in this study, so the vertical stress is only briefly discussed in this article.

3.2.2. Horizontal principal stresses

The two horizontal principal stresses have evolution trend similar with pore pressure near wellbores or in stimulated zones, but they change in obvious strips along their respective directions, as shown in Fig. 5b and 5c. Variations in pore pressure will cause variations in effective stress and in turn reservoir rock deformation. But in places where the pore pressure is stable but the two horizontal principal stresses vary in different ways, the variation trend of the two horizontal principal stresses may differ from that of the pore pressure ^{[28, 30]}. For example, the pore pressure increased near the wellbore of injector J1-2 during water injection, while the effective stress decreased, and the formation rock in the depleted zone expanded, squeezing surrounding rock mass. The zone between J1-2 and J1-1 was mostly squeezed in the minimum horizontal principal stress direction, making the minimum horizontal principal stress there increase significantly. Similarly, the zone between J1-2 and J2-2 mainly had squeeze in the maximum horizontal principal stress direction, so the maximum horizontal principal stress there increased noticeably, leading to divergence between the change trends of the principal horizontal stresses and pore pressure.

Three representative sections of the main production layer Chang 6₃¹, AA° (along the horizontal section of Well PH2), BB° (the section of J1-1, J1-2, and J1-3), and CC° (the X-direction section over J1-2), were taken as examples to analyze the evolution of the principal horizontal stresses during waterflooding (Fig. 7).

Figs. 8a and 9a show the variations of two horizontal principal stresses along the horizontal section of Well PH2 (AA° section). It can be seen the minimum horizontal principal stresses at the 3 injectors changed little after 2013, but the maximum horizontal principal stress increased year by year. At this position, the water injection would offset the decrease of maximum horizontal prin-cipal stress caused by production, and even make the horizontal principal stress increase over its initial value. In the section of injectors (BB° section), in the zones between the injectors not swept by pore pressure, the maximum horizontal principal stress decreased, while the minimum horizontal principal stress increased year by year and even exceeded the initial value (Figs. 8b and 9b). In the X-direction section through Well J1-2 (CC° section), significantly affected by water injection in adjacent injectors, the horizontal principal stresses at producers PH1 and PH2 changed in different ways, in the stimulated zones of them, the minimum horizontal principal stress decreased with the decrease of pore pressure, but changed little outside the stimulated zones. In contrast, in the stimulated zones, the maximum horizontal principal stress began to rise noticeably after 2013, as shown in Fig. 8c and 9c. In comparison, producers PH4 and PH5 are farther from the injectors, and the water injection had little effect on the horizontal principal stress near these two wells. The variations of horizontal principal stress near these two wells were mainly affected by the pore pressure gradient; the greater the pore pressure gradient, the more significant the stress variation was, and even stress reversal could occur ^{[28, 43-44]}. But as the horizontal principle stress difference in this reservoir was relatively large, and the pore pressure gradient near the producers was not large enough due to the improvement of matrix permeability by hydraulic fractures, the horizontal stress field at the producers didn’t reverse ^[30].

Fig. 10 shows the directions of minimum and maximum horizontal principal stresses of the study area in December 2016. The principal horizontal stresses were orthogonal. By December 2016, the horizontal stresses in this area shifted 0° to 30° in direction, the shift in direction was especially obvious near the injectors, where the maximum horizontal principal stress was distributed radially. During the production and injection, the stress field variation was mainly affected by poro-elasticity effect, thermo-elasticity effect, and hydraulic fracture opening ^{[8, 28-30]}. For porous medium like rock, the increase of pore pressure would make the effective stress decrease, the rock swell, and the normal and tangential stress decrease, so the stress direction diverted due to extrusion. Moreover, the more significant the pore pressure variation, and the greater the pore pressure gradient, the more significant the stress variation and stress direction diversion would be. This has been confirmed in other articles ^{[28, 30, 45-48]}.

3.2.3. Difference of horizontal principal stresses

Fig. 11 shows the distribution of horizontal principal stress difference in formation Chang 6₃¹ in December 2016 (after 60 months of production and injection). The initial horizontal principal stress difference ranged from 7 to 9 MPa. Affected by variations in horizontal principal stresses, the horizontal principal stress difference at the horizontal sections of producers squarely across the injectors rose most significantly to 12.4-13.8 MPa, but changed little in other horizontal sections. Meanwhile the zones between injectors not swept by pore pressure dropped most significantly in horizontal principal stress difference (to 1.2-5.8 MPa by December 2016). These places were most likely to have stress reversal. In contrast, the horizontal principal stress difference near the injectors had little variations.

During the five-year waterflooding of tight sandstone reservoir in Yuan 284 Block, the pore pressure decreased by up to 9.0 MPa near the producers, but increased by up to 22.8 MPa near the injectors. Accordingly, the horizontal principal stress difference increased by up to 52.7% along the horizontal wellbore close to the injectors and decreased by up to 84.6% in the zones between the injectors not swept by pore pressure. Clearly, long-term waterflooding caused significant changes of the stress field, as a result, water fingering and invalid injection became increasingly severe, and it is necessary to take measures to address these problems. During long term waterflooding, stress diversion will lead to changes in hydraulic fracture propagation direction; the minimum horizontal principal stress directly affects the fracturing pressure, and asymmetrical distribution of minimum horizontal principal stress on both sides of the wellbore can induce frac-hit, resulting in asymmetrical distribution of hydraulic fracture on two sides of the wellbore^{[46-47, 49-51]}. In addition, the stresses in this block conform to the normal faulting regime, so the horizontal principal stress difference would affect the geometry of branch fractures generated in fracturing. To make up the production loss, a scheme including re-fracturing of horizontal producers and conversion of injectors to producers was proposed based on 4D stress evolution of the reservoir.

4.1.1. Optimization of re-fracturing scheme for horizontal producers

The horizontal wells in Yuan 284 Block were fractured for the first time at average pumping rate of about 3.2 m³/min with fracturing fluid of about 160 m³. That means the fracturing was small in scale, so there are several large zones between the injectors not swept by water where the horizontal principal stress difference is small, these zones are the best sites for fracturing. Therefore, these zones should be given priority in horizontal well re-fracturing. The fracturing scale should be increased in order to create more complex branch fractures. Meanwhile, affected by water injection, the stresses at the injectors changed in direction, and the maximum horizontal principal stress is distributed radially. This would cause fractures generated in refracturing of horizontal sections near the injectors to divert towards the injectors^{[45, 52]}. Based on the simulation results of reservoir stress field, geometric parameters of hydraulic fractures were calculated by the fracture modeling software. On this basis, the optimal pumping rate of 3.5 m³/min, optimal fracturing fluid volume for the horizontal well re-fracturing stages in between the injectors of 800-900 m³, and the fracturing fluid volume of the stages close to the injectors of less than 700 m³ were selected. Additionally, the fracturing fluid volume for the stages near injectors should be controlled properly based on MS data to prevent channeling.

4.1.2. Optimization fracturing scheme of injectors to be converted to producers

Due to long-term waterflooding, the evolution of stress field in Yuan 284 Block has changed the stress circumstance of the reservoir considerably. Therefore, we proposed to convert several water fingering and invalid water injectors to producers after volume fracturing to increase effective drainage volume around the injectors, adjust the injection profile and stress field reversely, depress fingering caused by early production, and increase oil production of the block. The simulation results of 4D stress evolution showed that the horizontal principal stress difference at the horizontal sections close to the injectors was larger. Accordingly, the fracturing fluid volume of less than 700 m³, and the optimal pumping rate of 6.0 m³/min were selected to stimulate the reservoir fully and prevent fractures from propagating to the high horizontal principal stress zones to cause frac-hit and channeling.

According to the scheme above, starting from December 2016, the five horizontal wells in the study area were re-fractured successively, and the eight injectors were converted to producers after fracturing. Volume fracturing was adopted. The fracturing used a mixture of slick water and guar gel as fracturing fluid and double-packer isolation pipe string. Some of the horizontal wells were re-fractured on the original fracturing stages after adding perforations. The fracturing operations in the horizontal wells were at an average pumping rate of about 3.4 m³/min, and used 600 to 860 m³ fracturing fluid (760 m³ on average), and 30 to 60 m³ proppant (on average 54 m³) for each stage. The injectors were fractured at an average pump rate of about 5.5 m³/min with about 675 m³ fracturing fluid (obviously less than that used in the horizontal wells) and about 66 m³ proppant for each stage.

Stress evolution would affect propagation direction and shape of fractures generated in horizontal well re-fracturing. For instance, in Well PH5 re-fractured from April 20 to May 13, 2017 at 10 stages with new perforation clusters added in the fifth and tenth stages, according to the distribution of in-situ stresses, smaller fracturing fluid dosage was used for the stages at the heel and toe ends of the horizontal section, while larger fracturing fluid dosage was used for the stages in the middle. The re-fracturing parameters and MS monitoring results of Well PH5 are shown in Table 3, and Fig. 12 shows distribution of MS events of this well. The fractures in stages 1-6 are asymmetrical, specifically, the fractures on the west side of the wellbore are much longer than those on the east side in these stages. In contrast, the fractures on both sides at stages 7-10 are basically symmetrical. Fig. 13 shows the minimum horizontal principal stress in the middle of the re-fractures and on both sides of the wellbore. At the toe end of the wellbore, the minimum horizontal principal stress on the east side of the wellbore is about 2.0 MPa greater than that at the west side; while at the heel end, the horizontal principal stress on the west side and east side of the wellbore have a small difference of about 0.3 MPa. The hydraulic fractures on two sides of the wellbore are asymmetrical when the difference of the minimum horizontal principal stresses on two sides of the wellbore is greater than 1.0 MPa. Therefore, the hydraulic fractures at the toe of the well propagate longer toward the west; while the fractures at the heel propagate symmetrically. Fig. 14 shows the horizontal principal stress difference at the same place as that in Fig. 13. It can be seen the difference of horizontal principal stresses on both sides of PH5 wellbore was small, but the difference at the heel and toe was about 8 MPa—significantly greater than the 6 MPa in the middle section. The width of the MS event points in the middle stages of the well was about 65-90 m, greater than the 30-50 m at the heel and toe. The hydraulic fractures extended much wider when the horizontal principal stress difference is less than 6.5 MPa. In addition, the fractures generated in re-fracturing of this well are about NE83°-NE93° at stages 1-2 and about NE62°-NE66° at stages 7-10. Compared with the initial azimuth of the maximum horizontal principal stress of NE75° in the target zone, the fractures at the toe and heel of PH5 obviously divert to the east injectors (J5-1 and J5-2) due to the stress re-orientation, with a maximum rotation angle of 18°.

The hydraulic fractures in the other horizontal wells at the stages between the injectors were 280-560 m long, about 440 m on average, and the half fractures on two sides of the wellbore had length differences of 30-250 m. But as horizontal principal stress difference on two sides of the wellbore of Well PH1, PH2, and PH3 is smaller, of 1.2-3.8 MPa, the widths of hydraulic fracture propagation ranges in these wells reached 80 to 250 m, much larger than those in Well PH4 and PH5. In addition, the fractures in the horizontal well sections against injectors diverted by 8° to 21°.

Without MS monitoring for injectors in this area, it is difficult to tell the sizes of hydraulic fractures in the injectors. But the pumping pressures during their fracturing didn’t significantly reduce, and the BHP of the adjacent wells didn’t increase, indicating that the hydraulic fractures haven’t reached the adjacent wells and the fracturing operations met the designed requirements.

Fig. 15a shows the oil production rates of the horizontal wells in this area before and after re-fracturing (December 2011 to July 2021). The horizontal wells after re-fracturing had oil production rates about 45%-135% (60% on average) of those after the initial fracturing, which were 3 times higher than before refracturing. These wells had an average oil production rate of about 1.78 t/d in stable production period, and effective stimulation period of over 1200 days, showing good stimulation effect. As a result, their production decline in the late period has kept slow. Fig. 15b shows the oil production rates of the producers converted from injectors after fracturing (December 2016 to July 2021). These wells after fracturing had an average initial oil production rate of about 1.07 t/d, and average oil production rate about 0.57 t/d in stable production period, similar with directional production wells in nearby blocks. Moreover, they had stable oil production rates after fracturing, and some of them even had steady rise of oil production rates, for example, J2-1. The re-fracturing of horizontal producers and conversion of injectors to producers have not only enhanced oil production of the block, but also eliminated the risk of water fingering in the horizontal wells, resulting in good effect.

After five years of waterflooding, the three principal stresses of the reservoir in Yuan 284 Block of the Ordos Basin have increased by 6-13 MPa near the injectors and decreased by 2-6 MPa near the producers. The principal horizontal stresses are distributed in obvious strips along their respective stress directions. In the injector row direction, the horizontal principal stress difference has dropped by 2-8 MPa in the zones between the injectors; and the maximum horizontal principal stress has diverted 0 to 30°, this is especially obvious near the injectors, where it is distributed radially.

Stress evolution during waterflooding affects the shape and growth direction of fractures generated in horizontal well re-fracturing. Hydraulic fractures tended to grow longer in the side with smaller minimum horizontal principal stress when the minimum horizontal principal stresses on two sides of the wellbore had differences greater than 1 MPa, resulting in asymmetrical growth of fractures on two sides of wellbore. The hydraulic fractures had larger propagation width of 65 to 250 m in zones where the horizontal principal stress difference was less than 6.5 MPa. In addition, the growth direction of fractures generated in re-fracturing of the horizontal wells diverted and pointed toward the nearby injectors affected by the water injection in adjacent injectors, with a maximum rotation angle of 21°.

According to the 4D stress evolution pattern of the reservoir, the plan of horizontal producer re-fracturing and fracturing of producers converted from injectors had been made: The fracturing fluid for the horizontal well stages away from the injectors should be increased to enhance the length of fractured range from 210 to 440 m; the fracturing fluid used for the injectors and the horizontal well stages close to the injectors must be controlled at less than 700 m³. After implementation of the plan, the horizontal wells increased by three times in oil production rate; the converted producers had stable production rate equivalent to directional production wells in the nearby blocks; and the fingering risk of the horizontal wells has been eliminated.

B_o, B_w—oil and water volume factor, m³/m³;

E_i—anisotropic elastic modulus vector, where i =1, 2, 3, Pa;

F_i—body force tensor of rock mass, where i =1, 2, 3, N/m³;

g—acceleration of gravity, m/s²;

g₁, g₂—boundary function;

G_ij—anisotropic shear modulus of rock mass, where i, j=1, 2, 3, Pa;

h—formation thickness, m;

H—vertical depth, m;

K_i—anisotropic absolute permeability tensor of matrix, where i =1, 2, 3, m²;

K—matrix permeability, m²;

K₀—initial matrix permeability, m²;

K_o, K_w—oil and water permeability, m²;

K_ro, K_rw—relative permeabilities of oil and water;

n—serial number of time step;

p—average pore pressure, Pa;

p_cow—capillary pressure, Pa;

p_o, p_w—oil and water pressures in the pore space, Pa;

q_o, q_w—oil and water flow rate, m³/s;

r—arbitrary variable;

r_w—wellbore radius, m;

s—serial number of iterative step;

S_o, S_w—oil saturation and water saturation;

t—time, s;

V—grid cell volume, m³;

u_i, u_j—rock mass displacement vector, where i, j=1, 2, 3, m;

α—Biot’s effective stress coefficient;

β—conversion factor, dimensionless;

δ_ij—Kronecker delta;

ε_ij—strain tensor, where i, j=1, 2, 3;

ε_v—bulk strain;

μ_o, μ_w—oil viscosity and water viscosity, Pa^.s;

ρ_o, ρ_w—oil density and water density, kg/m³;

σ_ij,σ_ij^'—stress tensor and effective stress tensor, where i, j=1, 2, 3, Pa;

υ_ij—anisotropic Poisson's ratio tensor, where i, j=1, 2, 3;

ϕ—matrix porosity, m³/m³;

ϕ₀—initial matrix porosity, m³/m³.