Well Pattern and Well Spacing Optimization of Large Volume Water Injection in a Low-Permeability Reservoir with Pressure Sensitivity

Even with the fractured wells, the primary oil recovery of low-permeability reservoirs is still poor in Block X of Shengli Oil ﬁ eld. To further enhance the oil recovery, water is injected into the reservoir. Di ﬀ erent from the conventional injection scheme, the maximum daily injection rate of the proposed scheme by Shengli Oil ﬁ eld reaches 2000 m 3 , and the average daily injection rate is around 1500 m 3 . Thus, the conventional well spacing of certain well pattern is not suitable for the novel injection scheme. In the paper, the optimal well pattern and well spacing for the large volume water injection scheme to develop a pressure-sensitive low-permeability reservoir is investigated. Firstly, the CMG is employed to build the basic reservoir model developed by fractured vertical wells. To ﬁ nely depict the pressure sensitivity, the dilation-recompaction geomechanical model is introduced to couple with the basic reservoir model. Based on the established coupled model, the optimal well spacing for the inverted 5-spot well pattern and the inverted 9-spot well pattern is investigated with a total of 80 sets of numerical experiments. The numerical experiments indicate that the optimal well spacing for the inverted 5-spot well pattern is 850 m/ 350 m and the optimal well spacing for the inverted 9-spot well pattern is 550 m/450m. To further screen the well pattern, the normalized index of oil production per unit area of each well pattern is proposed. And it is found that the oil production per unit area of the inverted 5-spot well pattern is higher than the inverted 9-spot well pattern. For the reservoir developed with fractured vertical wells coupled with large volume water injection, compared with the inverted 9-spot well pattern, the inverted 5-spot well pattern is better, and the corresponding optimal well spacing is 850m/350 m. The paper proposes an e ﬃ cient simulation and optimization work ﬂ ow for the development of pressure-sensitive low-permeability reservoirs with fractured vertical wells coupled with large volume water injection, providing practical guidance for the e ﬃ cient and sustainable development of pressure-sensitive low-permeability reservoirs.


Introduction
There are abundant low-permeability reservoirs in China and worldwide, which have become a key component in the global energy system. For instance, 46% of China's oil and gas resources are of low quality, which are mostly lowpermeability reservoirs [1][2][3][4]. Due to the nature of the lowpermeability reservoir, the reservoir does not perform well if the EOR/EGR technology, such as fracking and acidizing, has not been applied. Over the years, the technology of developing low-permeability reservoirs has been significantly improved. But water injection is still the primary method to improve the performance of low-permeability reservoirs [5][6][7][8]. Shengli Oilfield proposes an innovative waterflooding scheme with large volume injection to further improve the low-permeability reservoir performance in Block X, which has been depleted for years with low productivity [9]. There are two factors accounting for the improvement of reservoir performance. On one hand, the large volume injection will increase the swept volume. On the other hand, the residual oil located in the region with high seepage resistance can be further mobilized due to the established high-pressure driven system induced by the large volume of injected water so that the displacement efficiency can be further improved. To some extent, the reservoir is rejuvenated due to the novel injection scheme. Different from the conventional injection scheme, the maximum daily injection rate of the proposed scheme by Shengli Oilfield reaches 2000 m 3 , and the average daily injection rate is around 1500 m 3 . Therefore, the conventional well spacing of certain well pattern is not suitable for the novel scheme, large volume water injection coupled with fractured vertical producers. The inverted 5-spot well pattern and the inverted 9-spot well pattern are widely implemented in the field. Meanwhile, the optimal well pattern and well spacing under different field conditions have been investigated based on machine learning (data-driven modeling) and numerical simulation [10][11][12][13]. But few studies have been reported on the optimization of well pattern and well spacing of the novel scheme proposed by Shengli Oilfield to develop the pressuresensitive low-permeability. In the paper, we build up the coupled reservoir models to represent different scenarios (different well spacing for both the inverted 5-spot well pattern and inverted 9-spot well pattern of the novel scheme to develop a pressure-sensitive low-permeability reservoir) via the advanced reservoir simulator, CMG. With a total of 80 sets of numerical experiments, the optimal well spacing for both the inverted 5-spot well pattern and inverted 9-spot well pattern of the novel scheme is quantified. Then, with the proposed normalized index of oil production per unit area of each well pattern, the optimal well pattern between the inverted 5-spot well pattern and inverted 9-spot well pattern for the novel scheme is studied, as shown in the following flowchart (Figure 1). The study proposes an efficient simulation and optimization workflow for the development of pressure-sensitive low-permeability reservoirs with fractured vertical wells coupled with large volume water injection. With the methodology presented in the study, not only the field test can be further optimized, leading to more efficient and sustainable development, but also the novel scheme proposed by Shengli Oilfield can be promoted and widely applied in China and worldwide.

Simulation Methodology and Integrated Workflow
The governing equations of the numerical model are the mass balance equations, including accumulation term and convection term and sink/source term [14, 15]. Darcy's law, which states that fluid flow rate is directly proportional to the pressure gradient, is applied to the fluid flow in the matrix of the model. As to the fluid flow within the hydraulic fracture, the Forchheimer model with the non-Darcy coefficient is implemented to simulate a turbulent flow, accounting for the inertial effects [16][17][18]. Meanwhile, local grid refinement with logarithmic spacing, discretizing the reservoir to a finer degree region around hydraulic fractures and more coarsely further away from the hydraulic fractures, is coupled with the Forchheimer model to accurately depict the detailed transient fluid flow around the hydraulic fractures [19][20][21]. To obtain the fluid properties, the Peng-Robinson equation of state is employed. And the oil-water relative permeability of the model is generated by the analytical correlation using the endpoint data [14].
To meet the needs of finely simulating the dynamic evolution of reservoir properties for the pressure-sensitive low-permeability reservoirs, the dilation-recompaction geomechanical model is introduced to couple with the basic reservoir model. Compared with conventional flowgeomechanical coupling models [22], where complex coupling schemes, expensive computational cost, and massive input data, such as rock mechanical data and in situ stress data, get involved, the methodology employed in the study characterizes the dominating mechanism of the physical process while keeping the modeling and computational cost low. The methodology has been applied and validated by previous work to both accurately and efficiently simulate the process [9,[23][24][25][26].
The dilation-recompaction model finely depicts the relation between the porosity and reservoir pressure as a piecewise and path-dependent function, illustrated in Figure 2 [9, [27][28][29]. Different value is given to the compressibility according to the range of reservoir pressure. For instance, small compressibility is given to the line segment ab. With the gentle slope, from point a to point b, the rock experiences an elastic small change of porosity due to the change of pressure, which is reversible. As to the steep line segment bc, big compressibility is assigned, leading to the intense change of porosity induced by reservoir pressure, which indicates that the reservoir undergoes irreversible dilation, usually accompanied by the opening of fissures. If the pressure drops at some point during the dilation phase, two phases of compaction will occur. If the pressure remains above the recompaction pressure (P R ), reversible elastic compaction occurs in the reservoir. If the pressure continues to drop until it is below the recompaction pressure (P R ), the reservoir enters the irreversible recompaction phase, which has a larger slope than the elastic compaction. In other words, significant compaction occurs in the reservoir during the recompaction phase. The maximum porosity (ϕ max ) in the dilation-recompaction model is correlated with the r at , which is the maximum proportional increase allowed in porosity. The residual dilation fraction (f r ) accounts for the proportion of the total dilation which is permanent and irreversible. If the lower limit of 0 is assigned to the residual dilation fraction, the increase in pore volume as a result of dilation can be fully removed. Conversely, the 2 Geofluids increase in pore volume induced by dilation is permanently preserved if the residual dilation fraction takes its maximum value of 1.
There is a correlation between porosity and permeability. Since the porosity evolves with the pressure, so does the permeability, which is also the feature of the low-permeability reservoir with pressure sensitivity [30,31]. The analytical correlations for the dynamic porosity and permeability are as follows: where c is the compressibility; P r is reference pressure; and ϕ r is the porosity at the reference pressure.
where K 0 is the original permeability; K MUL is a user-defined permeability multiplier; and ϕ 0 is the original porosity. Based on the above simulation methodology, orthogonal numerical experiments can be conducted to obtain the optimal well spacing of each well pattern. Comparison between two types of objects (different well patterns) cannot be performed directly by the numerical simulation. To obtain the 3 Geofluids optimal well pattern, the analytical method of reservoir engineering is introduced to couple with the numerical simulation to normalize the performance of each well pattern with optimal well spacing. That is why the normalized index is developed. The process of the analytical calculation of the normalized index is as follows. Firstly, based on the reservoir engineering method, the actual cumulative oil production of each well pattern can be determined with different weighting factors assigned to the corner well and side well, indicating the actual contribution of each well to the group [32]. Then, with the optimal spacing of each well pattern, the corresponding area of each well pattern can be acquired, which is also the area controlled by the injector. Thus, the proposed normalized index of oil production of each well pattern over the corresponding area is quantified so that the optimal well pattern can be determined. With the integrated workflow, combining the analytical method of reservoir engineering with numerical simulation, the well pattern and well spacing can be optimized simultaneously. The detailed analytical calculation will be presented in the results and discussion part.

Reservoir Model
Based on the CMG, the reservoir model is developed with the data of Block X. The dimensions of the numerical model are 1750 m × 1450 m × 8 m, corresponding to the length, width, and thickness of the reservoir, respectively. For the inverted 5-spot well pattern, there are one vertical injector and four vertical fractured producers, as shown in Figure 3. For the inverted 9-spot well pattern, there are one vertical injector and eight vertical fractured producers, as shown in Figure 4. d stands for the distance between wells. b stands for the distance between each row. All the wells are perforated from the top to the bottom of the reservoir. The half-length of the hydraulic fracture of the corner producer is 125 m, and the half-length of the hydraulic fracture of the side producer is 75 m. The conductivity of hydraulic fracture is 3.05 mD·m. And the cumulative water injection is 6:0 × 10 4 m 3 . The detailed injection scheme is shown in Figure 5, which is the constraint for the injector of the numerical model. The producer is operated with a minimum bottom-hole pressure of 200 kPa, to fully harness the formation energy. Due to the large volume of injected water in each slug within a few days, the bottom-hole pressure is built up rapidly, leading to the increase of the injection pressure. To stabilize the injection pressure at the wellhead within the safe operation limit, the injection is terminated for several days between each slug to facilitate the pressure diffusion outward from the injection spot. Instead of continuous injection, the water slug injection mode is employed in the field. The specific parameters used in the models are listed in Tables 1 and 2.

Results and Discussion
Based on the above established model, 40 simulation scenarios are developed with different well spacing for each well pattern, respectively, as shown in Tables 3 and 4. There is a total of 80 sets of numerical experiments. The simulation outcomes are illustrated in Figures 6 and 7, generated with MATLAB.
If the well spacing is too small, it will lead to higher water production with lower oil production. If the well spacing is too large, the supplemental energy by the injection cannot be utilized efficiently to improve the reservoir performance. Based on the cumulative oil production of the well group with different well spacing, the optimal well spacing for the inverted 5-spot well pattern is 850 m/350 m, and the optimal well spacing for the inverted 9-spot well pattern is 550 m/ 450 m, as shown in the following figures.
With the above optimal well spacing of each well pattern, the optimal well pattern will be determined between the    Cumulative oil production 10 4 m 3 Figure 6: Cumulative oil production for the scenarios with different well spacing (inverted 5-spot well pattern).  ) 50 MPa Initial reservoir pressure (P 0 ) 28 MPa Permeability multipliers (I/J/K) (K MUL ) 5 0 / 6 Geofluids inverted 5-spot well pattern and the inverted 9-spot well pattern with the proposed normalized index of oil production per unit area. The normalized index is determined based on the actual cumulative oil production of a well group and the area of a well group. As to the actual cumulative oil production of each well group, different weighting factors will be assigned to the different wells of the group. 1/4 will be the weighting factor of the corner well to account for the oil production contribution to the group. And 1/2 will be assigned to the side well. The analytical correlation for the normalized index is as follows. The detailed info is listed in Table 5. The oil production per unit area of the inverted 5spot well pattern is higher than the inverted 9-spot well pattern. For the reservoir developed with fractured vertical wells coupled with large volume water injection, based on the normalized index, the inverted 5-spot well pattern is better.
Inverted 5-spot well pattern: Inverted 9-spot well pattern:  Figure 7: Cumulative oil production for the scenarios with different well spacing (inverted 9-spot well pattern).

Conclusion
In the study, based on the efficient modeling method, the numerical model of pressure-sensitive low-permeability reservoirs developed with fractured vertical wells coupled with large volume water injection is established. With the coupled model, the well pattern and well spacing optimization workflow is developed with the proposed normalized index, oil production per unit area of a certain well pattern. Based on the integrated optimization workflow, with a total of 80 sets of numerical experiments, it is found that the reservoir developed with fractured vertical wells is coupled with large volume water injection; compared with the inverted 9-spot well pattern, the inverted 5-spot well pattern is better, and the corresponding optimal well spacing is 850 m/350 m. The insights obtained from the paper will shed light on the development of low-permeability reservoirs with the novel scheme proposed by Shengli Oilfield.

Data Availability
Data is available upon request.

Conflicts of Interest
The authors declare no conflict of interest.