Numerical simulation and field application of biological nano-technology in the low- and medium-permeability reservoirs of an offshore oilfield

As a result of deep burial depth, small pore throat, poor connectivity between pores, different clay mineral contents in reservoirs, and strong reservoir sensitivity, injection wells often have problems such as rapidly increasing water-injection pressure and insufficient water-injection quantity in the process of water-injection development. The main measures used to solve the difficulties of water injection in low-permeability reservoirs include fracturing, acidizing, and surfactant depressurization and injection increase, all of which have some disadvantages of high cost and environmental damage. In recent years, depressurization and injection-increase environment-safe bio-nano-materials have been introduced into low-permeability reservoirs and have achieved good application results in China. On the other hand, although there have been many researches on EOR (enhanced oil recovery) of nano-materials, the numerical simulation field of nano-depressurization and injection-augmenting technology is still a blank that the wettability mechanism of nano-materials and EOR nano-materials used in bio-nano-depressurization and injection-augmenting technology are almost completely opposite, and the influence of adsorption on formation is almost completely opposite. The adsorption of nanoparticles in other EOR studies will reduce the porosity and make the reservoir more hydrophilic. Nanoparticles used in biological nano-technology will produce hydrophobic film near the well, which will reduce the seepage resistance through the slip of water phase. In this study, a set of water flooding model of numerical simulation technology for depressurization and injection-augmenting of biological nano-materials considering adsorption characteristics and reservoir physical properties was established, the sensitivity analysis of key injection parameters was carried out, and the application effect prediction chart of biological nano-technology was drawn, and the model and prediction chart were verified by real oilfield data. As far as we know, this is the first numerical simulation study on biological nano-technology that has been applied in oil fields.


Introduction
Due to long-term exploitation and expanding oil demand, many onshore oil fields in China have begun to turn to largescale exploration and development of gas reservoirs (Anees et al. 2022a, b). At the same time, many offshore oilfields are gradually developing medium-and low-permeability reservoirs that permeability less than 50mD and greater than 5mD. However, in the process of water-injection development, offshore low-permeability reservoirs are prone to problems such as rapidly increased water-injection pressure, insufficient water-injection capacity, and a rapid decline of oil-well production. This is the result of factors such as deep burial, small pore throat, poor connectivity between 1 3 pores, different clay mineral contents, and strong reservoir sensitivity (Anderson et al. 2010;Cao et al. 2014;Chen et al. 2016a, b;Moghadasi et al. 2019;Wang et al. 2020Wang et al. , 2021. At present, solutions to the difficulty of water injection in low-permeability reservoirs include fracturing, acidification, and surfactant depressurization and injection increase (Qin 2016;Yuan and Wang 2018;Qu et al. 2021;Kalam et al. 2021;Omari et al. 2021;Motta et al. 2021). Although these methods have helped overcome some of the problems of water-injection development, they still have certain disadvantages of high cost and environmental damage (Chen et al. 2016b;Yang 2017;Zhao et al. 2017;Feng et al. 2019;Lei et al. 2019;Ren et al. 2021). By contrast, the nano-materials can expand the effective radius of formation pores and have a super-hydrophobic effect, which can reduce injection resistance and improve water-phase permeability (Di et al. 2015;Qu et al. 2021). Additionally, they have anti-swelling properties, can isolate contact between injected water and clay minerals on the rock surface, and can reduce the blockage of formation pores caused by particle migration, allowing them to achieve the effect of depressurization and injection increase (Gu et al. 2007;Alaskar 2013;Kazemzadeh et al. 2019;Liu et al. 2020;Qu et al. 2021;Davin et al. 2022).
In recent years, some new type nano-materials of depressurization and injection increase has been introduced into low-permeability reservoirs and has achieved good application results (Zhou et al. 2004;Chen et al. 2016a, b;Silei 2018;Dai et al. 2018;Zhai et al. 2019;Ding et al. 2020;Li 2021). After injecting nano-polysilicon, nanoparticles are coated on the surface of clay, which can prevent the immersion of injected water and prevent clay from swelling, thus achieving the purpose of reducing well head pressure and increasing injection, it has achieved good application results in Wuqi oilfield (Li 2014). The recent research found that the water-based nano-polysilicon depressurization and injectionaugmenting agent manufactured by in situ surface modification technology has a good wettability change effect, and its practical application effect is good (Liu et al. 2017(Liu et al. , 2018. Using experiments such as nano-agent stability, a China researcher optimized the water-based nano-agent SL-3 to make it suitable for an ultra-low-permeability reservoir in the Shishen A area; the three wells this was applied to were successful and had good results for depressurization and injection increase (Yi 2017). In addition, adding surfactant to nanoparticle solution can greatly improve the permeability of water phase. The field application results of three wells showed that the average pressure-reduction rate was 30.2%, the average daily water injection was increased by 84.4%, and the average validity period was 195 days, thus meeting the needs of long-term depressurization and injection increase for water-injection wells in offshore oil fields (Xu et al. 2019). In Cupiagua oilfield, Castilla oilfield, and Chichimene oilfield, nanofluids have successfully improved the reservoirs damaged by precipitation (Adil et al. 2016;Zawrah et al. 2016;Franco-Aguirre et al. 2018).
The transportation process of nanoparticles underground is very complicated, which is related to concentration, pH, permeability, flow velocity, particle (such as size, size distribution, shape, surface charge, and so on) (Alaskar et al. 2012;Fadili et al. 2022). When nanoparticles move in the reservoir, they are captured and absorbed into the pores (Irfan et al. 2019). Therefore, it is important to consider the net loss rate of nanoparticles, which is given by Ju et al. after modification, the model studies the effects of water saturation, nanoparticle concentration, porosity, and permeability on the flow of nanoparticles in porous media (Ju et al. 2002;Ju and Fan 2009). Chen et al.(2016a, b) established a drag reduction model for hydrophobic nanoparticles, which was numerically simulated by IMPES method, and predicted the changes of core surface wettability and well displacement. In heterogeneous porous media, Salama model confirmed the deposition of nanoparticles by considering the filtration theory Abdelfatah et al. 2017). Agista(2017) established a modified linear adsorption model, which can accurately simulate the adsorption process of nanoparticles, but can't well simulate the desorption behavior caused by scouring in the subsequent waterinjection process. El-Amin et al.(2015) used a dimensional analysis method based on a nanoparticle transport model to investigate the effect of each parameter used in the model governing equations. After performing dimension analysis, different dimensional numbers can be introduced into the system, such as the Darcy number, the Capillary number, and the Bond number Irfan et al. 2019).
In oil and gas reservoirs, production is closely related to mineral composition, fracture, permeability, and porosity that are characterized by geological models in the field numerical simulation, and there are also many excellent treatment methods (Thai Ba et al. 2020;Liu et al. 2020;Thanh and Sugai 2021;Ashraf et al. 2021;Ma et al. 2021;Shen et al. 2022). There are many and perfect numerical simulation studies on these aspects of conventional chemical flooding(Yadali Jamaloei 2011; Ansah et al. 2020;Li et al. 2021a;Ma et al. 2021;Shen et al. 2022). There is, however, a lack of research on the numerical simulation of depressurization and injection-increase technology, especially biological nano-depressurization and injection-increase technology. The difference between nano-technology of decreasing pressure and increasing injection and EOR (enhanced oil recovery) numerical simulation of nano-materials lies in that biotechnology increases the formation seepage channel and reduces the seepage resistance through the adsorption of hydrophobic nano-materials in the formation, while conventional nano-materials of EOR mostly use hydrophilic nanoparticles and seldom consider the adsorption of nanoparticles in the formation and the change of formation physical properties. In this research, a new numerical simulation model of depressurization and injection-augmenting is established for the biological nano-technology which has been successfully applied in the field, and the sensitivity analysis of the influencing factors of its application effect is carried out, and the established model is verified by the actual production data. Compared with the previous EOR study of nanoparticles, this study not only considers the cumulative oil increase, but also pays attention to the well head pressure of injection well.

Mechanism of depressurization and injection-augmenting of biological nano-solution
In the field of petroleum engineering, there are two methods to depressurization and injection-augmenting: one is to treat the formation such as acidification and fracturing to enlarge the pore throat of the formation and increase the seepage area; the other is to reduce the friction between pore throat and seepage fluid (Li et al. 2021b).
Before nanoparticle injection, acid will be used to perform limited plug removal around the bottom hole. After nanoparticles are injected into the formation, they will be adsorbed to the formation to form a thin (nanometer thickness) hydrophobic film (Lu et al. 2003;Lashari and Ganat 2020;Foroozesh and Kumar 2020), as shown in Fig. 1(a). In the process of nanoparticles adsorbing to the wall to form a hydrophobic nanoparticle film, the water film on the rock wall will fall off, which will lead to the increase in seepage channels, thus also leading to the increase in porosity and permeability (Liu et al. 2017;Li et al. 2021b) to make the treatment more convenient, the two methods are combined here to treat the porosity increase, which can be expressed as follows: where Φ is the porosity after biological nano-technology; Φ 0 is original porosity; ρ s is the density of inorganic scale; ρ np is the density of nanoparticles; c np is the concentration of nanoparticles.
The permeability of formation can be improved by nanoparticle plugging removal, which is described by the Carmen-Kozeny law (Zhang et al. 2017): where K is the permeability after biological nano-technology; K 0 is original permeability.
After forming a hydrophobic film composed of nanoparticles, the slip effect will occur on nanoparticle film of porous walls ( Fig. 1(b)), which will reduce the water flow resistance and improve the water-injection capacity (Gu et al. 2007(Gu et al. , 2020Liu et al. 2018). The adsorption of nanoparticles is described by the Langmuir isotherm (Alafnan et al. 2021): where ĉ np is the concentration of nanoparticles adsorbed in the rock pores; A and B are the Langmuir adsorption constant. (1)

3
After nanoparticle adsorption, the original water-wet pore throat surface is gradually changed to neutral or even oil-wet, and the relative permeability of the water phase is increased, which can be characterized by the change of seepage-resistance coefficient, relative permeability be calculated by where K rwnew is the relative permeability of formation water phase after biological nano-technology; K rw is the relative permeability of formation water phase before biological nano-technology.

Multiphase and multicomponent mathematical model
The biological nano-materials coupling model of the water-drive reservoir is mainly composed of three parts: the waterflood reservoir-seepage model, nanoparticlemigration model, and reservoir porosity and permeability change model. The seepage equation of oil, water, and gas components in water-drive reservoirs is the same as that of the black-oil model (Todd and Longstaff 1972;Shoaib and Hoffman 2009;Nojabaei and Johns 2016;Du and Nojabaei 2020), and its basic assumptions are as follows: (1) The seepage flow in the reservoir is isothermal.
(2) There are three phases of oil, gas, and water in the reservoir, and the seepage flow of each phase fluid conforms to Darcy's law.
(3) The model considers three components: the oil component, gas component, and water component. (4) The mass exchange of gas components occurs in the oil gas phase and the water gas phase. (5) The phase balance is completed instantly. (6) The water component only exists in the water phase, and there is no mass exchange with the oil gas phase. (7) The reservoir rocks are compressible and anisotropic. (8) The reservoir fluid is compressible, and the influence of gravity and capillary force in the seepage process should be considered.
Nanoparticles are regarded as a component dissolved in the water phase. According to the principle of conservation of matter, the continuity equation can be written as: where c np is the concentration of nanoparticles; P is the formation pressure, x is the distance from the bottom of the injection well; S w is the water saturation; q w is the waterinjection volume; d np is the diffusion coefficient of nanoparticles; F np is the percentage of pore surface in contact with the water phase.
The main components of the coupling model of depressurization and injection increase in biological nanoparticles in a water drive are given below. There are also some auxiliary equations, such as the relationship between saturation and the capillary force, the PVT physical-property equation, and the nanoparticle physicochemical equation: where S o is the oil saturation; S g is the gas saturation; p o is the capillary pressure of oil; p w is the capillary pressure of water; p g is the capillary pressure of gas; p cow is the between the p o and p w ; p cog is the between the p g and p o .
This system has the initial conditions given by For this conceptual model, it is an oil reservoir with a closed boundary.
Finally, due to the existence of the well, there are the following conditions where Q v is the well yield; p wf is the well bottom-hole pressure; δ (x,y,z) is the Dirichlet-like function, if (x,y,z) corresponds to well coordinates, δ (x,y,z) = 1, otherwise δ (x,y,z) = 0. In other words, it is not considered whether there is water supply in other places of the reservoir except the well.

Model and method
Taking the basic data of well group Q1 in the Bohai oilfield as an example, a water-drive conceptual model was established, as shown in Fig. 2. The main parameters were as follows: (1) The reservoir thickness was 30 m, porosity was 20%, and permeability was 30 mD. (2) The injection-production well spacing was 350 m, and the wells were arranged in reverse seven points. (3) The liquid production of a single well was controlled, the production well was 20 m 3 /d, and the water-injection well was 120 m 3 /d. (4) The properties of crude oil were that of conventional thin oil, and the formation pressure was 28 MPa. (5) The formation depth was 2600 m, and the temperature was 55 °C. (6) The compressibility and relative permeability of the rock fluids were as shown in Fig. 3. Lastly, the number of model grids was 100 × 100 × 10.
Water flooding model Figure 4 shows simulation results of the water flooding with and without plugging. Without considering the formation damage caused by impurities in injected water, we can know from the without plugging curve in Fig. 4 that the oil production of the production wells was relatively stable, and the oil production began to decline gradually after the water breakthrough in 2022. The injection volume of the water-injection well remained constant, and the well head injection pressure and bottom-hole flow pressure remained basically constant, with only a slight decrease, which shows that the injection and production were basically maintained in a balanced injection-production state without considering the damage of the water-injection formation.
Based on the normal water flooding model without plugging, the blockage of the water-injection well near the well was simulated by adding an inorganic scale-formation reaction in the water phase. The well head pressure and wellbottom flow pressure could be obtained through numerical simulation. We can know from the plugging curve in Fig. 4 that the formation of the inorganic scale led to a decrease in porosity near the well, the injection volume gradually decreased when the injection well reached the maximum injection pressure, and the under-injection situation became increasingly serious.

Biological nanoparticle model
There are two main mechanisms for the on-site water injection of bio-nanoparticles to reduce pressure and increase injection: (1) dissolving inorganic scale and drive water film; (2) nano-materials adsorbing to form nano-film to reduce seepage resistance.
We assume that the number of nanoparticles in the two mechanisms is equal, and the injected nanoparticles react to generate nanoparticle components corresponding to the following two mechanisms: The first mechanism of action of some nanoparticles is to dissolve the inorganic scale; the chemical reaction is nanoFH → 0.5nanoSH + 0.5nanoXF  shown below. We can know that the inorganic scale CaCO 3 dissolves and then transforms into S-nano component in liquid, resulting in a increase in porosity. Adding the Carmen-Kozeny formula option in the variable-permeability option could simulate the improvement of the water-phase permeability.
The second mechanism of action of some nanoparticles is described by the Langmuir isotherm adsorption option on the pore wall, as shown in Fig. 1a.
On the basis of the water-drive simulation, the above two mechanisms were added, and the mole fraction of the nanocomposite system was 0.0002 after one year of under-injection. When the injection time was two months, the injection rate remained unchanged. According to the simulation results, as shown in Fig. 5a, we can know that injecting the bio-nano-solution greatly delayed the under-injection time of the water-injection well, and reduced the well head pressure and bottom-hole flow pressure. The biological nano-solution was able to prolong the effective injection-allocation time of scaling and plugging wells. The minimum well-bottom flow pressure of the production well was set at 21.5 MPa. According to the simulation results, as shown in Fig. 5b, compared with the change in the whole oil production, we can know that the oil production declined later after the injection of the biological nanosolution, and the decline rate was significantly lower than that of the water drive without measures.

Sensitivity analysis of injection parameters of biological nano-technology
There are two aspects to evaluate the construction effect of biological nano-technology. It is the decreasing amplitude of well head pressure of injection well, which is the difference Fig. 4 Water flooding simulation results of the CMG conceptual model: a the production performance of production wells and b the production performance of injection wells Fig. 5 Comparison of water flooding with plugging and biological nano-technology simulation results of the CMG conceptual model: a the production performance of production wells and b the production performance of injection wells 1 3 between well head pressure before and after construction. The other is the increasing oil quantity of a well group, which is the difference between the cumulative oil production with and without biological nano-technology.
The depressurization effect of the biological nano-technology was evaluated by the maximum decline of the well head pressure after injection, and the final recovery was characterized by depressurization and injection increase as well as the cumulative oil increase.

Influence of operation time
To analyze the sensitivity of injection timing, the degree of under-injection in the formation is defined by referring to the water absorption efficiency of the formation (Zhao et al. 2017). Define the formation under-injection proportions are where α under-injection rate of formation; Q t is the actual water-injection volume of the water-injection well; and Q f is the theoretical water absorption of the formation, which is 120m 3 /d here. Figure 6 shows the influence of operation time on decreasing amplitude of well head pressure and increasing oil quantity of well group. We can know from Fig. 6 that the effect of biological nano-technology first increases and then decreases with the increase in the under-injection rate of formation. This can be explained as follows: the higher the degree of under-injection, the more organic scale and thicker water film in the formation. After the operation of biological nano-technology, the porosity and permeability will be improved significantly. However, when the degree of underinjection is too high, many channels have been completely blocked, and biological nanoparticles can't enter the channels, so the effect is weakened.

Influence of injection volume
There are many ways to measure injection volume, such as injection volume and pore volume multiple (PV). For the sake of universality, we use pore volume multiple to measure injection volume here. Figure 7 shows the influence of injection volume on decreasing amplitude of well head pressure and increasing oil quantity of well group. We can know from Fig. 7 that the decreasing amplitude of well head pressure and the increasing oil quantity have been increasing with the increase in injection volume. This can be explained as follows: the more biological nanoparticles are injected, the more strata will be affected, so the better the application effect of biological nano-technology.

Influence of injection intensity
Different injection rates will also change the behavior of nanoparticles after injection. If the injection rate is too fast, it will lead to formation blockage (Davin et al. 2022). As shown in Fig. 8.
For reservoirs with different thicknesses, the injection rate leading to formation plugging will be different, so it can be measured by the injection rate per unit thickness. It can be expressed as where β is the injection intensity, m 3 /d/m.  Figure 9 shows the influence of injection intensity on decreasing amplitude of well head pressure and increasing oil quantity of well group. From Fig. 9, we can know that with the increase in injection intensity, the decreasing amplitude of well head pressure is smaller and smaller, while the increasing oil quantity first increased and then decreased. This can be explained as follows: when the injection intensity is too high, the biological nanoparticles can't be uniformly transported to the formation area, which will lead to the decrease in permeability near the well for some time after the injection of biological nanoparticles. Therefore, increasing the injection intensity will reduce the well head pressure drop. With the passage of time, biological nanoparticles are gradually and evenly distributed in the reservoir, so the oil increase will increase. However, if the injection strength is too large, the migration process will take a long time, so the oil increase will decrease with the increase in injection intensity. Figure 10 shows the influence of injection intensity on decreasing amplitude of well head pressure and increasing oil quantity of well group. From Fig. 10, we can know that with the increase in concentration, the oiling effect is getting better and better, but there is a maximum value, while the effect of well head pressure drop is significant but first increases and then decreases. This can be explained as follows: with the increase in concentration, the area of action will be enlarged correspondingly, but after reaching a certain level, too high concentration of nanoparticles may lead to the decrease in permeability near the well. On the other hand, when the concentration of nanoparticles is too high, the poor improvement effect will also reduce the decreasing amplitude of well head pressure.

Design plate of injection parameters of the biological nano-technology
Using the previous sensitivity analysis as the sample point, the injection parameters of injection volume, injection intensity, and injection concentration were represented by x 1 , x 2 , and x 3 , respectively. Then, multivariate binomial fittings were carried out for the depressurization and injectionincrease amplitude y 1 and cumulative oil increase y 2 .
Through binomial fitting, the response equation of the depressurization amplitude within the range of the simulation parameters at 30 mD was obtained as follows: The response equation of the cumulative oil increase in the range of the simulation parameters is as follows: According to the intersection of the predicted results and the actual results, as shown in Fig. 11, the binomial-fitting correlation coefficient exceeded 90%, indicating that the fitting formula had high fitting accuracy. As such, it can be (13) used for reference in the prediction of design parameters for similar well-group biological nano-solution depressurization and injection-increase schemes. Figure 12 shows the response surfaces of the decreasing amplitude and increasing oil quantity obtained by Eqs. 10 and 11. When we get the injection rate, PV, and injection concentration, then find the corresponding coordinates on the template in Fig. 12, and read out the color values of the corresponding coordinates, so as to predict the value of pressure drop and oil increase after using biotechnology according to the color chart in Fig. 12.

Field application
To verify the accuracy of the method in Sect. 5.1, the actual well-group data is used for verification ( Table 1). The pressure drop is verified by the actual production data (Fig. 13a), while the oil increase is verified by the data obtained by numerical simulation software (Fig. 13b). The numerical simulation software used in this study is CMG (Computer Modelling Group ltd., Canada).
Well R22 is a water-injection well in oilfield, which began to be injected in November 2015, with an initial injection allocation of 450 m 3 /d. In the process of injection, the well head pressure rose rapidly to about 12 MPa in a short time, and the actual water-injection rate gradually decreased from 400m 3 /d to about 50m 3 /d. With the water absorption capacity of the reservoir getting worse and worse in the injection process, the injection pressure rose to 15 MPa in the first half of 2018, and the daily injection rate rose to about 190m 3 /d. After acidizing measures were taken in May 2018, the daily injection rate rose to 250 m 3 /d, but the water-injection rate dropped rapidly. The phenomenon of high injection pressure and under-injection is becoming more and more serious. Before the biological nano-technology, the water-injection pressure is as high as 15 MPa or so, so that the injection allocation of 250 m 3 /d can be completed.
Biological nano-technology was used in Well R22 from July 1, 2019, to September 1, 2019. After the measures were implemented, the well head pressure dropped to about 6 MPa, which was about 6.5 MPa lower than that before the measures (Fig. 13a). The corresponding daily injection rate increased from 30 m 3 /d to 125 m 3 /d, which was 95m 3 /d higher than that before the measures. The injection effect of water-injection wells was remarkable. Since the implementation of the measures to the current date, the injection rate has remained relatively stable, and the injection-increase effect is still good, which can meet the injection-allocation requirements of water-injection wells. Results of biological nano-technology application: a reality well head pressure before and after application of biological nano-technology and b predicted cumulative oil with and without biological nano-technology by simulation 10.97 Well head pressure before biological particle (MPa) 15 Injection strength of biological particle (m 3 /d/m) 5.595 Injection concentration of biological particle (ppm) 1200 Injection volume of biological particle (PV) 0.000298 Figure 13a shows the actual well head pressure before and after application of biological nano-technology, we can know that the well head pressure is reduced by 6.5 MPa after using biological nano-technology, which is not much different from the calculated result in Table 2 by using the method described in Sect. 5.1. Figure 13b shows the comparison of the results of predicted cumulative oil by CMG (numerical simulation software). We can know by Fig. 13(b) that the cumulative oil production increased by 48,280 m 3 after the application of biological nano-technology, while the result in Table 2 obtained by the method in Sect. 5.1 was 43,947 m 3 , which was close to each other, indicating that the method obtained in Sect. 5.1 was reliable.

Summary and Discussion
At present, the research on the application of nanoparticles in oil fields is still mostly in the laboratory theoretical stage. This study is the first time to analyze the sensitivity of biological nano-technology that has been applied in the mine, and get the influence of each construction parameter on the two most concerned parameters in the practical application process, namely, the amplitude of pressure reduction and the amount of oil increase. This study can provide a certain reference for theoretical research and field development scheme of similar nano-technology.
Although some preliminary original numerical simulation study has been done for biological nano-technology in the low-and medium-permeability reservoirs, there are still some defects that need further research in the future. Hence, some discussion and remarks are listed as following.
(1) The model adopts the conventional Carmen-Kozeny law to describe the porosity and permeability increasing after injecting biological nano-solution. Although we have incorporated some mechanisms of inorganic scale plugging removal and water film decreasing, the influence of some mechanism needs further evaluation by conducting experiments and simulation studies.
(2) The nanoparticle adsorption is described by using Langmuir isotherm equation, and the relevant water relative permeability decrease is simply characterized by the change of seepage-resistance coefficient. How-ever, the adsorption amount and the resistance coefficient are difficult to obtain, which should be further investigated by doing more experiments.
(3) The new model can reasonably predict the depressurization and injection-augmenting performance of biological nano-solution. More model validation with coreflood experiments and well pattern cases need to be performed for improvements, especially for the parameters adjustment for actual field prediction application. Once the model is validated, the coupled model can be modified to simulate hybrid EOR process related to nanoparticle effect, such as nanoparticle-assisted polymer flooding process in the reservoirs of an offshore oilfield. (4) In this study, in order to simplify the migration of nanoparticles in the reservoir, it is assumed that the diffusion of nanoparticles is along the radial direction. However, in the actual production process, many directional wells and horizontal wells will be used in offshore oil fields, which will lead to a small number of vertical diffusion of nanoparticles. Therefore, the simulation of directional wells and horizontal wells still needs to consider the influence of well types on the diffusion of nanoparticles in the actual use process.

Conclusions
This paper systematically studied the influence of injection parameters, such as operation time, injection volume, injection intensity, and concentration, on foam well head pressure and cumulative oil after the application of biological nanotechnology. In addition, according to the binomial-fitting method, the chart for calculating the amplitude of depressurization and oil increase is obtained, and it is compared and verified by using the oilfield production data. Based on the results of this study, the following conclusions can be drawn.
(1) The production performance of Well R22 shows that biological nano-technology can effectively reduce the well head pressure of injection wells and increase the cumulative oil production of well groups.
(2) For different injection parameters, the sensitivity of biological nano-technology is different. The operation time, injection intensity, and concentration are more effective than injection volume in terms of decreasing amplitude and cumulative oil. (3) The nearly perfect predicted decreasing amplitude and increasing oil quantity could be achieved by the design plate of injection parameters of the biological nanotechnology. However, there are some disadvantages of the method as follows: In this study, the adsorption of nanoparticles was described by Langmuir equation, but a lot of experiments were not carried out to obtain the adsorption capacity and adsorption coefficient. At the same time, the adsorption process in this study is calculated based on vertical wells, so the simulation accuracy of non-vertical wells, especially horizontal wells, needs to be further verified and improved.
In this study, Carmen-Kozeny law is mainly used to simulate the change of nanoparticles on reservoir porosity and permeability, but the effect of nanoparticles on porosity and permeability still needs to be further evaluated and verified by experiments.
Funding This work was supported by the National Natural Science Foundation of China (Grant No. 51804048).

Conflict of Interest:
The authors declare that they have no conflict of interest.
Ethical statements I certify that this manuscript is original and has not been published and will not be submitted elsewhere for publication while being considered by Journal of Petroleum Exploration and Production Technology. And the study is not split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time.
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