Quantitative Characterization Method of Additional Resistance Based on Suspended Particle Migration and Deposition Model

Abstract

The phenomenon of pore blockage caused by injected suspended particles significantly impacts the efficiency of water injection and production capacity release in offshore oilfields, leading to increased additional resistance during the injection process. To enhance water injection volumes in injection wells, it is essential to quantitatively study the additional resistance caused by suspended particle blockage during water injection. However, there is currently no model for calculating the additional resistance resulting from suspended particle blockage. Therefore, this study establishes a permeability decline model based on the microscopic dispersion kinetic equation of particle transport. The degree of blockage is characterized by the reduction in fluid volume, and the additional resistance caused by particle migration and blockage during water injection is quantified based on the fluid volume decline. This study reveals that over time, suspended particles do not continuously migrate deeper into the formation but tend to deposit near the wellbore, blocking pores and increasing additional resistance. Over time, the concentration of suspended particles near the wellbore approaches the initial concentration of the injected water. An increase in seepage velocity raises the peak concentration of suspended particles, but when the seepage velocity reaches a certain threshold, its effect on particle migration stabilizes. The blockage location of suspended particles near the wellbore is significantly influenced by seepage velocity and time. An increase in particle concentration and size accelerates blockage formation but does not change the blockage location. As injection time increases, the fitted injection volume and permeability exhibit a power-law decline. Based on the trend of injection volume reduction, the additional resistance caused by water injection is calculated to range between 0 and 3.85 MPa. Engineering cases indicate that blockages are challenging to remove after acidification, and the reduction in additional resistance is limited. This study provides a quantitative basis for understanding blockage patterns during water injection, helps predict changes in additional resistance, and offers a theoretical foundation for targeted treatment measures.

1. Introduction

Water flooding has become one of the effective tertiary recovery techniques for many oilfields to enhance oil recovery rates [1,2]. However, research and field production have shown that some oilfields experience water injection blockages, which significantly impact the effectiveness of water flooding and hinder production capacity. This issue has gradually gained the attention of field engineers and researchers [3,4]. How to clearly identify water blockage patterns and further improve oil recovery rates after water flooding has become a crucial research topic.

There are many factors that affect the clogging of water injection wells. Primarily, impurities in the water cause suspended particles to migrate and block the rock pore throats, leading to pore blockage [5,6]. High clay content can also lead to clogging due to clay hydration and dispersion, blocking the screens and formation. Additionally, acid precipitates formed by acidizing can block pore throats, making water injection difficult. Among these factors, statistics show that of the 113 water injection wells in the Bohai SZ36 and PL19 blocks, 51% of injection difficulties are caused by suspended particle migration clogging. Therefore, studying the issue of suspended particle migration clogging pores, which causes water injection difficulties, is particularly important [7,8].

In fact, large amounts of oxides, clay minerals, humic substances, and other colloidal particles in the underground environment have a high specific surface area and are sensitive to chemical and physical environments. These properties enable them to strongly adsorb pollutants and move with the water flow, potentially accelerating the diffusion of contaminants. Due to the high velocity of injected water, suspended particles easily fill aquifer spaces during infiltration, causing blockages. According to an experimental study by Sun on artificial recharge in southern Australia, over 92% of total clogging is due to suspended matter [9]. The injection difficulties caused by clogging have become an extremely challenging issue for water injection in oilfields.

In current studies on water injection clogging, given the complexity of the injection process, the methods for calculating additional injection resistance remain general. For example, Yu developed a model for calculating additional pressure in injection wells by considering formation pressure, injection pressure, wellhead pressure, and the injectivity index [10]. This model estimates that additional pressure in injection wells generally ranges between 1.5 and 3.0 MPa. Although such methods provide theoretical references for increasing wellhead pressure to counteract clogging, they are too generalized in calculating additional pressure and lack insights into particle migration-induced clogging mechanisms. Consequently, the internal mechanisms of particle migration remain underexplored, and studies are still in the experimental phase. Thus, theoretical knowledge in this area is scarce, and there is limited understanding of the patterns of pore blockage due to particle migration.

Suspended particle concentration is the primary factor affecting the degree and timing of suspended particle deposition, while seepage velocity is a key factor influencing particle migration. Changes in seepage velocity alter the driving force on suspended particles, thus changing the particle migration characteristics in the medium [11,12,13]. Low flow rates promote suspended particle deposition, especially for larger particles, leading to severe surface clogging, while high flow rates facilitate particle migration deeper into the medium. Yao et al. conducted experiments adjusting seepage velocity while keeping particle size and concentration constant. They found that increased seepage velocity led to a greater particle migration rate [14]. Starr et al. used tracers to study particles smaller than 5.00 μm and found that at low flow rates, particle migration speed could exceed water flow speed; as flow rates increased, particle migration speed approximated the flow rate [15]. Avella used resin sand as suspended particles and observed that seepage velocity changes affected particle migration and deposition [16]. Their study showed that for large particle size ratios, suspended particle penetration rate was proportional to seepage velocity, with velocity having a significant impact on penetration rate. By varying the seepage velocity, they found that velocity affects the deposition location of particles: at low velocities, most particles deposit near the surface, while at higher velocities, particles deposit deeper. Besides velocity, particle size ratio also affects deposition location; with smaller particle size ratios, particles tend to deposit near the surface.

Wang et al. also used resin sand as suspended particles and observed changes in particle migration and deposition caused by varying seepage velocity, demonstrating that large particle sizes increase clogging likelihood [17]. They proposed a particle migration and clogging model, where the key parameters need to be determined experimentally. However, the model does not provide insights into the regularities of particle migration and clogging processes, limiting its practical engineering applications. In fact, the processes of particle migration, deposition, clogging, capture, and release have been studied in soil mechanics. As early as 1988, Mitra and Goeppert proposed a particle migration model in porous media [18,19,20]. The model considers the effects of time, particle flow velocity, particle diameter, and other factors, but it does not account for the initial particle concentration. Based on this model, Katzourakis, V. E. et al. developed a highly complex numerical model for particle deposition and blockage. This model considers the influence of initial concentration and also predicts the deposition of suspended particles in pores [21]. The model incorporates calculations of the blockage locations within the porous medium at different times following suspended particle migration, further refining the migration and deposition processes of suspended particles in porous media. However, this formula is solved using highly complex numerical simulations, making the process intricate and time-consuming, and thus unsuitable for the theory of pore blockage in oilfield water injection.

Developing a model for suspended particle migration and blockage in water injection and using the model results to predict permeability reduction and increased additional pressure in the formation is key to addressing the difficulty of water injection.

Therefore, based on existing models of suspended particle migration, deposition, and blockage in porous media, this paper establishes a model for the migration, deposition, and pore blockage of suspended particles in water injection by considering factors such as time, particle concentration, flow velocity, and particle size. Based on this model, we further describe the reduction in formation injection volume due to pore blockage and the additional resistance caused by decreased injection volume. By predicting changes in additional resistance during water injection, the model estimates the potential increase in injection pressure.

2. Laboratory Study

2.1. Physical Model

The migration of suspended fine particles in porous media is primarily due to convection and dispersion. In a one-dimensional steady-state flow, the migration of suspended particles in saturated, homogeneous porous media is commonly described by the convection–dispersion equation [22,23,24,25,26], as shown in Figure 1.

$${\displaystyle \frac{\partial C(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}C(x,t)}{\partial x^2}}-v{\displaystyle \frac{\partial C(x,t)}{\partial x}}-{\displaystyle \frac{\rho }{\varphi }}{\displaystyle \frac{\partial \sigma }{\partial t}}$$

(1)

In the equation, C(x,t) represents the concentration of deposited particles in the porous medium as a function of time (mg/L); x is the particle transport distance (cm); v is the average flow velocity through the cross-sectional area (cm/s), D is the particle diameter (μm); $\rho$ is the fluid density (kg/m³); $\sigma$ represents the ratio of the particle volume deposited on the surface of the porous medium to the solid volume of the porous medium, dimensionless; and $\varphi$ is the porosity (%).

The particle deposition dynamic equation considering the dispersion effect. The classical particle migration–deposition model can be represented as follows:

$${\displaystyle \frac{\rho }{\varphi }}{\displaystyle \frac{\partial \sigma }{\partial t}}=kC-k{\displaystyle \frac{D}{v}}{\displaystyle \frac{\partial C}{\partial x}}$$

(2)

In the equation, k is the permeability coefficient (D).

Substituting Equation (2) into Equation (1) gives

$${\displaystyle \frac{\partial C(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}C(x,t)}{\partial x^2}}-(k{\displaystyle \frac{D}{v}}-v){\displaystyle \frac{\partial C(x,t)}{\partial x}}-kC(x,t)$$

(3)

Assuming $D_1=k{\displaystyle \frac{D}{v}}-v$, and simplifying the above equation:

$${\displaystyle \frac{\partial C(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}C(x,t)}{\partial x^2}}-D_1{\displaystyle \frac{\partial C(x,t)}{\partial x}}-kC(x,t)$$

(4)

The corresponding initial boundary conditions are

$$C(x,t)=0,0\le x<+\infty$$

(5)

Take the Laplace transform of both sides of Equation (4) with respect to t, and use the initial condition (5):

$$D{\displaystyle \frac{{\partial }^{2}C(x,t)}{\partial x^2}}-(D_1-v){\displaystyle \frac{\partial C(x,t)}{\partial x}}-kC(x,t)=0$$

(6)

Equation (6) is a second-order partial differential equation. To solve the equation and apply the Laplace transform, the analytical solution for the concentration distribution of suspended particles in the porous medium can be expressed as

$$C(x,t)={\displaystyle \frac{C_i}{2}}\left\{\begin{array}{l}\exp \left[{\displaystyle \frac{vx}{2D}}(1-{\varphi }_{init})\right]erfc({\displaystyle \frac{x-vt{\varphi }_{init}}{2\sqrt{Dt}}})\\ +\exp \left[{\displaystyle \frac{vx}{2D}}(1+{\varphi }_{init})\right]erfc({\displaystyle \frac{x+vt{\varphi }_{init}}{2\sqrt{Dt}}})\end{array}\right\}$$

(7)

In the equation, C(x,t) represents the concentration of deposited particles in the porous medium as a function of time (mg/L), C_i is the initial concentration of suspended particles in the porous medium (mg/L), and ${\varphi }_{init}$ is the initial porosity (%). The function erfc(x) is the complementary error function.

As suspended particles deposit, the pore spaces in the formation change with the accumulation of particles, leading to changes in porosity after water injection [27,28].

$${\varphi }_p={\varphi }_{init}{\displaystyle \frac{{\rho }_s-C_i}{{\rho }_s-C_s}}$$

(8)

In the equation, ${\varphi }_p$ represents the porosity after pore blockage (%). ${\varphi }_{init}$ is the initial porosity (%). ${\rho }_s$ is the fluid medium density (kg/m³). $C_s$ is the concentration of particles deposited in the pore space, which is determined by C(x,t) (mg/L).

The permeability of the porous medium after particle deposition can be expressed as follows [29,30]:

$$K_p=K_i{\displaystyle \frac{{({\rho }_s{\varphi }_p-C_s)}^3{(1-{\varphi }_p)}^2}{{{\varphi }_p}^3({\rho }_s+C_i)\left[{\rho }_s(1-{\varphi }_p)+C_i+C_s\right]}}$$

(9)

In the equation, $K_p$ is the permeability after water flooding, mD. $K_i$ represents the initial permeability of the porous medium, mD.

When the permeability decreases, the volume of injected water also decreases. The decline in fluid volume after water injection is obtained as

$$R_q=1-{\displaystyle \frac{Q_p}{Q_{init}}}=1-{\displaystyle \frac{K_p}{K_i}}$$

(10)

In the equation, $R_p$ represents the magnitude of the liquid volume decline, which is dimensionless. $Q_p$ represents the current injection rate (m³/day) and $Q_{init}$ represents the initial injection rate (m³/day).

As the formation porosity becomes blocked, the skin factor of the formation increases [31]:

$$s=({\displaystyle \frac{K_p}{K_i}}-1)\ln ({\displaystyle \frac{R_s}{R_w}})$$

(11)

In the equation, $R_s$ represents the radius of suspended particle migration and blockage (m), and $R_w$ represents the wellbore radius (m).

As the skin factor of the formation increases, the additional injection resistance also increases [32]:

$$\Delta p_s={\displaystyle \frac{\mu B{Q}_p}{0.543 K_{p}h}}s$$

(12)

In the equation, $\Delta p_s$ represents the additional injection resistance (MPa), $\mu$ is the fluid viscosity (mPa·s), $B$ is the volume factor (dimensionless), and h is the reservoir height (m).

By substituting Equations (4) and (5) into Equation (6), the post-blockage pore permeability can be obtained. After blockage, as the permeability decreases, the relationship between the post-blockage permeability and the injection rate can be derived. This allows for the determination of the skin factor changes after blockage, and subsequently, using Equation (9), the additional resistance after water injection can be calculated.

Using the suspended particle deposition blockage model, the flowchart for determining the additional resistance of the injection well is shown in Figure 2.

2.2. Indoor Experiment

An experimental system for simulating sand layer migration and deposition, developed by the National Key Laboratory of Efficient Development of Offshore Oil and Gas, was used to conduct tests on suspended particles with different particle sizes and porous media of various dimensions. This study focuses on the impact of particle size variation on the migration and deposition characteristics of suspended particles.

2.2.1. Experimental Equipment

To address the core issue and simplify the experiment, a radial geotechnical model around a well was selected for simulation. The cylindrical experimental tube is made of acrylic, with a length of 100 cm and an inner diameter of 10 cm, as shown in Figure 3. A certain concentration of suspended particles from the syringe was injected into the sand column sample. Different mesh screens with varying pore sizes were arranged at the inlet and outlet of the device, based on the sample’s particle size, to ensure uniform flow through the entire cross-section of the sand column.

2.2.2. Experimental Plan

A collection pipe was set up at the end of the experimental tube to collect the effluent during the experiment. The deposition of suspended particles at different particle sizes and flow velocities was then tested. For each injection, water samples were collected every 2 to 3 min (depending on the flow velocity). The turbidity of each collected water sample was measured. By continuously injecting high-purity deionized water into the water sample containing a certain concentration of suspended particles for dilution, and corresponding turbidity measurements, the relationship between particle concentration and turbidity in the water could be established. The relationship between concentration and turbidity is well documented in the literature and will not be repeated here.

In the 100 cm long acrylic tube, the particle travel time within the tube was determined based on the flow velocity. The distance traveled during each time segment was recorded, and the particle concentration in each segment could then be determined. Finally, by analyzing the relationship between the turbidity of the suspended liquid and concentration, the changes in particle concentration in the effluent water samples over time were analyzed. The adaptability of the suspended concentration migration model under different conditions was also studied.

During the experiment, the permeability changes in some experiments could also be observed. The permeability coefficient of the sand column was calculated using Darcy’s law, where K is the permeability and K₀ is the initial permeability. The change in permeability is expressed as K/K₀, which was used to analyze the blockage characteristics of the infiltrated medium.

2.2.3. Experimental Materials

To address the core issue and simplify the experiment, natural quartz sand was used as the porous medium. The SiO₂ content of the quartz sand was no less than 99.6%, and its density was 2.50 g/cm³. The sand’s particle size was analyzed using the sieve method, and the results show that the median particle size of the quartz sand was 100 μm. As shown in

Table 1. Physical parameters of the quartz sand used in the experiment.

Material	Particle Size (µm)	Median Particle Size (µm)	Porosity (%)	Uniformity Coefficient
Quartz sand	60–120	100	30	1.3

.

In this experiment, clay was selected as the suspended material. The clay was sieved to obtain suspended particles of different particle sizes. Particle size analysis of the suspended particles was conducted using the Bettersize 2000 laser particle size analyzer. The median particle sizes of the clay suspensions were 5.0 μm, 10 μm, and 15 μm, respectively. Suspended particles in the injection water are detailed in

Table 2. Analysis of fine particles in injection water.

Injection Water	Suspended Solids mg/L	Median Particle Size µm
Value	1–15	5, 10, 15

.

3. Results and Discussion

Suspended particle migration and clogging are particularly important in the process of injection-induced blockage. Therefore, through experimental methods, the accuracy of the suspended particle migration model was verified. This section compares experimental values with theoretical values. The parameters used in the theoretical calculations are shown in

Table 3. Basic parameters for the suspended particle migration and deposition model.

Flow Velocity (cm/s)	Suspension Concentration (mg/L)	Permeability (μD)	Porosity (%)	Median Particle Size (µm)
0.36	1.0	10	30	20

.

3.1. Migration and Clogging Radius of Suspended Particles at Different Times

The migration time of suspended particles is crucial for their movement and clogging. As time progresses, the suspended particles gradually migrate deeper into the formation and clog the pores. Based on the theoretical and experimental methods introduced earlier, the particle concentrations at different time intervals were collected, and the particle migration position was determined according to the running time. Using the theoretical model in Formula (2), particle concentrations at different times were calculated, as shown in Figure 3.

From the figure, two distinct migration curves are observed during the particle migration with water flow. The first case, shown in Figure 4a, demonstrates clogging between 50 cm and 200 cm, where the deposition curve peaks, and the suspended particle concentration approaches the initial particle concentration of 1.0 mg/L. It is worth noting that at a distance of 200 cm from the inlet, a bimodal phenomenon appears as time progresses. This is because, as the distance increases, the suspended particles undergo a process of migration, deposition, and remigration. As a result, there is a process of particle accumulation, migration, re-accumulation, and remigration, leading to the appearance of a bimodal phenomenon. However, it is currently unclear at what distance from the inlet this bimodal issue will occur. This type of phenomenon is discussed in the literature [21]. Figure 4 in the literature [21] clearly shows that microparticle migration exhibits fluctuation phenomena. The particle migration and deposition process can lead to multiple peaks.

In the second case, shown in Figure 4b, far from the inlet, the suspended particles do not undergo significant deposition, with the deposition curve showing no obvious peak. The suspended particle concentration remains close to 0 mg/L–0.2 mg/L.

This indicates that suspended particles do not migrate indefinitely with the fluid. In most cases, they deposit in the near-well zone, blocking the pores and thereby causing additional resistance.

3.2. Suspended Particle Distribution Under Different Concentration Conditions

Under the same conditions, the concentration of suspended particles was varied, ranging from 1 mg/L to 5 mg/L, 10 mg/L, and 15 mg/L. The particle deposition concentrations at different positions were observed. Based on the theoretical model in Formula (2), the particle concentrations at different positions were calculated, as shown in Figure 4. The calculation results were compared with experimental data. From the comparison, it can be observed that with increasing concentration, the maximum particle deposition position did not change significantly and remained approximately at 62 cm from the inlet, where the suspended particle concentration was highest.

It is worth noting that at particle concentrations of 5 mg/L, 10 mg/L, and 15 mg/L, a bimodal phenomenon appears during the particle migration process. It can be observed that as the concentration increases, the suspended particles are more likely to deposit, then migrate, and deposit again. As shown in Figure 5.

As the concentration of suspended particles increased, the deposition concentration in the simulated formation also increased. Over time, the concentration approached the initial concentration value. As the injection time increased, the porous medium gradually became filled with the suspended quartz powder, leading to an increase in the relative concentration of suspended particles. This concentration eventually reached a peak at a certain moment, and as the suspension continued to be injected, the peak concentration was maintained for a period.

3.3. Suspended Particle Distribution Under Different Permeability Velocities

Under the same conditions, the permeability velocity of the suspended particles was varied, ranging from 0.09 cm/s to 0.18 cm/s, and 0.36 cm/s. Based on the theoretical model in Formula (2), the particle concentrations at different positions were calculated. When the concentration of the injected suspended particles remained constant, the peak particle concentration increased with the increase in permeability velocity. The higher the flow velocity, the more the particles migrated deeper into the formation. As shown in Figure 6.

As the permeability velocity increased, the peak particle concentration also increased. However, when the permeability velocity reached a certain value, the effect of flow velocity on particle migration started to level off. In reality, at higher flow velocities, the hydrodynamic forces acting on the particles become stronger, which weakens the deposition effect of the suspended particles, leading to an increase in the number of particles flowing out of the system. However, at higher permeability velocities, the frequency of collisions between the suspended particles and the pore surfaces of the matrix also increases. Consequently, the outflow rate of suspended particles decreases relative to the flow velocity of the infiltrating water, resulting in a significantly higher likelihood of clogging at higher permeability velocities. Similarly, as the seepage velocity increases, a bimodal phenomenon appears in the particle migration process at a seepage velocity of 0.36 cm/s. It can be observed that as the seepage velocity increases, the suspended particles first migrate, then deposit, and migrate again.

3.4. Suspended Particle Distribution Under Different Particle Size Conditions

Under the same conditions, the particle size of the suspended particles was varied, with sizes ranging from 15 μm to 10 μm, and 5 μm. The particle deposition concentrations at different positions were observed. Based on the theoretical model in Formula (2), the particle concentrations at different positions were calculated, as shown in Figure 7. The calculated results were compared with the experimental data. From the comparison, it can be seen that as the particle size decreased, the maximum deposition position of the particles did not change significantly. However, the concentration of particles gradually decreased.

In reality, the larger the particle size, the fewer the number of pores in the porous medium that are larger than the particle size, increasing the likelihood of the particles being captured by pore throats. Since some of the suspended particles deposit on the surface of the porous medium, the porosity of the medium gradually decreases, leading to a reduction in permeability. Furthermore, as the volume of the injected suspension continues to increase, the relative concentration of suspended particles shows a decreasing trend.

3.5. Different Suspended Particle Sizes and Their Impact on Permeability Decline After Blockage

Under the same conditions, the particle size of the suspended particles was varied, with the median particle sizes of the suspended particles ranging from 15 μm to 10 μm, and 5 μm. The measured experimental permeability and the ratio of the initial permeability changed with time. After blockage by 15 μm particles, the permeability coefficient decreased to approximately 85.43% of the initial permeability. For 10 μm and 5 μm particles, the permeability coefficient decreased to approximately 95.58% and 98.51% of the initial permeability, respectively, as shown in Figure 8.

The permeability coefficient shows a significant decrease, gradually lowering and stabilizing over time. Under the same conditions, the larger the particle size of the infiltrating medium, the higher the permeability coefficient. Larger particles block later, and to reach the same level of blockage, larger particles take longer. The smaller the particle size of the suspended particles, the further they migrate, and the greater the depth at which the relative permeability significantly decreases in the infiltrating medium. On the other hand, larger suspended particles are more likely to be retained on the surface of the sand column, with a smaller impact on the permeability of the deeper medium, resulting in a smaller decrease in permeability. This part of this study has been verified in the literature [24].

4. Application Effect Evaluation

After the early water injection in the Bohai SZ oilfield, some well areas experienced clogging, resulting in a decline in liquid production. In some wells, the decline exceeded 60%, severely affecting the oilfield’s production capacity. The method presented in this study was used to quantitatively calculate the decrease in liquid production after water injection, characterizing the blockage pattern, and enabling the adoption of targeted remediation strategies.

SZ-1-34 Well: This well was designed for water injection, located in the Dongying Formation oil reservoir. The perforated thickness was 36.4 m, with the effective thickness of the fractured oil reservoir being 23.2 m. It used four stages of high-quality screen pipes for sand control and a hollow integrated injection tubing string. Initial Injection Rate is 566.67 m³/day, Suspension Concentration is 10.9 mg/L, Permeability is 750 μD, Porosity is 30%, Median Particle Size is 15 µm. As shown in

Table 4. SZ-1-34 injection well additional resistance calculation parameters.

Initial Injection Rate (m3/day)	Final Injection Rate (m3/day)	Suspension Concentration (mg/L)	Permeability (μD)	Porosity (%)	Median Particle Size (µm)
566.67	105.77	10.9	750	30	15

.

On 5 January 2015, the new well was put into operation. Initially, the injection rate was 150 m³/d, and over time, the injection volume increased. By 15 July 2015, the injection pressure rose to 10.0 MPa, with an injection rate of 321 m³/d. By 31 July, the injection rate began to fall short of the required volume, with the same injection pressure. By 6 August, the injection rate dropped to 120 m³/d. Pressure reduction operations were carried out, but even with increased pressure, the required injection volume could not be met.

• Water Injection Curve (Figure 9a)

The water injection curve shows the variations in injection volume and pressure over time, highlighting the challenges in maintaining the required injection rate.

• Analysis Using the Additional Resistance Model

Using the previously established water injection additional resistance calculation model, we analyzed the additional resistance for the SZ-1-34 well. The water quality analysis revealed that the concentration of suspended particles in the injected water was 10.9 mg/L. Using Formula (4), the suspended particle distribution in the Bohai SZ oilfield was determined (Figure 9b), with a maximum suspended particle concentration of 8.3 mg/L. The blockage position was identified at 50 cm. It is assumed that the pore blockage reaches its maximum value within a certain period of time.

• Permeability Change Trend (Figure 9c)

The permeability change trend was calculated using Formula (6). Through calculation, the permeability decay coefficient is −0.0051. Based on this permeability decay coefficient, the process of gradual decline from the maximum permeability is obtained, and the permeability changes over time follow a power law distribution.

• Decrease in Injection Volume (Figure 9d)

Based on the reduction in permeability, Formula (7) was used to calculate the decrease in injection volume. Initially, the injection volume was 566.67 m³, and after 25 days, the injection volume decreased to 105.77 m³.

• Skin Factor and Additional Resistance (Figure 9e)

The skin factor and additional resistance were determined over time. The skin factor ranged from 0 to 6, while the additional resistance increased from 0 to 2.29 MPa. This indicates an inverse relationship between permeability reduction and additional resistance, with the overall trend following a power function relationship.

These results demonstrate the effectiveness of the model in assessing the water injection performance and identifying trends in clogging and resistance in water injection wells. This, in turn, helps inform remediation strategies aimed at improving water injection efficiency.

Figure 9. Analysis of additional resistance in the first acidizing stage for well SZ-1-34. (a) Water injection pressure and injection volume curves; (b) Water injection blockage location; (c) Permeability change after blockage; (d) Injection volume change after blockage; (e) Skin factor and additional resistance changes after blocking.

Figure 9. Analysis of additional resistance in the first acidizing stage for well SZ-1-34. (a) Water injection pressure and injection volume curves; (b) Water injection blockage location; (c) Permeability change after blockage; (d) Injection volume change after blockage; (e) Skin factor and additional resistance changes after blocking.

After acidizing, the formation blockage by suspended particles was reduced but not completely cleared. Using the previously established water injection additional resistance calculation model, the additional resistance after the second acidizing treatment for well SZ-1-34 was analyzed. Water quality analysis indicated a continuous injection water concentration of 10.9 mg/L. Based on Equation (4), the blockage position was determined to be at 50 m. The permeability trend over time, calculated by Equation (6), follows a power function relationship, as shown in Figure 10a.

Based on the permeability reduction, the decrease in water injection volume was determined using Equation (7). The initial injection volume of 302 m³ decreased to 107.88 m³ after 25 days, as illustrated in Figure 10b.

Further analysis of the skin factor and additional resistance over time shows that the skin factor ranges from 0 to 6, with additional resistance increasing from 0 to 1.44 MPa. This demonstrates an inverse relationship between permeability change and additional resistance, following an overall power function trend, as seen in Figure 10c.

Well SZ-1-12 was designated as an injection well, targeting the Dongying Formation oil layer, with an effective perforated thickness of 61.6 m. It features four sections of high-quality screen pipes for sand control and a hollow integrated injection string. Suspension Concentration is 10.9 mg/L, Permeability is 742 μD, Porosity is 30%, Median Particle Size is 15 µm. As shown in

Table 5. SZ-1-12 injection well additional resistance calculation parameters.

Initial Injection Rate (m3/day)	Final Injection Rate (m3/day)	Suspension Concentration (mg/L)	Permeability (μD)	Porosity (%)	Median Particle Size (µm)
700	207	10.9	742	30	15

.

On 9 June 2014, the well was newly commissioned with an initial injection rate of 200 m³/day. Subsequently, the injection volume was dynamically adjusted upwards, along with a rapid increase in injection pressure. By 1 January 2015, the injection rate reached 700 m³/day, with pressure rising to 10.0 MPa. Injection shortages began on 28 January. By 1 June 2015, injection pressure was still at 10.0 MPa, but the injection rate had dropped to 207 m³/day, far short of the required 548 m³/day. Layered flow tests conducted between 3 June and 6 June 2015 indicated shortages across all well layers. On 28 June, a tractor acidizing operation was performed. Post-acidizing, the injection rate on 1 July increased to 638 m³/day at 9.0 MPa, meeting the target of 637 m³/day. However, by 10 July, the injection pressure had again reached 10.0 MPa, with an injection rate of 597 m³/day, signaling the start of another injection shortage. The injection curve is shown in Figure 10a.

Using the previously established additional resistance calculation model for water injection, the additional resistance in well SZ-1-12 was analyzed. Water quality analysis indicated an injection water concentration of 10.9 mg/L. Using Equation (4), the distribution of suspended particles in Bohai SZ Oilfield was determined (Figure 11a), with a maximum particle concentration of 9.9 mg/L. The blockage was located at 68 cm. It is assumed that the pore blockage reaches its maximum value within a certain time, as shown in Figure 11b. The permeability trend over time, calculated by Equation (6), indicates that the permeability decay coefficient is −0.0079. Based on this decay coefficient, the process of gradual reduction from the maximum permeability is obtained, and the permeability changes over time follow a power law distribution, as shown in Figure 11c.

Based on the reduction in permeability, Equation (7) was used to calculate the reduction in injection volume. The initial injection volume of 742.34 m³ dropped to 205.67 m³ after 18 days, as illustrated in Figure 11d.

Further analysis of the skin factor and additional resistance over time indicates that the skin factor ranges from 0 to 7, with additional resistance increasing from 0 to 2.35 MPa. This relationship between permeability reduction and additional resistance follows an inverse trend, with an overall power function relationship, as seen in Figure 11e.

After acidizing, the formation blockage by suspended particles was partially cleared but not entirely removed. Using the previously established water injection additional resistance calculation model, the additional resistance for water injection in well SZ-1-12 after the second acidizing stage was analyzed. Water quality analysis indicated a continuous injection water concentration of 10.9 mg/L. According to Equation (4), the blockage position was determined to be 50 m. The permeability variation trend was derived from Equation (6) and shows a power function relationship with time, as illustrated in Figure 12a.

Based on the reduction in permeability, Equation (7) was used to calculate the decline in water injection volume, showing an initial injection volume of 648 m³ that decreased to 84 m³ after 26 days of injection, as shown in Figure 12b.

Further analysis of the skin factor and additional resistance over time reveals a skin factor range between 0 and 6, with additional resistance increasing from 0 to 3.85 MPa. This indicates an inverse relationship between permeability variation and additional resistance, following an overall power function trend, as shown in Figure 12c.

The comparative analysis of wells SZ-1-12 and SZ-1-36 indicates that the initial high injection rates confirm higher formation permeability. Following acidizing, the injection rate decreased compared to the initial rate, indicating that formation blockage cannot be effectively removed and the acidizing range is limited. Additionally, the lack of significant increase in additional resistance after acidizing suggests that blockage is difficult to alleviate. Therefore, acidizing offers only limited potential for increasing pressure space through blockage removal.

5. Conclusions

Based on previous particle transport and deposition models, this paper establishes an analytical solution model for the transport and deposition of suspended particles in pore spaces. The model simplifies the calculation process of suspended particle deposition over time. Building on this model, the decay characteristics of injection volume, porosity, and permeability are determined. Additionally, the blockage resistance caused by suspended particles during water injection is quantified.

The blockage location of suspended particles in the near-well zone is significantly influenced by flow velocity and time. An increase in particle concentration and particle size accelerates particle blockage but does not affect the blockage location. Engineering examples show that after acidizing, the blockage material is difficult to remove, and the reduction in additional resistance is limited.

Porosity and permeability are critical factors in determining whether oil wells will experience water blockage. When designing water flooding, it is important to pay attention to differences in formation physical properties across different well areas, their compatibility with water injection, and the variability in injection concentrations. This method can serve as a reference for similar oilfields.