A New Methodology for the Multilayer Tight Oil Reservoir Water Injection Efficiency Evaluation and Real-Time Optimization

School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China Institute of Porous Flow & Fluid Mechanics, Chinese Academy of Sciences, Langfang 065007, China PetroChina Research Institute of Petroleum Exploration & Development, Beijing 100083, China Exploitation and Development Research Institute, PetroChina Xinjiang Oilfield Company, Karamay, Xinjiang 834000, China The No. 2 Oil Production Plant, PetroChina Xinjiang Oilfield Company, Karamay, Xinjiang 834000, China College of Petroleum Engineering, Yangtze University, Wuhan, Hubei 430100, China


Introduction
Waterflooding operations are performed to improve the oil recovery of reservoirs worldwide [1]. In some oil fields with many wells and a long production history, the wide expansion of the injection water and the continuous changes in the flow field in the layer results in water channeling and dead oil areas [2,3]. These problems are mainly due to two factors. First, the physical properties of the reservoir are affected by the sedimentary environment, tectonic movement, and diagenesis; there are intralayer and interlayer heterogeneities in the reservoir.
Second, unreasonable injection and production systems have exacerbated the phenomenon of injection water fingering and bottom water coning. Therefore, based on the existing reservoir development knowledge, it is important to identify the injection-production correspondence, to accurately determine the dominant flow path, and to evaluate the injection efficiency of the injectors [4,5]. This forms both the key issue and the basic work to increase the high economic value oil production and to determine a reasonable injection-production system [6,7].
Due to the relatively poor physical properties of tight oil reservoirs, the following measures were used to optimize the development effect currently, including fracturing [8], asynchronous injection alternating production [9], water flooding huff-puff of horizontal wells [10], and CO 2 huff and puff of horizontal wells [11]. Flow paths can reflect the flow behavior of fluid in a period of time, so it is often used to evaluate the development effect of reservoir under a certain working system. Commonly used methods for distinguishing flow paths include the tracer method, reservoir numerical simulations [12][13][14], and data-driven models. The tracer test incurs significant time to acquire results, and some operations during the test have a certain impact on the normal reservoir production. In multilayer waterflooding reservoir, the capillary forces may have a significant effect to change the flow pathways [15]. Therefore, although this method has high accuracy, it is difficult to popularize and to apply on a large scale in a reservoir. Reservoir numerical simulation methods are restricted by model accuracy, and it is difficult to achieve accurate real-time analysis of the entire reservoir. Data-driven models, as part of a new class of dominant flow path identification methods, do not require complex geological modeling and can determine information such as fluid flow dynamics and water injection efficiency through the analysis of the production data. There are currently two kinds of data-driven models, namely, the CRM (capacitance-resistive model) and the INSIM (interconnection-based numeric simulation model). The CRM was first used by Yousef et al. [16] to determine the connectivity between wells in 2005. Following that, the model was used in waterflooding reservoirs [17], large-scale reservoirs [18], and analysis of the formation damage from injected water [19]. Zhao et al. [20] proposed the INSIM in 2015. By simplifying the characterization of the reservoir as a network of interconnected nodes that consider a series of complex geological features such as wells, water bodies, and faults, an interwell connection network is established. It is characterized by two parameters, connected volume and transmissibility, to fit the actual production dynamics, to correct the interwell connection parameters, and to make the model conform to the actual connectivity of the reservoir. Because a complex geological modeling process is not considered, the INSIM and its derived INSIM-FT (interconnection-based numeric simulation model with front-tracking algorithm) [21][22] and INSIM-FPT (interconnection-based numeric simulation model with the flow-path tracking strategy) [23] reduce the modeling time significantly, while considering reservoir properties and maintaining the calculation accuracy. Based on the distribution of sedimentary facies in each layer, this study identifies the influence of different sedimentary environments on the interwell connection relationship of INSIM. The pressure, saturation, and water cut of each node are solved by the connection parameters corrected by production data combined with the principle of material balance. Then, the vertical/plane dividing coefficient and water injection efficiency of the injector are obtained. Based on this, the water injection efficiency of each layer is calculated and combined by the optimization principle to optimize the injection and production systems [24,25].
Since the main factors affecting the accuracy of the water injection efficiency evaluation method are INSIM's modeling data and the model's historical fitting results, the method has good general applicability to the reservoirs applicable to INSIM. At present, the INSIM model has been widely used in sand reservoirs, carbonate reservoirs, fracture-cavity reservoirs, etc. This method has certain practical significance for evaluating the water injection efficiency of these types of reservoirs. Compared with the traditional reservoir numerical simulation method [26,27], the calculation speed of the water injection efficiency is significantly improved. It is of great significance to accurately identify the injection-production correspondence relationship and increase oil production at a lower cost.

Water Injection Efficiency Evaluation
Method and Optimization Theory  (Figure 1), the material balance equation of the fluid in the connected unit can be obtained by The implicit difference discretization of Equation (1) can solve the bottom hole pressure of each well, then saturation and water cut of each node in the connected network can be calculated with the saturation-tracking method. Fitting the actual production data to correct the connection parameters is conducted to align the production result of the model with the actual production. Let the feature parameter matrix b denote the reservoir connectivity parameter, applying the stochastic perturbation approximate gradient algorithm (SPSA) to optimize the objective function OðbÞ causes it to have a minimum value.
Using percolation theory and interwell connection parameters, the production index in the cross-well connectivity unit can be expressed as Combining the bottom hole pressure and the production index, the phase mobility in the connected unit is determined by the upstream weight method: Based on the above method, the total production index of well i can be determined: Therefore, the vertical dividing coefficient of well i in layer k is determined by the ratio of the sum of the production index of well i in layer k to the total production index of well i: Meanwhile, the plane dividing the coefficient between wells i and j is determined by the transmissibility between the nodes and the pressure difference: Compared to the dividing coefficient calculation method proposed in this study, the reservoir engineering method does not consider changes in pressure and uses only the percolation theory to calculate the dividing coefficient from the well properties.
A n ijk =

Water Injection Efficiency Calculation and
Optimization. After clarifying the dividing coefficient in each layer, the layered water injection efficiency of the injectors can be solved further, that is, the ratio of the amount of oil displaced by the injector to the surrounding producers in this layer and the water injection amount of this layer. The ratio of the total oil production of layer k to the total water injection of layer k is the average water injection efficiency of layer k and is expressed as After obtaining the water injection efficiency of each well in this layer, it should be compared with the average water injection efficiency of this layer to determine the liquid volume adjustment measures. When the water injection efficiency is higher than the average, the injection is increased; otherwise, the injection is reduced, and the liquid volume of each well in this layer is determined by Equations (12) and (13).
In Equation (13), η max and η min are assigned to the values of -0.5 and 0.5, respectively. Because the fractional part value is less than or equal to 1, it means that the liquid volume adjustment range for the next time step will not be greater than 50% of the liquid volume in the previous time step. As the injection volume of the reservoir will not change significantly in a short time, to keep the injection volume of the entire reservoir constant, it is necessary to adjust the injection volume of each layer according to Equation (14). The well water injection volume is equal to the sum of the injection volume of each layer, as shown in Equation (15).
After determining the liquid volume adjustment of the injectors, assuming that the reservoir injection-production ratio in the optimization stage is constant, the liquid volume adjustment of the production well is calculated from the injection volume and the dividing coefficient, as shown in Equation (16).

The Conceptual Model
A two-layer model including five injectors and four producers is established to introduce the basic parameters of the INSIM and the workflow for calculating injection efficiency (IE). The producer-injector spacing is 200 m, and the thickness of each layer is 10 m. The range of permeability is 165.0 mD in the 1st layer and 171.6 mD in the 2nd layer, while the permeability field is shown in Figure 2. The initial oil saturation is 0.8. The viscosity of the formation water is 1.0 mPa s and the viscosity ratio of oil to water is 20. The daily water injection rate of each injector is 40 m 3 /d. Before 3000 days, the daily fluid production rate of each producers is 50 m 3 /d. From 3000-6000 days, the working system from P1 to P4 changes to 80 m 3 /d, 60 m 3 /d, 40 m 3 /d, and 20 m 3 /d, respectively, and the FrontSim streamline field is shown in Figure 3. The initial value of the interwell connection parameters could be calculated from the well property data. Then, using the reservoir production data combined with the optimization theory, the initial value of the interwell connection parameters is corrected after the fitting indicators meet accu-racy requirements. The final value of the interwell connection parameters is shown in Figure 4. The line between the wells shows the transmissibility of the reservoir; the red line indicates strong connectivity, the blue line indicates the second strongest connectivity, and the black line indicates weak connectivity. Upon comparing the distribution of strong connectivity with high permeability zones, these were found to be consistent. It shows that after correction, an INSIM could accurately characterize the reservoir properties and provide a reliable basis for calculating the dividing coefficient of injection water and evaluating the water injection efficiency.
In this study (Equation (6)), the reservoir engineering method (Equation (8)) and the streamline numerical simulation method are used to calculate the vertical dividing coefficient of each layer. Figures 5 and 6 show the difference between the calculation results of the INSIM method and other methods. From the result shown in Figure 6, the vertical dividing coefficient of the reservoir engineering method does not change because it does not consider the flow field change caused by the adjustment of the production system. However, the dominant seepage channel in the reservoir forms gradually and stabilizes with the development. Due to a superior physical property of the first layer in the model, the vertical dividing coefficient of the first layer gradually increases during the formation of the dominant seepage channel. Following the stabilization of the seepage channel, the vertical dividing coefficient also stabilizes. The method in this study and the streamline numerical method can reflect this law accurately.
Meanwhile, Equations (7) and (9) were used to calculate the plane dividing coefficient of the injection well at 6000 days. As shown in Figure 7, the plane dividing coefficient obtained based on this method is basically consistent with the results based on the streamline simulation, which verifies the reliability of this method. The reservoir engineering method cannot accurately reflect the plane dividing coefficient currently because it does not consider the changes in the reservoir production system. The streamline simulation method takes 25.54 seconds for one complete calculation. Compared to this method, the method in this study takes only 0.74 seconds for one complete calculation, which is faster by a factor of 35.
The single-well water injection efficiencies and average water injection efficiencies of the reservoir engineering method, streamline method, and method of this study are calculated using Equations (10) and (11), as shown in Figure 8. According to the reservoir engineering method, the water injection efficiency of each well is lower than the other two methods. Due to changes in the flow field, the division of injected water in each direction is different periodically. Some remaining oil is produced by adjusting the working system, and this volume of oil cannot be calculated using the reservoir engineering method. The method in this study solves this problem better by considering the change in the bottom hole pressure of each well, and the calculation results are basically consistent with the streamline method. By using the dividing coefficient and the water injection efficiency of the conceptual model in Equations (12), (13), (14), (15) and (16), the production system of each well in the optimization stage is calculated. 4 Geofluids

Geofluids
The production system of the original plan and the adjusted plan were brought into the Eclipse model to predict the oil production and the water cut of the reservoir after the implementation of the plan. As shown in Figure 9, five months after the implementation of the adjusted plan, the daily oil production of reservoir is increased, the comprehensive water cut of reservoir is decreased, and the adjusted plan had an optimized effect on the overall reservoir development.  oil wells were opened. The daily liquid production was 2837 t, the daily oil production was 370 t, and the comprehensive water cut was 86.9%. The recycle of injection water is inefficient in such a high water cut reservoir. After history matching the INSIM (Figure 10), the field interwell connection obtained by fitting the production dynamics is shown in Figure 11, in which the red line repre-sents a strong connectivity, the green line represents the second strongest connectivity, and the black line represents weak connectivity.

Practical Applications
Based on the sedimentary facies' types of reservoirs and the INSIM, an interwell connectivity classification standard is established, which reflects the similarities and differences in the various sedimentary facies. The orange area represents the channel bar, and the light yellow and bright yellow areas represent the river channels, as shown in Figure 12. Overall, the connectivity from sandstone to conglomerate in the vertical direction deteriorates, and the permeability decreases. In       (b) Field water cut (FWCT) Figure 9: The comparison of optimized and original production systems. 8 Geofluids the plane, affected by source direction, the interwell connectivity from the southwest to the northeast improves and the permeability increases.
Combining the interwell connectivity with the sedimentary facies at the well nodes, the distribution pattern of the characteristics is obtained as shown in Table 1, which divides the interwell connectivity into four major categories and twelve subcategories. The four types include the middle channel, the side of channel, the middle of the channel bar, and the side of the channel bar. Since the porosity of different sedimentary facies of the same section in this tight oil reservoir is not much different, the connection volume is mainly affected by the well spacing, and different types of connection units cannot be accurately distinguished. Therefore, when   10 Geofluids dividing the types of interwell connection units, only choose the transmissibility and divide it into three types as the criteria for connectivity classification. The three types of connectivity classification include T ij > 3, 1 < T ij < 3, and T ij < 1, which, respectively, indicate strong, medium, and weak connectivity of the interwell transmissibility based on the proportion of each subtype in the different types of sedimentary facies. In the middle channel, the interwell is well connected and exhibits anisotropy. In the side of the channel, the interwell has a partial connection, with significant anisotropy. In the middle of the channel bar, the interwell is connected, and there is a large difference between the channel bars. In the side of the channel bar, the interwell is connected in some directions, and the physical properties are relatively worse. The sedimentary structure of the middle channel is shown in Figure 13(a). The middle channel has a thick channel sediment with good properties. When the two wells are connected parallel to the source direction, the connectivity between the wells is good and may develop a dominant seepage channel. As shown in Figure 13(b), the sedimentary period changes quickly to the side of the channel and the sediment thickness of a single period decreases. Correspondingly, the flow barriers between sand bodies are more developed. Furthermore, the flow barriers cause the properties of the reservoir to become worse than those of the middle channel. The interwell connectivity is without change, and the dominant seepage channel is infrequent. In general, the connectivity of the parallel source direction is better than that of the vertical source direction.
As shown in Figure 14(a), the central area of the channel bar is dominated by parallel interlayers. The parallel interlayers of sand bodies between wells in the central area are uniformly distributed, with good connectivity in all directions. Additionally, the dominant seepage channels are developed.       As shown in Figure 14(b), the side of the channel bar is affected by river erosion. Locally inclined interlayers develop, which worsen the connectivity between wells. The water injection has a poor development, and substantial oil remains in this area. In the parallel source connectivity direction, when the interlayer distribution is more consistent with the source direction, the interwell connectivity is better, and the seepage channel is more developed. When the interlayer distribution is inconsistent with the source direction, the interwell connectivity is poor. In the vertical source direction connectivity, the interwell connection becomes poor due to the changes in the sand body contact relationships and permeability. There is no obvious dominant seepage channel in this area. Based on the study of the remaining oil enrichment area and the connectivity characteristics of each layer, the A2 area in Figure 15 with higher remaining oil saturation and complex connection relationship is preferred as the adjustment area for the water injection efficiency evaluation. The water injection efficiency of injectors in A2 area is shown in Figure 16.

The Optimization of the Production System and Implementation Effect.
Targeting the problem of dominant seepage channels in some directions of the well group in area A2 that caused the dividing coefficient to be concentrated, this study combined the automatic optimization method of the injector and producer working system to form three sets of adjustment plans.
The first plan is a liquid volume adjustment plan for water injection volume. Comparing the average water injection efficiency with the efficiency of different layers in different wells, there is an increase in water injection volume in the higher efficiency layer and conversely, the injection is reduced. For the connected producers, calculating the change in injected water volume in each direction according to the dividing coefficient and summing all as the liquid volume adjustment of the producers. The second plan is a reservoir property adjustment plan. The measures include profile control for inefficient injectors, as well as sealing, backflow, and well function reassignment for strong connectivity, large dividing coefficient, high water cut producers.
To ensure accurate and effective measures, the dividing direction and tracer test data of key well groups were verified.
In Figure 17, the plane dividing coefficient of injector T1 shows that the producers P1 and P2 of the same well group are the main dividing directions, with 45% and 27% of injected water diverted on the sandstone section, respectively, and 32% and 42% on the conglomerate section, respectively. Table. 2 shows a good correspondence among the three wells; the tracer test data from the T1 well group shows that P1 has the longest sustained dose time (41 days) and the highest peak tracer concentration (148.66 ng/ml); P2 has a continuous tracer time of 30 days and the peak tracer concentration 126.31 ng/ml; no tracer reaction was seen in P3; P4 continued to be seen for 36 days, and the peak tracer concentration was 97.04 ng/ml.
The water injection efficiency of T1 in the sandstone section was 0.15, and the water injection efficiency of the conglomerate section was 0.07; the water injection efficiency in the sandstone section was higher than the average water injection efficiency. The injection volume of T1 in the sand and conglomerate sections changed to 4.2 m 3 and 7.5 m 3 , respectively.
In Figure 18, the horizontal dividing coefficient of T6 shows that the producers, P7 and P8, in the same well group are the main dividing directions, with 36% and 27% of the injected water diverted in the sandstone section, and 33%    Table 3 shows a good correspondence among the three wells. The tracer test data of the T6 well group shows that P7 lasted 26 days, and the peak tracer concentration was the highest (97.01 ng/ml). Two kinds of tracers were seen in P8, and the peak tracer concentrations were 52.4 ng/ml and 82.41 ng/ml, while no tracer reaction was seen in P5 and P6.
The water injection efficiency of T6 in the sandstone section was 0.17, and the water injection efficiency of the conglomerate section was 0.13. The water injection efficiency in both was higher than the average water injection efficiency. The injection fluid volume of the T6 increased to 5.0 m 3 in the sandstone section and increased to 3.5 m 3 in the conglomerate section.
After a three-year production simulation of the adjusted production system in Figure 19 and with comparisons to the original system, the daily oil production in the field increased by 15.5%; the cumulative oil production in the field increased by 4:24 × 10 4 t; the comprehensive water cut decreased by 1.39%, and the effect of the adjustment is obviously shown in Figure 20.

Conclusion
A layered water-injection efficiency evaluation method based on INSIM is proposed in this study by accurately adding virtual well points to characterize the actual geological connectivity of glutenite reservoirs and a well-to-well conductivity of the fitted connectivity model. Combined with the location of the sedimentary facies at the well point, a classification standard for interwell connectivity of glutenite was established, and the interwell connectivity was divided into four major categories and twelve subcategories. On this basis, to determine the injection water splitting quantity, the layered water injection efficiency of the injection well which is the basic knowledge to adjust the injection-production schedule was evaluated. The method determines the vertical and horizontal water injection split coefficients by calculating data, such as connectivity parameters, pressure difference, and production index, and identifies the injection-production correspondence relationship and the effect of water injection. The calculation speed and the calculation result of the water injection efficiency are more reliable than the reservoir engineering method.
Based on this method, three types of liquid volume adjustment plans were formulated for an actual oilfield: injectionproduction structure adjustment, injection-production measurement adjustment, and comprehensive adjustment. It is predicted that after two years of implementation of the liquid volume adjustment measures for 20 water injection well groups in the A2 area, the daily oil production rate in the block will increase by 15.5%; the cumulative oil production in the block will increase by 4:24 × 10 4 t; the comprehensive water cut will decrease by 1.39%, and the effect of increasing oil precipitation will be obvious.

A:
Dividing coefficiency b: Feature parameter matrix b r : Initial estimation of feature parameter matrix C t : Total compressibility, MPa -1 e: Water injection efficient e x : Average water injection efficient f w : Water cut G B : Covariance matrix of the model parameters G D : Initial estimation covariance matrix of the model parameters h: Average interwell reservoir thickness, m hðbÞ: Initial numerical simulation observation values J: Production index K: Layer average permeability, 10 -3 μm 2 K: Interwell average permeability, 10 -3 μm 2 K r : Relative permeability, 10 -3 μm 2 k obs : Observation values L: Well spacing, m N l : Number of reservoir layers N Ic : Number of connected injectors N w : Number of connected wells O: Objective function for history matching p: Flowing bottom hole pressure, MPa q: Production or injection volume, m 3 r: Wellbore radius, m s: skin factor S w : Water saturation T: Transmissibility, m 3 /d·MPa -1 V: Pore volume, m 3 η: Liquid volume adjustment coefficient λ: Mobility, 10 -3 μm 2 /(mPa·s) μ: Viscosity, mPa·s Subscripts i: Well-node index ij: Parameters between well i and well j j: Well-node index k: Layer index max: Parameter maximum min: Parameter minimum o: Oil phase w: Water phase Superscript N: Timestep.

Data Availability
Data is available on request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.