Development and Application of Well-Test Model after Injection Biological Nanomaterials

Nanomaterials are gradually shining in the petroleum industry and have made great achievements in EOR, pipeline coating, fracturing, and other aspects. In practical application, they have the advantages of long validity period and more environmental friendliness. At the same time, due to the short e ﬀ ective period of conventional plugging removal measures such as acidizing and fracturing, which are not friendly to the environment, nanomaterials are gradually applied to reduce pressure and increase injection of injection wells, and good application results are obtained. In this study, a new biological nanomaterial with long-term injection enhancement characteristics was evaluated. To have a better understanding of the reservoir performance of water injection wells after pressure reduction and injection increase measures, it is necessary to establish the well test model after the step-down and injection increase measures of biological nanomaterials that are taken for water injection wells. In this paper, the in ﬂ uence of the injection amount of bionanomaterials on the permeability is studied, and the rule of the permeability around the injection well with the distance from the bottom of the well is analyzed. On this basis, the ﬂ ow mathematical model of three zones (inner zone, transition zone, and outer zone) of bionanosolution is established. Then, using the basic principles of ﬂ uid mechanics and the Laplace transform principle, a mathematical model of the bottom hole pressure response of water injection wells in the third zone of the bionanosolution reservoir is obtained, and the in ﬂ uence of the key process parameters of the bionanosolution water injection on the bottom hole pressure is analyzed. The established well test interpretation model can be used to calculate the near wellbore permeability, reservoir skin factor, reservoir sweep radius, and reservoir resistance coe ﬃ cient. Based on application examples, it was determined that the ﬁ tting results of the interpretation chart are in good agreement with the ﬁ eld test data, and the reservoir parameters obtained from the interpretation are reasonable and reliable. The ﬁ ndings of this study can help for better understanding of the dynamic change law of reservoir after injection of biological nanomaterials or similar plugging removal measures (the improvement e ﬀ ect varies with the distance from the bottom hole). To the best of our knowledge, it is the ﬁ rst paper on the application of biological nanomaterials in reducing pressure and increasing injection in injection wells.


Introduction
Water injection is a common means of secondary and tertiary oil recovery [1][2][3][4]. According to research and statistics, in the process of water injection in most of China's oilfields, due to the low original permeability of the reservoir, substandard injection water quality, and high salinity, the pres-sure of the water injection wells generally increases with time, resulting in an excessive load on the water injection pump, a reduced water injection efficiency, and a highpressure during injection [5][6][7][8][9]. At the same time, offshore oilfields are facing increasingly stringent environmental protection standards. To reduce pressure and increase injection without polluting the environment, some oilfields in China have developed a new type of bionanomaterial, which is environment-friendly and has a valid period exceeding that of ordinary pressure-reducing and injection-increasing materials. However, after multiple rounds of these measures, the amount of rock particles available for dissolution in the reservoir decreases, and the period of successful conventional plugging removal measures becomes shorter and shorter [10][11][12][13]. To improve the effective period of water injection well measures, a new nanomaterial with a longterm injection increase characteristic was studied, a field test was carried out, and good application results were obtained. However, at present, the reservoir performance after the depressurization and injection increase measures are implemented in the water injection well is still unclear. In addition, the seepage of the water injection well is more complex than producing well [14][15][16][17][18], which causes many difficulties and challenges in well tests after biological nanomeasures are applied to water injection wells.
Many scholars have carried out relevant research on well test interpretation methods for various well conditions [16,[19][20][21][22][23][24][25][26]. Compared with previous scholars, Ju et al. increased the factors of permeability heterogeneity and established a mathematical model for well test interpretation of water injection wells in low permeability two-layer composite reservoirs, which considered the well test theory of water injection wells from another aspect [27]. Based on the law of conservation of mass and the principle of relative permeability normalization, Liu et al. established a function for the relationship between the water saturation and the relative permeability and then combined it with Darcy's law of oilwater two-phase flow to establish a well test model of a water injection well, which simultaneously considered the influences of the formation and fluid properties [28]. Jiang et al. established the pressure drop well test model of water injection wells based on BL equation and studied the influence of crude oil viscosity on well test curve, which provided a certain basis for injection increasing measures of water injection wells [16,17]. Wang et al. established a mathematical model of water injection well test considering the influence of WIF (waterflood-induced fracture), applied it in Changqing Oilfield, and achieved good interpretation results [29,30].
At present, with the application of various nanomaterials in China's oil fields, many articles explaining nanomaterials have been published, but most of them are limited to experimental tests, and there is no relevant seepage theory, especially the well test theory of water injection wells established after pressure reduction and injection increase measures. This study innovatively considers that the improvement effect caused by the adsorption of nanomaterials in the formation becomes worse with the distance from the bottom of the well, the nanomaterials need to reach the critical concentration in the formation to improve the effect, and when the concentration of nanomaterials exceeds a certain concentration, the improvement effect will not increase. The formation is divided into inner area (around the well, the concentration is high, which is the effective area of bionano materials), transition area (slightly away from the well, the concentration of bionanomaterials decreases rapidly, and the improvement effect also decreases with the distance), and outer area (farthest from the well, the concentration of bionanomaterials is too low to show improvement effect). The relationship between concentration and zoning is shown in Figure 1, where the depth of red indicates the concentration of biological nanomaterials. Then, based on the divided areas, the mathematical model of seepage is established, and the influence of the key process parameters of the bionanosolution water injection on the bottom hole pressure is analyzed. The model established on the application surface is reliable and can be applied to plugging removal measures (such as adsorption) reservoir performance analysis.
This paper is organized as follows: first, the mathematical model and solutions are presented. Then sensitivity about the reservoir's physical parameters and the fluid parameters are discussed, followed by an application example. The final chapter is the conclusion, summarizes the research results briefly, and puts forward the limitations of this article.

Physical Model of the Biological Nanosolution Injection
Stage. Before establishing the model, we used bionano to treat the core to explore the influence of bionanoinjection on reservoir physical properties. In the actual test process, PV number, a dimensionless number, was used instead of bionanoinjection to minimize the influence of the original formation physical properties on the experimental results.
In Figures 2 and 3, after increasing the injected PV number, we can easily find the relationship between the improvement degree of reservoir permeability and the amount of bionanoinjection; with the increase of bionanoinjection, the reservoir permeability will also increase rapidly, and there is a linear relationship between the permeability and the logarithm of injected PV number (corresponding transition zone), but after the amount of bionanoinjection reaches a certain level, its permeability increase is no longer obvious; that is, there is an upper limit to the increase in permeability caused by biological nanomaterials (Corresponding outer zone).

Geofluids
That is, within a certain PV number range, the following relationship is satisfied: where k is the formation permeability (10 -3 μm 2 ), A is slope of the regression line, and B is intercept of the regression line.
This experiment can also be approximately regarded as the improvement of formation permeability around the injection well, and the injected PV number has a negative linear relationship with the distance from the formation to the bottom of the injection well. Therefore, it can be seen from the above figure that in the actual radial formation, the permeability improvement effect is obvious within a range from the wellhead. With the increase of the wellhead distance, the permeability shows a downward trend, and    3 Geofluids then, after falling to a certain critical concentration, the biological nanoaction disappears.
where R is the distance from the formation to the bottom (m). Therefore, after simple transformation, there is the following equation: where μ is the viscosity (mPa·s); R 1D is the dimensionless radius of the inner region, M 12 is the mobility ratio at the interface between the inner zone and the transition zone, r D is the dimensionless radius, θ 1 is the mobility variation parameter, and it is related to the concentration of biological nanomaterials, 0 ≤ θ 1 ≤ 2. One of the mechanisms by which biological nanoparticles can increase permeability is to increase seepage channel and porosity. Therefore, considering the Kozeny model [31], there is the following equation: where φ is the porosity (%), τ is the tortuosity, and r c is the capillary bundle radius. Therefore, the well storage coefficient has the following equation: where C t is the comprehensive compression coefficient (mPa -1 ), F 12 is the storage capacity coefficient ratio at the interface between the inner zone and the transition zone, and θ 2 is the change parameter of the storage coefficient; it is related to the concentration of biological nanomaterials, In practical work, the effect of too low bionanomaterials is basically negligible; that is, the formation physical properties will not be affected by bionanomaterials after a certain distance from the bottom of the injection well (corresponding inner zone). Then, the change rule of mobility and storage coefficient can be shown in Figure 4: In view of the above situation, the physical model after treatment with a biological nanosolution is shown in Figure 5 and makes the following assumptions. Its assumptions are as follows before establishing the model: (1) The formation is a three-zone radial composite reservoir with a horizontally consistent thickness (2) The gravity and capillary force in the formation can be ignored, and the fluid is a single-phase fluid (3) The fluid seepage in each area is isothermal seepage and conforms to Darcy's plane radial seepage law (4) The water injection volume of the well is constant, and the water injection is carried out from t = 0. Before the water injection, the pressure balance in the formation is equal to p i (5) There is additional resistance at the interface between each seepage area (6) The influences of the wellbore storage effect and the skin effect are considered In this study, the three-zone radial composite reservoir model after nanoparticle adsorption only differs from the multi zone radial composite reservoir model in terms of the treatment methods of the formation and the fluid's physical properties in the transition zone. The former is characterized by continuous change, while the latter is characterized by step-like changes.

Three-Zone Radial Composite Seepage Mathematical
Model of a Biological Nanosolution Reservoir. According to the previously described physical models and assumptions and based on the theory of seepage mechanics, a well test Inner zone Inner zone Transition zone Outer zone Figure 4: Schematic diagram of the mobility and storage coefficient distribution. 4 Geofluids mathematical model of a three-zone radial composite reservoir with a biological nanosolution was established.
(1) The dimensionless seepage differential equations are as follows: Inner region is as follows: Transition region is as follows: Outer region is as follows: In Eqs. ((6)-(8)), p 1D is the dimensionless pressure in the inner zone, p 2D is the dimensionless pressure in the transition zone, p 3D is the dimensionless pressure in the outer zone, t D is the dimensionless time, η is the ratio of the pressure conductivity coefficients of the inner zone and transition zone, R 1D is the dimensionless inner radius, R 2D is the dimensionless transition radius, and R eD is the dimensionless outer radius.
(2) The inner boundary conditions are calculated as follows: Wellbore storage effect is as follows: Skin effect is as follows: In Eqs. (9) and (10), C D is the dimensionless well storage coefficient, p wfD is the dimensionless well bottom pressure, pφD is the dimensionless pressure of the formation near the wellbore, and S is the skin factor.
(3) The outer boundary conditions are calculated as follows: Infinite outer boundary is as follows: Circular constant pressure boundary is as follows: Circular closed boundary is as follows: (4) The interface continuity conditions are calculated as follows: Equal interface pressure is as follows: Equal interface velocity is as follows: In Eqs. ((14)-(17)), S 1f is the inner skin coefficient, S 2f is the inner skin coefficient, and M 13 is the mobility ratio of the inner zone to the outer zone.
(5) Initial conditions are as follows: At the initial time, the pressure in the formation is equal:  5 Geofluids transform about the dimensionless time was carried out, and the general solution of the seepage differential equation in Laplace space was obtained as follows: In Eqs. ((19)-(21)), I 0 ðxÞ and K 0 ðxÞ are the first and second variants of the zero order Bessel functions, respectively, I v ðxÞ and K v ðxÞ are the first and second variants of the v order Bessel functions, respectively, and A 1 -A 6 are coefficients, which are determined by the internal and external boundary conditions and the interface connection conditions. Dimensionless well bottom pressure calculation equation in the Laplace space is as follows: Eq. (22) gives the dimensionless well bottom pressure Laplace space solution of the three-zone composite reservoir with exponentially changing physical properties for the case of an infinite boundary, closed circular boundary, and circular supply boundary. Then, the Stehfest numerical inversion can be carried out to obtain the solution in the real space (there is a detailed solution process and information and code of Stehfest numerical inversion in the appendix). Figure 6 shows a typical dimensionless well bottom pressure curve for a three-zone composite reservoir with exponentially changing physical properties in the transition zone under the conditions of an infinite boundary, closed circular boundary, and circular supply boundary.

Analysis of a Typical Well-Test Curve Chart
As can be seen from Figure 6, the well bottom pressure response can be divided into the following main flow stages by comparing the position from the bottom hole. It should be noted that the division order of this part is based on the distance between the formation affecting each flow stage and the bottom of the well. For example, a flow stage is affected by the physical properties of the formation closest to the bottom of the well, while the affected area of H flow stage is the formation area with the maximum distance from each flow area to the bottom of the well. The dimensionless well bottom pressure response in a three-zone composite reservoir with exponentially changing physical properties in the transition zone is different from the conventional three-zone radial composite reservoir due to the influence of the parameter θ 1 , which is mainly reflected in flow stage (c). That is, the seepage in the transition zone, which is characterized by exponentially changing physical properties, is no longer a horizontal straight line on the derivative curve. For a conventional three-zone composite reservoir, if R 2D − R 1D is large enough, the radial flow stage may occur in the transition zone, the derivative curve is a horizontal straight line segment, and there is a transition flow stage from the inner zone to the transition zone before the straight line segment.

Sensitivity Analysis of the Reservoir's Physical
Parameters. Figure 7 shows the influence of the storage capacity parameter θ 2 on the well bottom pressure performance. As can be seen from Figure 7, θ 2 mainly affects flow stage (c) and the seepage in the transition zone. Although θ 2 does not affect the start time and duration of flow phase (c), it does affect the shape of the pressure derivative curve. If θ 2 is larger, the storage capacity coefficient at r D in the transition zone is smaller than that in the inner zone (φc t ) 1 . For a steeper pressure derivative curve, the slope of the curve will increase in the early stage of the flow period and then gradually decrease. Conversely, the smaller the pressure derivative θ 2 is, the smoother the pressure derivative curve is. Figure 8 shows the influence of the dimensionless inner radius R 1D on the well bottom pressure performance. As can be seen from Figure 8, when the other parameters remain unchanged, the larger the radius R 1D of the inner zone, the later the end time of radial flow (b) in the inner zone and the start time of flow stage (c) in the transition zone, and the earlier the start time of transition flow stage (h) from the transition zone to the outer zone. That is, the larger R 1D , the longer the duration of flow stages (b) and (h), and the shorter the duration of flow stage (c). Figure 9 shows the influence of the transition zone radius R 2D on the well bottom pressure performance. As can be seen from Figure 9, when the other parameters remain unchanged, the larger the radius R 2D of the transition zone, the longer the duration of transition zone flow stage (c), and the shorter the duration of transition flow stage (h) from the transition zone to the outer zone. Conversely, the smaller the transition zone radius R 2D , the shorter the duration of transition zone flow stage (c), and the longer the duration of transition flow stage (h) from the transition zone to the outer zone. Figure 10 shows the influence of the mobility change parameters in the transition zone θ 1 on the well bottom pressure of the   Figure 11 shows the influence of the mobility ratio M 12 at the interface R 1D between the inner zone and the transition zone on the well bottom pressure of the injection well. The change in M 12 affects the mobility ratio M 23 at the interface between the transition zone and the outer zone R 2D ; so, flow stages (c) and (h) are mainly affected by M 12 . If M 12 ≠ 1, then there is a sudden change in the mobility at the interface R 1D between the inner and middle regions. As can be seen from the pressure response curve, there is a transition stage (i) before flow stage (c). The larger M 12 is, the greater the slope of the derivative curve and the longer the duration of transition stage (i) are, the later the start time and the shorter the duration of transition stage (h) are, and the higher the position of flow stage (c) is. However, the time reflecting the flow in the transition zone and the transition stage remains almost unchanged. Figure 12 shows the relationship between the mobility ratio M 13 of the inner zone and the outer zone on the well bottom pressure performance. As can be seen from Figure 12, the mobility ratio M 13 mainly affects flow stages (d) and (h), that is, the reflection of the fluid seepage in the outer zone. The smaller the mobility ratio is, the shorter the duration of transition stage (h) of the fluid seepage from the outer zone to the inner zone is, and the smaller the slope of the pressure derivative curve is; while the earlier the outer zone fluid radial flow stage (d) appears, the lower the position of the derivative curve is, and its value is M 13 /2.

Application Examples
Well A1 is a water injection well in oilfield W, which was put into operation on March 1, 2016. The production horizon is E2s3UI + II + IV, the upper part of E2s3MI, the lower parts of oil formations I, and the parts of oil formations II, which are divided into four sand control sections. The vertical depth in the middle of the oil layer is 2567.35 m, the well completion depth is 2989 m, the completed vertical depth   Table 1 presents the parameters of the water injection well and the reservoir. Biological nanoplugging removal and injection increase measures were implemented in well A1 in July 2019. After these measures were implemented, the actual pressure data were measured and fitted to the results of the three-zone radial composite model described above. Firstly, wavelet transform and appropriate threshold are used to reduce the noise of well test data, and then particle swarm optimization is used to fit the well test data [32]. The near well zone permeability, reservoir skin coefficient, reservoir sweep radius, and reservoir resistance coefficient after the measures were taken were obtained ( Figure 13 and Table 2). The fitting results of the interpretation chart are in good agreement with the field test data, and the reservoir parameters obtained from the interpretation are reasonable and reliable.

Summary and Conclusions
(1) Based on the dynamic characterization mathematical model of how a biological nanosolution changes the reservoir seepage resistance, seepage mathematical models of the biological nanosolution action area, the biological nanosolution action transition area, and the external nonaction area were established, the analytical solution was obtained in Laplace space, and the solution was obtained in real space using the Stehfest numerical inversion. The typical pressure performance curve for a three-zone homogeneous radial composite reservoir with exponentially changing physical properties in the transition zone was drawn, and we divide it into eight seepage stages (2) The dimensionless well bottom pressure response of a three-zone composite reservoir with exponentially changing physical properties in the transition zone is different from the conventional three-zone radial composite reservoir due to the influence of the parameter θ 1 . This is mainly reflected in the flow stage in the transition zone with the changing physical properties; that is, the seepage in the transition zone with exponentially changing physical properties is no longer a horizontal straight line on the derivative curve (3) The established well test interpretation model can be used to calculate the near well zone permeability, reservoir skin coefficient, reservoir sweep radius, and reservoir resistance coefficient. The fitting results of the interpretation chart are in good agreement with the field test data. The reservoir parameters obtained from the interpretation are reasonable and reliable. The results of this study have important guiding significance for the fine evaluation of the use of nanomaterials to improve the oil reservoir development effect  11 Geofluids injected, the water saturation around the water injection well will still change after bionano is injected. This model does not take into account the change of water saturation of multiphase seepage around the water injection well, which can be improved in the follow-up work Then, we can get