Study of sand particle transport characteristics and different critical velocities in sand-producing wells via indoor experiments

Abstract

Due to the widespread application of new technologies such as increased production and injection, the output of many old wells has improved, but this has also led to varying degrees of sand production. Increased sand production has affected oil well production, increased equipment wear, wellbore plugging, and safety risks. To increase the sand particle transport performance within wellbores, indoor experiments have been conducted to study the sand transport characteristics, transformation relationships, and critical flow velocities for fluid carrying sand. The experimental results indicate that sand particle transport within wellbores can be classified into four modes: stationary, rolling, skipping, and suspended. The flow pattern characteristics of the sand particles were further subdivided into eight categories, and the transformation relationships between these flow patterns were established. Experiments on the critical flow velocities for different grain sizes of sand in inclined wellbores have shown that the fluid velocity required for lifting sand particles is greater than that for rolling and sliding. When the wellbore inclination angle is greater than or equal to 70°, the fluid velocity required for lifting sand particles is approximately 1.9 times greater than that for rolling, and the rolling velocity is approximately 1.6 times greater than that for sliding. In vertical wellbores, the critical velocity for the sand particle suspension is approximately 0.81 times the terminal settling velocity of the sand particles in water. This research provides important scientific evidence for improving and optimizing sand removal techniques in oil wells, as well as a systematic approach and experimental foundation for further studies on sand transport in complex wellbores.

Introduction

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In the process of petroleum extraction, oil wells frequently encounter the challenge of sand intrusion, which poses a significant threat to stable production and the normal operation of equipment¹. According to relevant data, the sand production issue in China’s oil wells has directly or indirectly led to a reduction of approximately 20% in annual oil production^2,3. Throughout the entire production lifecycle, over 50% of oil wells need to consider sand production and implement appropriate sand control measures⁴. As many domestic oilfields have entered the middle to late stages of water flooding development⁵, coupled with the widespread application of new technologies aimed at enhancing oil recovery and injection⁶, an increasing number of oil wells are experiencing varying degrees of low pressure and sand production issues⁷. Owing to the relatively low formation pressure within the well, the produced fluid struggles to effectively transport sand particles out of the wellbore, leading to the accumulation of sand within the well and its gradual increase⁸. Once the accumulation of sand within a well reaches a certain level, it not only reduces the production efficiency of the oil well but also accelerates the wear and damage of equipment, thereby increasing the frequency of maintenance and replacement of tubing and other well devices⁹. Excessive accumulation of sand within a well can lead to wellbore plugging, thereby extending the downtime of the oil well. These issues not only affect the production of individual wells but also inherently increase the overall operational costs of oil wells^10,11. Therefore, in-depth research on deriving the transport characteristics of sand particles within the wellbore, analyzing the different critical flow velocities during fluid–sand transport, enhancing the sand transport performance within the wellbore, reducing wellbore sand settling, mitigating the risk of formation sand burial, and achieving further cost reduction and efficiency improvement remain critical issues that urgently need resolution in current domestic oilfield production engineering¹².

Sand production in oil wells refers to a phenomenon that occurs during the production and development of oil wells, where natural or human-induced factors lead to destruction of the reservoir formation structure, causing the detachment of rock matrix particles and the existing mobile sand particles in the reservoir to flow into the wellbore along with the produced fluid. This also leads to a series of operational issues, such as sand plugging and pump sticking, damage to sucker rods and tubing, and even the potential burial of the produced formation by sand¹³. This phenomenon affects the normal production and economic efficiency of oil wells. A thorough analysis of the causes of sand production can be categorized into three main factors: geological factors, exploitation factors, and completion factors. Among these factors, geological factors¹⁴ (also referred to as natural factors) are intrinsic causes, primarily including the mode and strength of formation cementation, fluid properties, and structural stresses in the formation. When the cementation strength of the formation rock is weak, it is more prone to lose stability during production, leading to sand production. Additionally, the properties of the produced fluid, such as the presence of hydrates, may also exacerbate the risk of sand production. Exploitation factors¹⁵ and completion factors¹⁶ are collectively referred to as artificial factors, which are external causes leading to sand production. Exploitation factors primarily involve changes in the formation pressure drop and production pressure differential, water injection operations, steam huff and puff operations, and an increase in the water cut. These operations can potentially disrupt the original formation equilibrium, thereby inducing sand production. Particularly during high-pressure differential production, the rapid movement of fluids may further weaken the stability of the formation structure¹⁷. Completion factors primarily include the quality of cementing and perforation parameters. The quality of cementing directly affects the stability of the oil well wall. Poor quality may lead to the expansion of pores and ultimately result in sand production. Unreasonable perforation parameters, such as inappropriate selection of perforation density and diameter, can also have adverse effects on sand production¹⁸. Given the complex and diverse factors influencing sand production in oil wells, completely avoiding this issue during actual production is challenging. Therefore, it is practically impossible for any oil well to completely avoid sand production throughout its entire lifecycle¹⁹.

Oil well sand production poses specific hazards during the oil extraction process, which can be summarized into five categories. First, sand production exacerbates the wear of downhole equipment²⁰, as sand particles flowing at high speeds cause significant wear to the downhole tubing, pumps, and other devices, thereby reducing their service life. Second, oil well sand production causes structural damage to the formation structure²¹. This occurs because as sand particles are carried out, the supporting particles within the formation decrease, leading to formation collapse or a decrease in porosity, thereby affecting the well’s long-term production capacity. Third, sand production reduces production efficiency²². This is because sand particles clog the wellbore or production pipelines, reducing the flow area for fluids and resulting in decreased well production rates. Additionally, frequent sand clearance operations not only increase operational costs but also significantly reduce production efficiency. Fourth, oil well sand production can easily lead to environmental pollution²³. This is because improper handling of the discharged sand can contaminate the surrounding soil and water sources. This not only increases environmental protection costs but also may have negative impacts on the local ecosystem and residents’ lives. Fifth, sand production increases safety risks²⁴. This is because the presence of sand particles makes pressure control in the wellbore more challenging, thereby increasing the risk of accidents and endangering the safety of the well and personnel working on site.

To minimize sand production within the wellbore and avoid the aforementioned hazards, various sand control technologies have been proposed both domestically and internationally. The core of sand control lies in employing appropriate technological means to effectively prevent sand particles in the reservoir from entering the wellbore with the produced fluids. This approach safeguards the production equipment and pipelines, ensuring a stable production rate²⁵. After analyzing and categorizing existing sand control technologies, the currently available methods can be broadly classified into four main categories: mechanical sand control technologies²⁶ (such as gravel packing), chemical sand control technologies²⁷ (such as artificial wellbore lining), composite sand control technologies²⁸ (which combine two or more sand control methods), and other sand control technologies²⁹ (such as coking sand control). Despite the widespread application of these sand control technologies in numerous domestic and international oil wells, completely eliminating sand particles from entering the wellbore remains challenging³⁰. Given that the transport characteristics and critical migration velocity of sand particles are key aspects of their migration properties, conducting fundamental theoretical research, laboratory experiments, and numerical simulation analyses on sand particle migration within the wellbore is crucial for enhancing sand transport performance. This approach is also a significant pathway for optimizing existing sand control technologies.

Domestic and international scholars have conducted extensive analyses and research on sand particle migration within wellbores or particle migration in pipelines. In fundamental theoretical research on sand particle or granular migration, Durand³¹ initially proposed the basic concept of a “critical velocity” during particle migration, which is defined as the minimum flow velocity at which particles or solid phases cannot form a fixed bed in the fluid. Additionally, he established the world’s first predictive model for critical particle velocity, which laid the theoretical foundation for subsequent research. Doron et al.³² proposed a stratified flow model for solid‒liquid two-phase flow in horizontal circular pipes, which can be used to predict particle flow regimes and pressure drop variations within a pipeline. However, upon laboratory experimental verification, it was found that the model results deviated significantly from the observed phenomena. To address this issue, he improved upon the original model by separately developing three-layer solid‒liquid stratified flow models for horizontal and inclined circular pipes³³. Compared with the previous two-layer model, the new model introduces a starting velocity model for solid particles, significantly enhancing the accuracy of predictions. Dong et al.³⁴ proposed the concepts of the hydrostatic settling velocity, suspension velocity, and sand-carrying velocity for the fluid-driven sand transportation process in wellbores. Through sensitivity analysis and data fitting of these three characteristic velocities, empirical relationships between them were derived. Under the same conditions, the suspension velocity is approximately 0.8 times the hydrostatic settling velocity. When the sand-carrying velocity reaches or exceeds 3.73 times the suspension velocity, the fluid can achieve rapid sand transportation within the wellbore. Wang³⁵ conducted a theoretical analysis and laboratory verification of the flow patterns of cutting particle migration (the distribution and movement states of cuttings within the fluid during the process of fluid-borne cutting transportation) in downhole operations. The examination of various factors, such as fluid velocity, particle size, and pipeline inclination, revealed that the flow patterns associated with cutting migration can generally be classified into fixed cutting bed flow, fluctuating cutting bed flow, stacked dune flow, and dispersed cutting flow. Qu³⁶ conducted an in-depth analysis of the critical initiation migration of the surface particles of multilayer cuttings in wellbores. He identified three primary modes of cutting initiation and established mechanical equilibrium equations for these three modes under critical conditions. By calculating the shear velocity of the uppermost layer of the cutting bed and comparing it with the critical values derived from the mechanical equilibrium equations, one can determine the specific migration mode of the particles.

In the field of laboratory experimental studies on the migration characteristics of sand particles or granules, Dabirian³⁷ constructed a multiphase flow experimental platform. Using this platform, he conducted transport experiments with sand–gas–water mixtures (glass beads substituted for sand particles) and obtained critical velocities of sand particles under six different flow regimes. During the experiments, the critical flow velocity of the sand bed in the pipe increased with increasing sand particle concentration and diameter. Zhang et al.³⁸ utilized a multiphase flow simulation experimental apparatus to study the critical velocities of sand particles during the transport of gas‒water‒sand mixtures in inclined pipelines. They also further analyzed the effects of different wellbore inclinations, sand concentrations, and gas contents on sand particle migration. When the inclination angle was 90°, the sand particles exhibited rolling and jumping patterns of migration, with the critical flow velocity positively correlated with the inclination angle and sand particle diameter and negatively correlated with the gas content and sand concentration. Khan et al.³⁹ utilized an indoor experimental setup to investigate sand particle migration in horizontal circular pipelines, with a particular focus on analyzing the effects of the sand layer height, pipeline diameter, and flow rate on the critical flow velocity of sand particles. The study ultimately revealed that the critical flow velocity of sand particles increases gradually with increasing sand layer height and pipeline diameter. Zeng et al.⁴⁰ employed a fluid-carrying-sand simulation experimental apparatus to investigate sand particle migration in well sections with various inclinations. They identified the sensitivity factors influencing the fluid’s sand-carrying capacity, including the sand particle diameter, sand particle density, and fluid viscosity. An analysis of the experimental results revealed that the critical sand-carrying flow velocity of the fluid initially increases and then decreases with increasing wellbore inclination angle. Additionally, the wellbore is most prone to sand accumulation when the inclination angle is between 30° and 60°. Najmi et al.⁴¹ conducted experiments on the transport of gas-water‒sand three-phase mixtures in horizontal pipelines to determine the minimum flow rate required for fluid-carrying sand in the pipeline. An analysis of the experimental results revealed that the flow pattern of multiphase flow in a pipeline does not have a direct effect on the critical flow velocity. With increasing sand particle diameter, the critical velocity gradually increases, and the critical velocity of irregularly shaped sand particles is greater than that of spherical particles of the same size.

In a numerical simulation study of sand particle or grain migration characteristics, Ji et al.⁴² utilized CFD (Computational Fluid Dynamics)to establish a numerical simulation model for sand particle migration during sand washing in inclined well sections. Through this model, the effects of sand-washing fluid discharge, sand particle size, and wellbore inclination angle on sand particle sedimentation were analyzed, and orthogonal experiments were conducted on multiple influencing factors. The study ultimately revealed that sand particles could migrate smoothly in the wide annular gap, whereas in the narrow annular gap, the sand particles primarily moved in the forms of tumbling, meandering, and saltation. Hu et al.⁴³ employed a coupled CFD and discrete element method (DEM) model^44,45 to conduct numerical simulations of cutting migration in the annulus of extended-reach horizontal wells and analyzed the effects of drill pipe eccentricity, drill pipe rotation speed, and cutting size on cutting transport performance. An analysis of the simulation results revealed that an increase in drill pipe eccentricity and a decrease in drill pipe rotation speed both led to a reduction in the average migration velocity of cuttings. Shao et al.⁴⁶ employed a coupled CFD and DEM computational model to conduct numerical simulation analyses of the transport of cutting particles of different shapes (flake-shaped, cubic, and spherical) in horizontal well sections. The results revealed that the migration performance was the poorest for cubic particles and the best for spherical particles. Song et al.⁴⁷ established a simulation model for the transport of cutting particles within a wellbore and, in combination with the SIMPLEC algorithm and the realizable k − ε turbulence model, investigated the effects of different drilling fluid discharge rates, cutting diameters, and inclination angles on the transport of cuttings in horizontal wells. The final findings indicated that increasing the drilling fluid discharge rate, reducing the cutting diameter, and adjusting the inclination angle could significantly increase the transport capacity of cutting particles within the wellbore. Su et al.⁴⁸ utilized CFD to conduct numerical simulation analyses of the migration and deposition behavior of microsized sand particles. On the basis of these analyses, they proposed a method for determining the critical nonsettling water velocity of microsized sand particles in complex well sections (this method can be used to assess the condition of sand deposition within the wellbore). They also derived the critical nonsettling water velocities for microsized sand particles under three different particle sizes and three different sand concentrations.

In summary, although domestic and international scholars have achieved certain research outcomes in basic theoretical research, laboratory experiments, and numerical simulation analyses of sand or particle transport characteristics, most studies have focused primarily on the analysis of sand transport modes, the characteristics of individual or several flow patterns, and the establishment of models for critical sand-carrying velocities. However, few studies have been conducted on the overall flow pattern transformation rules and other critical velocities (such as the critical velocity for sand particle initiation and the critical velocity for sand particle suspension). Moreover, comprehensive and systematic research on sand transport is relatively scarce. To enhance the sand transport capability within the wellbore and address the aforementioned issues, this paper uses an indoor sand transport simulation experimental apparatus to analyze the transport modes, flow pattern characteristics, and conversion relationships of sand particles within the wellbore systematically, as well as the various critical velocities during the sand particle transport process. Specifically, the novelty of this study lies in systematically classifying and analyzing the transport modes and flow pattern characteristics of sand particles within the wellbore and qualitatively obtaining the rules of flow pattern transformation during the sand transport process. Additionally, this study provides a detailed analysis of the three critical velocities (the critical initiation velocity, the critical suspension velocity, and the critical sand-carrying velocity of the fluid) for sand particles of different sizes (sand average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, 0.18 mm, and 0.15 mm) during their transport in inclined wellbores at various angles. Through the analysis and summary of the experimental results, the study ultimately provides the following conclusions: four types of sand transport modes within the wellbore are categorized, the flow pattern characteristics during the sand transport process (divided into eight types) and the transformation rules of flow patterns during sand transport are illustrated, and specific values for the critical velocities when sand particles of different sizes begin to move (sliding, rolling, and lifting), the critical fluid velocity for sand particle suspension, and the critical velocity for fluid-carrying sand are provided.

Introduction to the sand transport simulation experimental apparatus

To investigate the different transport modes, flow pattern characteristics, and relationships among flow patterns in the process of sand transport within a wellbore, an indoor experimental device was used to conduct simulation experiments on sand transport in wellbores (this device is capable of carrying out sand transport experiments with water, foam fluid, and other fluids). The sand transport simulation experimental apparatus and specific working principles are shown in Fig. 1. The apparatus primarily consists of a fluid circulation system, solid‒liquid‒gas phase separation and mixing system, solid particle injection and collection system, and data acquisition and control system. The entire experimental setup is capable of simulating sand transport processes in wellbores under various well sections and operating conditions by adjusting parameters such as the flow rate, fluid properties, sand particle properties, and pipeline characteristics.

Before conducting experiments using the sand transport simulation experimental apparatus, it is essential to fully understand the key parameters and adjustment ranges of the main devices within the entire experimental setup (the specific values or ranges of the key parameters are detailed in Table 1). One must subsequently prepare the sand for injection (the particle size of the sand should be screened and weighed), determine the fluid carrying the sand (if foam, the foaming agent should be stored in a dedicated container first), and carefully inspect the connections and sealing integrity of all the pipelines. To facilitate the observation of the sand transport flow pattern, we decided to use clean water as the carrier fluid for this experiment and employ transparent tubing to simulate the wellbore. Once all the preparations are complete, the indoor sand transport experiment can commence. The detailed experimental steps are as follows:

Table 1 Main equipment parameters and adjustment range of the entire experimental setup.

Fig. 1

Physical object and schematic diagram of the sand particle transport simulation experiment apparatus

• 1. 1. As shown in Fig. 1(a), initially, the valve connecting the water supply line to the fluid tank, as well as the valve connecting the fluid tank to the mixing tank, is opened, allowing water to continuously enter the fluid tank. Simultaneously, the pumping system in the fluid tank was activated to transport the water into the mixing tank.

• 2. 2. Once the water level in the mixing tank reaches the preset level, the valves at both ends of the slurry pump and the transparent pipeline are opened, allowing water to flow into the transparent pipeline. The water then passes through subsequent piping, the gas separation device, and the solid‒fluid separation vibrator screen before returning to the fluid tank. After the transparent pipeline is filled with water, the slurry pump can be turned off.

• 3. 3. Next, the preprepared sand particles are poured into the sand addition device, and the valve connecting the sand addition device to the transparent pipeline is opened, allowing the sand particles to enter the pipeline smoothly. Once all the sand particles have been injected, the sand addition device and the valve connecting it to the transparent pipeline are closed.

• 4. 4. After waiting for a period of time, the slurry pump can restart, and the flow rate of water can be controlled by adjusting the valve opening size at the front end of the slurry pump. At this time, visual observation or camera recording can be used to observe the movement and flow pattern changes of the sand particles in the pipeline.

• 5. 5. As the flow rate of water increases, the sand particles begin to migrate out of the transparent pipeline outlet. These sand particles, along with the water, pass through the solid‒fluid separation vibration screen for separation. The separated fluid is subsequently returned to the fluid tank, while the separated sand particles are transported via a conveyor belt and weighed using an electronic scale for recording.

Through repeated experiments, various migration modes, flow pattern characteristics, and transition relationships during sand transport can be obtained. This provides experimental evidence for understanding the sand transport mechanism in a wellbore, which holds significant practical importance for enhancing sand production capacity in the wellbore.

Transport mode and classification of sand in the wellbore

By utilizing the aforementioned experimental setup and procedures, it is possible to conduct research on the specific methods of sand transport in the wellbore (the limitations of this experiment and how to reproduce the observed phenomena through numerical simulations are detailed in Reference⁴⁹). To facilitate clear observation of the specific modes of sand migration in the wellbore, river sand with particle sizes ranging from 1 to 3 mm was selected. The experiment also employed a transparent pipe with an inner diameter of 70 mm as a simulated wellbore, and the inclination angle of the transparent pipe was set at 90° (the transparent pipe was placed horizontally), thereby allowing observation and recording of sand transport in the wellbore.

By precisely adjusting the valve opening size and the flow meter readings, the actual flow rate of the fluid in the wellbore can be controlled, thereby indirectly regulating the flow velocity of the sand-laden fluid. With the continuous increase in fluid velocity, the sand transport mode gradually changes. A thorough analysis revealed that the sand migration patterns in a wellbore can be specifically categorized into four types: sand particles at rest, sand particles rolling (contact transport), sand particle saltation (jumping transport), and sand particle suspension (suspended transport).

Sand particles at rest

Before the experiment commenced, the sand injection device first introduced sieved 1–3 mm sand particles into the transparent pipe. Under the influence of gravity, the sand particles in the transparent pipe settle toward the bottom of the wellbore and eventually accumulate, forming a sand bed. After a certain period of waiting, the valve can be opened. By controlling the valve opening size, the flow meter was held at 10 L/s, and then the sand transport situation in the wellbore was observed. At this time, the sand-laden fluid in the wellbore hardly carries the sand particles along with it during its flow, and all the sand particles remain stationary, as shown in Fig. 2(a). A detailed analysis revealed that this is due primarily to the insufficient drag force, lift force, buoyancy, and corresponding moments at the support points experienced by the sand particles in the wellbore, which are not capable of overcoming the moments of gravity, friction, and their corresponding support points, thereby maintaining these sand particles in a stationary state. Figure 2(b) shows a schematic illustration of the stationary state of the sand particles within the wellbore when the fluid carries sand.

Fig. 2

Experimental and schematic diagrams of sand particles at rest when the fluid carries sand

Sand particles in contact transport

By further increasing the valve opening, maintaining the flow meter at 20 L/s, and continuing to observe the sand particle movement in the wellbore, the sand particles at the left end of the sand bed began to move along with the fluid. However, during movement, the sand particles remained in contact with the surface of the sand bed and hardly left it for migration. Upon observation, the sand particles primarily moved in a rolling manner, as shown in Fig. 3(a). A detailed analysis revealed that this is primarily due to the relatively low flow velocity in the wellbore at this stage. The sand particles on the surface of the sand bed start to move under the influence of the fluid drag force, lift force, buoyancy, etc. The dynamic moment acting on the sand particles is greater than the resistive moment, causing the sand particles to move in a rolling manner. As the flow velocity is not very high, the sand particles remain in contact with the surface of the sand bed and move along it. Figure 3(b) shows a schematic illustration of the contact migration state of sand particles in the wellbore when fluid carries sand.

Sand particles in jumping transport

With a further increase in valve opening, the flow meter was held at 25 L/s, and the movement of the sand particles in the wellbore continued. The sand particles at the left end of the sand bed were further transported by the fluid. At this point, the sand particles jumped off the surface of the sand bed, clearly indicating that they had left the sand bed surface for movement. After a short period, the sand particles were subsequently carried by the fluid and fell back onto the right end of the sand bed. The entire state of sand particle movement is shown in Fig. 4(a). A detailed analysis revealed that this phenomenon occurs primarily because, with the continuous increase in fluid velocity in the wellbore, the lift force acting on the sand particles also increases. As a result, the sand particles can be carried by the fluid and jump directly off the surface of the sand bed, moving forward together with the fluid. After being transported by the fluid for a certain distance, the sand particles, owing to their own gravity and the influence of vortices in the fluid, fall back onto the surface of the sand bed at the front end. The sand particles that fall back onto the sand bed surface also exert a certain impact on nearby sand particles (the magnitude of this impact force is related to the size of the sand particles, fluid velocity, and jump height of the sand particles). If the sand particles acquire sufficient momentum, they may be propelled upward again and continue to move in the fluid, sometimes even entraining nearby sand particles, causing them to join in the jumping movement. Conversely, if the sand particles receive only a small amount of momentum, they will remain on the surface of the sand bed and cease to jump. Figure 4(b) shows a schematic illustration of the jump migration state of sand particles in the wellbore when fluid carries sand.

Sand particles in suspended transport

If the valve opening is further increased, the flow meter reading is maintained at 35 L/s, and the sand particle movement in the wellbore is continued, the thickness of the sand bed becomes very thin, and a majority of the sand particles are carried by the fluid in a suspended state for transport. The movement state of the sand particles at this point is shown in Fig. 5(a). A detailed analysis revealed that with further increases in the velocity of the fluid in the wellbore, the turbulence intensity of the fluid also increases. At this point, various sizes, orientations, and rotating vortices are generated within the wellbore, prompting most of the sand particles on the sand bed to begin performing irregular movements. When the upward movement speed of the vortices far exceeds the settling velocity of the sand particles, the sand particles can be completely carried by the fluid for transport in regions of high flow velocity. At this point, the suspension force experienced by the sand particles is already sufficient to fully counteract their own gravity. If the suspended transport of sand particles is further categorized, it can be divided into two types: uniform suspended transport and nonuniform suspended transport. Figure 5(b) shows a schematic illustration of the suspended transport state of sand particles in the wellbore when fluid carries sand.

Analysis of the flow regime characteristics of the sand transport process in a wellbore

To observe the specific flow regime characteristics of sand particle transport within a wellbore more precisely and clearly, the experiment utilized sand particles with a particle size of 0.5 ~ 1 mm (these sand particles have the same density as real sand particles and are uniformly black in color, facilitating observation in clear water). Additionally, a transparent pipe with an inner diameter of 60 mm was employed as a simulated wellbore, and clear water was used as the carrier fluid for transporting the sand particles. During the experiment, as the flow rate of clear water increased, the transport characteristics of the sand particles in the wellbore gradually changed. After thorough observations and certain analyses of the specific flow regime characteristics of sand particle transport in the wellbore during the experiment, the flow patterns during sand transport could be subdivided into eight specific types: stable sand bed flow pattern, mobile sand bed flow pattern, stationary sand dune flow pattern, mobile sand dune flow pattern, dense sand line flow pattern, sparse sand line flow pattern, nonuniform sand suspension flow pattern, and uniform sand suspension flow pattern.

Stable and mobile sand bed flow patterns

Stable sand bed flow pattern: This pattern refers to the settling and accumulation of sand particles at the bottom of the wellbore under the influence of gravity after they enter it. Even if there is a certain flow rate of fluid in the wellbore, these sand particles remain stationary, forming a stable sand bed, as illustrated in Fig. 6(a).

Mobile sand bed flow pattern: This pattern occurs on the basis of the original stable sand bed, with an increase in the fluid flow rate within the wellbore, allowing the sand particles to begin contacting and moving. Overall, it appears that the entire sand bed is moving, as shown in Fig. 6(b).

Stationary and mobile sand dune flow patterns

Stationary sand dune flow pattern: This pattern is commonly observed in wellbores with significant inclination angles. Owing to the varying sizes of the sand particles, as the fluid flow rate within the wellbore increases, smaller-diameter sand particles can be carried out in the wellbore by the fluid, whereas larger-diameter sand particles remain stationary. This results in the formation of multiple stationary dune-like accumulations at the bottom of the wellbore, as illustrated in Fig. 6(c).

Mobile sand dune flow pattern: This pattern occurs on the basis of stationary sand dune flow, with an increase in the fluid flow rate in the wellbore, causing multiple clusters of sand particles to flow along with the fluid. Overall, it appears that several dunes are moving together, as shown in Fig. 6(d).

Dense and sparse sand line flow patterns

Dense sand line flow pattern: This pattern develops on the basis of mobile sand dune flow, with an increase in the fluid flow rate in the wellbore, causing multiple dune-like accumulations to gradually transform into denser sand lines, which continue to move along with the fluid, as illustrated in Fig. 6(e).

Sparse sand line flow pattern: This pattern further evolves from dense sand line flow. In the dense sand line flow pattern, smaller diameter sand particles can be directly carried out in the wellbore, leaving behind sand particles that can still move but are significantly sparser than the original dense sand lines, thus forming a sparse sand line flow pattern, as shown in Fig. 6(f).

Uniform and nonuniform sand particle suspension flow patterns

Uniform and nonuniform sand particle suspension flow patterns: These patterns occur on the basis of the sparse sand line flow, with an increase in the fluid flow rate in the wellbore, causing many sand particles to become suspended and migrate out of the wellbore along with the fluid. Owing to the varying diameters of the sand particles in the wellbore, the overall appearance of the suspension appears to be uneven, as depicted in Fig. 6(g). If the flow rate further increases, almost all the sand particles in the wellbore become suspended, resulting in a uniform suspension, as illustrated in Fig. 6(h).

Flow pattern transition analysis of the sand transport process in the wellbore

After conducting a theoretical analysis of sand transport in the wellbore (a detailed theoretical analysis is described in reference⁴⁹) and combining it with specific experimental results on sand transport, it was found that the two primary forces affecting sand transport in the wellbore are the weight of the sand and the drag force of the fluid. The dominant force depends primarily on the sand grain size and fluid velocity. Given a constant sand-carrying fluid discharge, larger-diameter sand particles typically exhibit rolling migration, whereas smaller-diameter particles exhibit jumping migration.

When the liquid phase flow is weak, gravity will dominate, causing the suspended sand particles to tend toward settling; when the liquid phase flow intensity is high, the drag force becomes dominant, and the generation of fluid flow vortices continuously pushes the sand particles on the sand bed surface into higher velocity flow zones, causing the sand particles to become suspended and transported. The movement patterns of sand particles within a wellbore are not independent of the flow regime; this relationship is influenced by various factors, such as the liquid flow intensity, sand particle characteristics, and engineering parameters. As the flow conditions change, the various movement patterns and flow regimes of the sand particles within the wellbore also change. The specific transformation process is illustrated in Fig. 7.

The Fig. 7 shows that as the fluid velocity within the wellbore continuously increases, the settled sand particles in the wellbore undergo a transformation from sand beds to sand dunes, then to sand lines, and eventually move as dispersed sand particles (alternatively, as the fluid velocity within the wellbore increases, the sand particle migration transitions from sand bed flow to transition flow and finally to dispersed flow for transport). Each state also includes two flow regime characteristics of sand particle migration (specifically, a stable sand bed flow regime, a mobile sand bed flow regime, a stable sand dune flow regime, a mobile sand dune flow regime, a dense sand line flow regime, a sparse sand line flow regime, a nonuniform sand particle suspension flow regime, and a uniform sand particle suspension flow regime), and the specific migration methods of the sand particles in each flow regime are different (specifically, sand particle quiescence, contact transport, saltation, and suspension). Furthermore, the modes of initiation for these sand particles are distinct (specifically, sliding initiation, rolling initiation, and lifting initiation; these modes are detailed in the subsequent sections).

Different critical flow velocity analyses in the sand transport process in the wellbore

When sand particles settle at the bottom of a wellbore, changing the fluid velocity in the wellbore is necessary to carry the sand particles out. Throughout the fluid sand-carrying process, there are three crucial fluid velocity values. The first fluid velocity is the critical velocity at which sand particles begin to move, i.e., the minimum fluid velocity at which sand particles start sliding, rolling, and lifting in the wellbore. The second fluid velocity is the critical velocity at which sand particles remain suspended in the fluid, i.e., the minimum fluid velocity at which sand particles are just suspended without movement. The third fluid velocity is the critical sand-carrying velocity of the fluid, i.e., the minimum fluid velocity at which the fluid can successfully carry sand particles out of the wellbore. These three critical velocities are key to analyzing the fluid sand-carrying process in a wellbore. Accurately determining these critical velocities during the fluid sand-carrying process can increase the sand transport capability in the wellbore and significantly reduce the risk of sand burying the producer.

Analysis of the critical fluid velocity for sand initiation transport

With increasing sand production within the wellbore, sand particles gradually accumulate at the bottom, forming a sand bed. An analysis of the critical state of force on the surface sand particles in the wellbore revealed that the primary methods of sand initiation transport are sliding, rolling, and lifting, as shown in Fig. 8. As illustrated in Fig. 8(a), the sliding initiation transport mode of the sand particles occurs when the drag force of the wellbore fluid exceeds the frictional resistance, prompting the particles to slide. Figure 8(b) shows that the rolling initiation mode occurs when the destabilizing moment around the particle’s pivot exceeds the stabilizing moment, causing the particle to roll. As depicted in Fig. 8(c), the lifting initiation mode of sand particles occurs when the lift force exceeds the buoyant weight component of the particle, leading to its lifting movement. Numerous studies have shown that the inclination angle of the wellbore directly influences the sand movement initiation mode. In low inclination sections, sand particles tend to initiate movement through lifting. In medium- or high-inclination sections, the rolling initiation mode is more common. In horizontal sections, the sliding initiation mode becomes predominant.

In the Fig. 8, G is the gravity force of the sand particle, N; $\:{F}_{f}$ is the buoyancy force exerted by the fluid on the particle, N; $\:{F}_{z}$ is the drag force exerted on the sand particle, N; $\:{\text{F}}_{\text{s}}$ is the lift force exerted on the sand particle, N; φ is the wellbore inclination angle, °; and $\:{M}_{s}$ and $\:{M}_{n}$ are both torques, m.

Through the selection of five groups of sand particles with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, and 0.18 mm for experimentation and with the experimental fluid confirmed as clean water, the inclination angle of the simulated wellbore in the experimental apparatus was adjusted to 90°. The aforementioned experimental setup was subsequently utilized to conduct group experiments. After collating the results obtained, the fluid velocities at which sand particles of different sizes begin to move in water were determined, as illustrated in Fig. 10. According to Fig. 10, as the average diameter of the sand particles increases, the fluid velocities for lift transport, rolling transport, and sliding transport of the sand particles all continue to increase. With increasing average diameter of the sand particles, the fluid velocity for lift transport clearly increases more significantly than that for rolling transport and sliding transport. As the average diameter of the sand particles decreases, the differences in fluid velocities among lift transport, rolling transport, and sliding transport also continue to decrease. When the average diameter of the sand particles remains constant, the fluid velocity for lift transport is greater than that for rolling transport and sliding transport, whereas the fluid velocity for rolling transport is greater than that for sliding transport. This further suggests that the critical fluid velocity for the initiation of sand particle movement is determined by the critical velocity during sliding transport.

By selecting sand particles with an average particle size of 0.43 mm and altering the inclination angle of the simulated borehole in the experimental setup, five sets of experiments were conducted at inclination angles of 70°, 75°, 80°, 85°, and 90°. After a certain amount of data were collated from the obtained experimental results, the influence of different borehole inclination angles on the fluid velocity at which sand particles initiate movement was determined, as shown in Fig. 10. As illustrated in Fig. 10, when the inclination angle of the borehole is constant, the fluid velocity required for sand particles to lift transport is significantly greater than that required for rolling transport and sliding transport (the fluid velocity for lifting transport is approximately 1.9 times greater than that for rolling transport). In addition, the fluid velocity required for rolling transport is greater than that for sliding transport (the fluid velocity for rolling transport is approximately 1.6 times greater than that for sliding transport). This finding indicates that sand particles face the greatest difficulty in initiating lift transport and further suggests that in highly deviated wells, sand particles are difficult to suspend and thus easily accumulate in sand beds. The critical fluid velocity at which sand particles initiate movement is determined mainly by the fluid velocity during sliding transport. As the inclination angle of the borehole increases, the fluid velocity of the sand particles during sliding transport gradually decreases, whereas the fluid velocity for lift transport gradually increases. This is primarily due to the increasing tendency of the sand particles to slide along the borehole wall toward the bottom under the influence of gravity in the former case and the increase in the gravity component of the sand particles perpendicular to the fluid direction in the latter case.

Analysis of the critical fluid velocity for sand suspension transport

Following the critical velocity experiments of fluids with different particle sizes using the aforementioned experimental setup, it was observed that when the inclination angle of the simulated wellbore was greater than 45°, observing sand particles in the suspended state was difficult. This is primarily because during the rolling initiation of sand particles in highly inclined wellbores, the fluid begins to carry the sand particles before they can reach a suspended state. To clearly observe the suspended state of the sand particles and accurately determine the critical velocity of the fluid when the sand particles are suspended, the inclination angle range of the simulated wellbore was set to 0°~45°. By adjusting the wellbore inclination angles to 0°, 15°, 30°, and 45° and selecting five groups of sand particles with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, and 0.18 mm for experimentation, the experiments were conducted. For the convenience of observing the transport state of the sand particles, the fluid in the experiment remained water. After appropriately organizing the experimental results, the effect of the sand particle diameter on the critical velocity of the fluid when the sand suspension was transported was determined, as shown in Fig. 11. Figure 11 shows that under a constant average diameter of sand particles, as the inclination angle of the simulated wellbore increases, the critical velocity of the fluid when the sand particles are suspended also increases continuously. When the inclination angle of the simulated wellbore is constant, as the average diameter of the sand particles increases, the critical velocity of the fluid when the sand particles are suspended also increases continuously, and the increase rate of this velocity also increases with increasing average diameter of the sand particles.

A comparison of the terminal settling velocity of sand particles with different particle sizes in water with the critical velocity of the fluid when sand particles are suspended in a vertical wellbore revealed that there is a certain numerical relationship between the two. To derive the specific relationship, multiple experiments were conducted (selecting sand particles with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, 0.18 mm, and 0.15 mm for repeated experiments). When mathematical methods are used to fit the obtained experimental data, a mathematical relationship between the fluid velocity when sand particles are suspended in a vertical wellbore and the terminal settling velocity of sand particles in still water is ultimately obtained, as shown in Fig. 12. Figure 12 shows that the fluid velocity when sand particles are suspended in a vertical wellbore has a linear relationship with the terminal settling velocity of sand particles in still water (with a correlation coefficient R² equal to 0.89), and the specific mathematical expression of this relationship is as follows:

$$\:{V}_{sx}{=0.81\times\:V}_{sj}$$

(1)

where $\:{V}_{sx}$ is the critical fluid velocity when the sand particles are in suspension in the vertical wellbore, m/s, and where $\:{V}_{sj}$ is the terminal settling velocity of the sand particles in still water, m/s.

Analysis of the critical sand-carrying fluid velocity for fluid carrying sand particles

To obtain the critical sand-carrying velocities of fluids for sand particles of different sizes, experiments were conducted using five groups of sand particles with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, and 0.18 mm. The fluid for the experiments was determined to be water, and the inclination angles of the simulated wellbores in the experimental setup were set to 0°, 30°, 60°, and 90°. Using the experimental apparatus described above, a series of experiments were conducted, and the results were compiled to determine the influence of different sand particle diameters on the critical sand-carrying fluid velocity, as shown in Fig. 14. As shown in Fig. 14, under a constant inclination angle of the simulated wellbore, the critical fluid velocity for sand transport increases with increasing average diameter of the sand particles, but the rate of increase in this velocity diminishes as the average diameter of the sand particles continues to increase. When the average diameter of the sand particles is constant, the critical sand-carrying velocity of the fluid is higher at a wellbore inclination angle of 60° than at 30°, higher at 30° than at 90°, and higher at 90° than at 0°. This finding indicates that the critical sand-carrying velocity of the fluid does not necessarily increase with increasing wellbore inclination angle when the average diameter of the sand particles is constant.

By selecting five groups of sand particles with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, and 0.18 mm and varying the inclination angles of the simulated wellbore in the experimental setup to 15°, 30°, 45°, 60°, and 75°, a series of experiments were conducted. After the experimental results were compiled, the influence of different wellbore inclination angles on the critical sand-carrying velocity of fluids was determined, as shown in Fig. 14. As shown in Fig. 14, under a constant inclination angle of the simulated wellbore, the critical fluid velocity for sand transport increases with increasing average diameter of the sand particles. When the average diameter of the sand particles is constant, the critical sand-carrying velocity of the fluid also increases as the inclination angle of the simulated wellbore increases from 15° to 60°. However, once the inclination angle exceeds 60° and continues to increase (from 60° to 90°), the critical sand-carrying velocity of the fluid decreases. The Fig. 14 clearly shows that the inclination angle of 60° makes it the most difficult for fluid to transport sand particles out of the wellbore, whereas an inclination angle of 15° makes it the easiest to transport sand particles out. This further illustrates that the medium- and large-scale deviation sections within the wellbore are most prone to sand particle deposition and can easily accumulate to form sand beds. A comparison of the terminal settling velocities of sand particles of different grain sizes in water with the critical sand-carrying velocities of liquids in vertical wellbores revealed that the velocity of the latter is approximately 3–4 times greater than that of the former.

Conclusion

• 6. 1. The sand transport simulation experiment apparatus was used to derive the specific characteristics of sand transport in the wellbore, which can be categorized into four modes: sand at rest, sand rolling (contact transport), sand saltation (hopping transport), and sand suspension (suspended transport). Additionally, the flow patterns during sand transport in the wellbore can be further subdivided into eight types: stable sand bed flow pattern, moving sand bed flow pattern, stationary dune flow pattern, moving dune flow pattern, dense sand line flow pattern, sparse sand line flow pattern, nonuniform sand suspension flow pattern, and uniform sand suspension flow pattern.

• 7. 2. As the fluid velocity within the wellbore continues to increase, the settled sand particles within the wellbore begin to transition from sand beds to sand dunes and subsequently to sand lines, ultimately moving as dispersed sand particles (essentially, with increasing fluid velocity in the wellbore, the sand particles transition from sand bed flow to intermediate flow and finally to dispersed flow for transport). Each state encompasses two types of sand transport flow patterns, and the specific transport modes of the sand particles vary in each flow pattern. Additionally, the mechanisms for sand initiation transport are different.

• 8. 3. An analysis of the critical flow velocities for different sand particle sizes (with average particle sizes of 0.43 mm, 0.30 mm, 0.25 mm, 0.21 mm, 0.18 mm, and 0.15 mm) during their initiation transport (sliding, rolling, and lifting) in inclined wellbores was performed. While keeping other conditions constant, the analysis revealed that the fluid velocity required for lifting transport of sand particles is greater than that for rolling and sliding transport. This further indicates that the critical flow velocity of the fluid during the initial transport of sand particles is determined by the critical velocity for sliding transport. When the inclination angle of the wellbore is large (greater than or equal to 70°), the fluid velocity required for lifting transport is approximately 1.9 times greater than that for rolling transport, and the fluid velocity required for rolling transport is approximately 1.6 times greater than that for sliding transport.

• 9. 4. After analyzing the critical flow velocities for sand particles of different sizes when suspended in inclined wellbores, the findings indicated that when the inclination angle of the simulated wellbore exceeds 45°, observing sand particles in a suspended state becomes difficult. By comparing the terminal settling velocity of sand particles in water with the critical fluid velocity when sand particles are suspended in vertical wellbores, a linear relationship was revealed between the two, and the value of the latter is approximately 0.81 times greater than that of the former.

• 10. 5. After the critical sand-carrying velocities of fluids for sand particles of different sizes in inclined wellbores were analyzed, the results indicated that as the inclination angle of the simulated wellbore increased (from 15° to 60°), the critical sand-carrying velocity of the fluid also increased. However, once the inclination angle exceeds 60° and continues to increase (from 60° to 90°), the critical sand-carrying velocity of the fluid decreases. This further indicates that the inclination angle of approximately 60° makes it the most difficult for fluid to transport sand particles out of the wellbore. A comparison of the terminal settling velocities of sand particles of different grain sizes in water with the critical sand-carrying velocities of liquids in vertical wellbores revealed that the velocity of the latter is approximately 3–4 times greater than that of the former.

Fig. 3

Experimental and schematic diagrams of sand particles in contact transport when fluid carries sand.

Fig. 4

Experimental and schematic diagrams of sand particles in jumping transport when fluid carries sand.

Fig. 5

Experimental and schematic diagrams of sand particles in suspended transport when fluid carries sand.

Fig. 6

Experimental diagram of specific flow pattern characteristics during sand transport in a wellbore

Fig. 7

Schematic diagram of transport modes, flow pattern characteristics, and transformation laws during sand particle transport in a wellbore

Fig. 8

Schematic diagram of the different initiation transport modes of surface sand in the sand bed

Fig. 9

Effect of the sand particle diameter on the velocity of the fluid carrying sand during sand initiation transport

Fig. 10

Effects of different well inclination angles on the velocity of the fluid carrying sand during sand initiation transport

Fig. 11

Effect of the sand particle diameter on the critical velocity of fluid during sand suspension transport

Fig. 12

Relationship between the critical fluid velocity when the sand particles are in suspension in the vertical wellbore and the terminal settling velocity of the sand particles in still water

Fig. 13

Effect of the sand particle diameter on the critical sand-carrying fluid velocity

Fig. 14

Effects of different well inclination angles on the critical sand-carrying fluid velocity.