For gravel-packing sand control wells, after being put into production, the formation sand tends to invade in the packed gravel zone, which leads to gravel-packing permeability decrease, production rate drop and sand control validity period shorten dramatically^[1,2]. Previous studies show that the process of sand retention is essentially the process of gradual plugging of the sand retention media by formation sand carried by fluid. During this process, the permeability of the sand-blocking media decreases gradually and reaches the final equilibrium, which causes well productivity loss after sand control job^[3,4,5,6,7]. The plugging mechanism of gravel-packing porous media is related not only to gravel-sand median size ratio (GSR), but also the viscosity and flow velocity of fluid^[8,9,10,11]. In recent years, the study of gravel plugging mechanism mainly focuses on the gravel size optimization and the influence of GSR on sand retention effect^{[8,9,10,11,12,13,14,15,16,17,18,19]}. In the early stage, Saucier^[12] put forward a typical gravel size design method considering of the influence of GSR through experimental study. Through numerical simulation and physical modeling, other researchers^{[8,9,10,11,12,13,14,15,16]} investigated the influence of GSR, sand uniformity coefficient, clay content, fluid viscosity, flow velocity and gravel-packing compactness on gravel plugging performance. But almost all of these studies are qualitative analysis of single factor on gravel plugging. Through gravel-packing plugging simulation experiments, DONG^[18] worked out an quantitative formula to describe the relationship between packed gravel plugging degree and production time, but didn’t examine the joint effect of flow velocity and viscosity of fluid. Other researchers^{[19,20,21,22,23]} optimized gravel size according to the sand-gravel size matching evaluation, but didn’t consider the flow factors of fluid either. In a word, the previous studies didn’t consider well producing conditions, such as oil viscosity, fluid producing intensity and flow velocity seriously, or only analyzed the influence of single factors on gravel plugging qualitatively.

As the mechanism and quantitative law of sand invading gravel is still not clear, there is still no direct basis and effective method reported to optimize the production rate for sand controlled wells put into production. In this work, a series of gravel-pack plugging modeling experiments were conducted with different sizes of formation sand, fluid viscosities and flow velocities. Based on the analysis of experiment results, the fluid viscosity-velocity index is put forward to characterize the influence mechanism and law of fluid properties and flow condition on gravel-pack plugging. On this basis, a systematic production optimization method for gravel packed well is developed, which involves the method to determine the maximum production rate and a multi-step promotion method to achieve the maximum production rate.

After the gravel-packing job, the bridge structure of screen pipe, gravel pack and formation sand is formed as shown in Fig. 1a. When the well is put into production, the sand produced from formation carried by fluid invades into the gravel, forming a sand-gravel mixed zone (Fig. 1b), which causes the permeability of gravel zone and well productivity to drop sharply. The drop degree of permeability and productivity depend on the invasion depth and permeability of the mixed zone^[2,9-10].

According to Fig. 1b, the process of sand invading into gravel is essentially the sand particles carried by fluid migrating and plugging in the throat space of the gravel porous media. The invading depth and volume of sand are closely related to GSR firstly, and to density, viscosity and flow velocity of fluid secondly.

An experiment apparatus was set up to simulate the process of sand retention and plugging in the gravel pack, which included a fluid tank, a pump, a sand mixer, a sand collector, main radial and linear displacement containers, a data acquisition system and a control system (Fig. 2). The linear displacement containers used in this experiment consists of a series of transparent cylinder joints with different diameters. The gravel can be packed into the cylinder joints freely to simulate the gravel-packing media.

During the experiment, formation sand of a certain proportion was mixed into the fluid through the sand mixer. The sand-fluid mixture flew through the packed gravel, modeling the invasion and plugging of formation sand particles of the gravel zone. As this experiment study focused on modeling the impact, invading and plugging of formation produced sand to packed gravel, the sand production from formation was simulated directly by the incoming flow of sand carried by fluid. The fluid viscosity, flow rate, sand size, sand content could be manually controlled. The amount and size of sand passing through gravel could be measured by the sand collector. The flow rate of fluid, differential pressure across the gravel layer and the packed geometric parameters were measured to work out the variation of permeability with time, and the invasion depth of sand into the gravel could be observed in real time. The gravel packing permeability under one-direction flow can be calculated based on experiment data with the formula below:

The fluid used in the experiments was prepared with clear water with viscosity of 1.0 mPa•s and guanidine gum solution with viscosity of 1-20 mPa•s. The formation sand used in the experiment was prepared with quartz sand with different sizes according to the size distribution curves of two typical sand samples from Shengli Oilfield. The two sand samples were 0.10 mm and 0.16 mm in median size, and 15% and 25% in clay content. The clay was prepared by mixing kaolinite, illite and montmorillonite at the ratio of 1:3:1. The solid packing materials were quartz sand and ceramsite of 0.6-1.2 mm in size. The GSR of them with formation sand ranged from 5.4 to 8.6, basically representing the common GSR range in oilfields. The detailed parameters of sand and solid packing materials used are listed in Table 1 and Table 2. It should be noted that the median size difference between the two types of solid packing materials is caused by the prime material sorting and processing. Moreover, this study ignored the sand retention and plugging difference related to the slight difference of the materials properties, and put emphasis on the influence of flow parameters on plugging performance.

S1 sand, T1 ceramsite and clean water with viscosity of 1.0 mPa•s were used to do sand retention experiment in the 2# linear displacement container shown in Fig. 2. The inside diameter of the cylinder container was 50 mm and the packed gravel length was 150 mm. The displacement time was about 46 min, and two flow rates were 0.7 m³/h and 1.2 m³/h. During the experiment, the flow rate and differential pressure across the packed gravel were measured and recorded, and the permeability variation curves of the two experiments could be calculated as shown in Fig. 3.

It can be seen from Fig. 3 that the tested gravel pack permeability gradually decreases from initial value and then reaches the final stable state, which is in accordance to the three stages of plug beginning, aggravating and balancing as mentioned in the reference [23]. After the plugging reaches equilibrium state, the mixed zone depth and permeability hardly change any more. In the experiments, the initial permeability of T1 ceramsite was about 335 μm², and the final gravel plugging permeability decreased to 45 μm² and 275 μm² at the flow rate of 1.2 m³/h and 0.7 m³/h respectively. The corresponding permeability ratio of them are 13.4% and 82.1%, which seem wide in difference. The results indicate that displacement rate (flow velocity) has considerable influence on gravel plugging performance. Under the same conditions, the higher the flow rate, the more serious the plugging of the sand to gravel will be.

After each experiment, the packed gravel sample was carefully removed out of the container and the mixed zone pattern formed by sand invasion was observed. The photographs of two mixed zones formed at two flow rates are shown in Fig. 4. The micro images of gravel pack and the mixed zone are shown in Fig. 5. Because of the distribution law of formation sand size and the heterogeneity of throats formed by gravel particles, even with the reasonable GSR, the invasion of sand into gravel zone is still unavoidable. Part of the sand grains with smaller size tend to enter the gravel throats with random size^[21], forming a sand-gravel mixed zone.

The final gravel plugging degree is characterized by the final plugging permeability (the apparent permeability of the whole gravel sample) and permeability ratio (the ratio of final plugging permeability to the initial permeability of clean gravel). The final plugging degree is mainly related to the sand invasion depth and permeability of gravel zone. In this case, the mixed zone depth formed at the flow rate of 1.2 m³/h is about 1.2 cm (Fig. 4a). Based on the total gravel length, initial gravel permeability and final plugging permeability, the real permeability of the mixed zone calculated is about 4.2 μm². In comparison, the mixed zone depth formed at the flow rate of 0.7 m³/h is about 4.2 cm (Fig. 4b) and the real permeability of the mixed zone calculated is about 192.1 μm².

It should be noted that, although the higher flow rate of 1.2 m³/h leads to lower final plugging permeability, but much shorter mixed zone depth (1.2 cm) than the lower flow rate of 0.7 m³/h. The reason is that under the same sand content of fluid, the higher flow rate indicates much quicker rate of coming sand particles, but due to the complicated throat structure of the packed gravel, the migrating velocity of sand in the throat is very low, so there is not enough time for the invaded sand to migrate deeply into the gravel throat, and the sand pile up on the gravel zone surface or shortly inside the gravel zone, blocking the channels for subsequent sand invasion, and forming the mixed zone with shorter length but lower permeability with a clear interface between the plugging and clean gravel finally (Fig. 5b).

In order to investigate the migration of sand inside the gravel, the packed gravel section is separated into three parts, A, B and C along the incoming direction of fluid as shown in Fig. 6. According to the analysis of experiments at different flow rates, the particle size median value distribution of sand-gravel mixture in the different sections were worked out as shown in Fig. 6, too.

In Fig. 6, section A is the location easy to be impacted and plugged by fluid and sand. As the displacement flow rate increasing, the median size of sand-gravel mixture in section A decreases obviously, which indicates the increasing amount of the invading sand in it. In contrast, the median size of mixture in section B decreases slightly, which indicates that only a small amount of sand reaches the middle of the gravel section. Correspondingly, the median size of sand-gravel mixture in section C almost hardly changed, showing hardly any sand reached the end of the gravel cylinder, in other words, the gravel zone functions well in blocking sand.

In order to study the influence of flow parameters on gravel plugging degree, a series of experiments were conducted with T1 ceramsite, S2 sand with clay content of 25%, and the displacing fluid with a viscosity of 1.0 mPa•s at different flow rates. The final plugging permeability (or permeability ratio), mixed zone permeability and depth of mixed zone at different flow rates are shown in Fig. 7.

Fig. 7 illustrates the influence law of flow rate (velocity) on gravel plugging degree. Under the same condition, the higher the flow rate, the lower the mixed zone permeability, and the lower the final plugging permeability and permeability ratio will be. When the flow rate increased from 0.7 m³/h to 1.0 m³/h, the permeability ratio of the gravel zone reduced from 0.818 to 0.154, that is 81.2%. It is worth noting that with the increase of displacement rate, the final depth (length) of the mixed zone gets shorter. The reason has been explained before.

A group of experiments were performed with T1 ceramsite and S2 sand with a clay content of 25% at the flow rate of 0.8 m³/h and fluid viscosities of 1, 5, 10, 15 and 20 mPa•s respectively to find out the effect of fluid viscosity on plugging degree. The results are shown in Fig. 8.

It can be seen from Fig. 8 that when the other flow parameters are the same, the higher the fluid viscosity, the lower the mixed zone permeability, and the lower the final plugging permeability and permeability ratio will be. Moreover, with the increase of fluid viscosity, the final depth of mixed zone decreases, the reason is similar to the variation of plugging degree caused by flow rate (velocity) mentioned above.

Another group of experiments were performed to explore the influence of fluid viscosity and flow velocity on gravel plugging degree further. These experiments were conducted with S1 sand and G1 quartz sand at different fluid viscosities and flow rates. The results are shown in Fig. 9. It can be seen that both flow rate (velocity) and fluid viscosity have directly effect on gravel plugging degree. The higher the flow rate and viscosity of the fluid, the stronger the sand-carrying capacity of the fluid will be, thus more sand particles will be carried into the gravel, making the final plugging permeability lower.

From the comparison of the experiment results of Fig. 7 to Fig. 9, it can be seen that flow rate and viscosity of fluid have similar influence on plugging degree.

Studies^[24,25] have proven that, for liquid-solid two phase flow, the solid particle carrying ability of fluid has approximate positive correlation with fluid viscosity and relative velocity between the fluid and solid. Due to the broad range of viscosity of produced fluid from numerous oil wells, it is very difficult to investigate the plugging degree of all the range of fluid viscosity. In order to characterize the influence of fluid viscosity and flow velocity on plugging degree, the “viscosity-velocity index” (v-v index) is put forward to represent the flowing conditions to avoid the uncertainty of single factor. The v-v index is defined as the product of flow velocity and viscosity of fluid:

To prove the joint influence of flow velocity and viscosity on gravel plugging and the rationality of characterizing flow conditions with v-v index, two groups of experiments were done as shown in Table 3. All of the experiments used S1 sand and G1 quartz sand. The v-v index of 3.33×10^-4 N/m and 10.0×10^-4 N/m were preset respectively to group A and Group B. In every group, the same v-v index was obtained by different combinations of fluid viscosity and flow velocity with the same product value of fluid viscosity and flow velocity. These groups of experiments were arranged to find out the gravel plugging degree variation law under different combinations of fluid viscosities and flow velocities, but with the same or close v-v index value.

The two groups of experiments were carried out according to the combinations in Table 3, and the mixed zone depth and gravel plugging permeability curves obtained are shown in Fig. 10. It can be seen that regardless of group A or B, the experiments with same or close v-v index tend to achieve the same or close final gravel plugging permeability with fluctuation range less than 15%, although they have different combinations of viscosity and flow velocity. Fig. 10b also illustrates the similar law of mixed zone depth with v-v index, and the variation range of mixed zone depth is less than 18%. These results indicate that although related to fluid viscosity and flow rate, the final gravel plugging degree has a better correlation with v-v index. For different combinations of viscosities and velocities, if the v-v index is the same or close, the final plugging degree would be the same or close, too. This means the v-v index can be used to characterize the comprehensive conditions of fluid flow very intuitively, and is the key factor affecting the gravel plugging degree.

To investigate further the relationship of gravel plugging degree and v-v index, 60 groups of experiments were done at the v-v index range of (1.0-17.0)×10^-4 N/m by arranging different fluid viscosities and flow velocities. The flow rates used varied from 0.7 m³/h to 1.2 m³/h (corresponding flow velocity from 0.099 m/s to 0.170 m/s) and fluid viscosities from 1 mPa•s to 10 mPa•s. The sand and solid material used were S1 sand and T1 ceramsite respectively. The relationships of final gravel plugging permeability and permeability ratio with v-v index are shown in Fig. 11.

In general, the larger the v-v index, the lower the final gravel plugging permeability will be. But with the increase of v-v index, the final gravel plugging permeability follows different variation pattern at three stages as shown in Fig. 11. (1) Stage Ⅰ: as v-v index increase to 1.7×10^-4 N/m, gravel permeability decreases rapidly from the initial value of 400 μm² to about 70 μm², the corresponding permeability ratio from 0.8 to 0.5. (2) Stage Ⅱ with v-v index changing from 1.7× 10^-4 N/m to 4.5×10^-4 N/m, with the increase of v-v index, the decrease rates of final gravel permeability and permeability ratio were much slower than stage Ⅰ, the gravel permeability decreased from 70 μm² to 40 μm², and the permeability ratio from 0.5 to 0.3. (3) Stage Ⅲ with v-v index exceeding 4.5×10^-4 N/m, gravel permeability and permeability ratio decrease very slowly or keeps stable, with an average of about 28 μm² and 0.105, respectively.

The fluid v-v index influences directly the final gravel plugging degree. According to the equivalence principle (the v-v index of actual well equals to that of experimental test), the proper working production could be determined, and the production scheme of gravel-packing well can be optimized by v-v index.

It can be seen from Fig. 11 that the higher the fluid v-v index, the lower the final gravel plugging permeability is. For a given oil well, the viscosity of produced fluid remains basically the same, so the controllable factor decides the v-v index is flow rate, in other words, the production rate of the well. By controlling the production rate properly, the final gravel plugging permeability will be higher, but this is conflict with seeking high production. To solve this problem, a new production optimization method, that is multi-step production enhancing method, which can not only control the reduction of gravel plugging permeability, but also meet the requirement of desired production rate, has been proposed.

To verify the validity of the above-mentioned method, S1 sand, T1 ceramsite and displacing fluid with a viscosity of 1.0 mPa•s were used to do three experiments at different combinations of flow rates (D, E and F) with the maximum flow rate of 1.2 m³/h. The curves of gravel permeability with displacement time in the three experiments are shown in Fig. 12.

In test D, the flow rate was kept as the target maximum value of 1.2 m³/h from beginning to end in one-step. Test E used two-step flow rate displacement. During the period from 0 to t₂ (about 1360 s), the flow rate was 0.8 m³/h, and the flow rate was increased to the target maximum value of 1.2 m³/h after t₂. In Test F, the flow rate changed in three steps. The flow rate was set at 0.8 m³/h from 0 to t₁ (about 750 s), 1.0 m³/h from t₁ to t₃ (about 2080 s), and 1.2 m³/h from t₃ to the end of displacement.

The maximum flow rates of the three tests were all 1.2 m³/h. But, just due to different combinations of flow rates, the gravel permeability variation during the displacement and the final gravel permeability were different widely. (1) In the test D taking one-step maximum flow rate, the gravel permeability decreased quickly at the early stage, and the final gravel plugging permeability was the lowest of about 33 μm². (2) In the test E taking two-step flow rate, a lower flow rate was used to displace for a period and then the flow rate was increased to the target maximum flow rate, the gravel permeability was kept higher during displacement, and the final gravel plugging permeability was higher than that of test D. (3) In test F with three-step flow rate, the gravel permeability during displacement and the final gravel plugging permeability are higher than those of the other two experiments.

Fig. 13 shows the gravel permeability in every displacement stage of the three flow-rate combinations. (1) During the startup of stage 1, the initial gravel permeabilities of three tests were similar to each other. As the displacement continued, gravel permeability decreased gradually. The higher the flow rate, the greater the permeability reduction was. (2) At the stage 2, the test D had higher flow rate, and higher gravel permeability drop than test E (0.8 m³/h) and F (1.0 m³/h), in general, the higher the flow rate, the greater the reduction of permeability. (3) In stage 3, test F had the lowest flow rate, so the gravel permeability dropped the least and kept the highest. (4) At the end of stage 4, although all of the three tests D, E and F reached the target maximum flow rate, they ended in widely different final gravel plugging permeabilities of 33 μm², 46 μm² and 60 μm² respectively. The final permeabilities of two-step displacement (test E) and three-step displacement (test F) were 39% and 82% higher than one-step displacement (test D) respectively. The permeability of three-step is about 30% higher than that of two-step.

The packed gravel zone is the inflow channel of oil and gas into the gravel-packing well. The gravel permeability determines the well productivity directly. Beside the enhancing manner of production rate, the target production rate is another crucial factor determining the final gravel permeability and well productivity. The above experiments and results give us an idea for production optimization of gravel-packing wells. For the oil well with proration production determined, based on the principle that the v-v index of actual well equals that of experiment test, the production rate of the well can be enhanced by steps to reach the proration production, this way, the final gravel plugging permeability would be higher than producing at the proration production directly, and the final production would be higher. The specific method is: The production is divided into 3 levels based on the proration production (40%-50%, 70%-80% and 100% of the proration production). Then the production rate can be increased by three steps one by one and the production period of the first two steps is 12-24 h. In consideration of field operation complexity, the production rate increase pattern of more than three steps isn’t discussed here.

Based on the relationships between final gravel plugging permeability and permeability ratio and v-v index, and the minimum gravel permeability required by gravel packing wells, the critical v-v index can be worked out, and then the production of well can be calculated by using fluid viscosity, and this value is the optimum proration production which can avoid excessive plugging of the gravel zone.

The specific method of optimizing production is illustrated here with a vertical well with a 10 m thick oil layer. Assuming the perforation density is 36 holes/m and size of perforation holes is 14 mm. The relationships of v-v index and liquid production rate at different fluid viscosities can be calculated by using flow velocity of the perforation hole and production rate (Fig. 14). If the minimum gravel plugging permeability is 40 μm², the corresponding maximum v-v index is calculated at about 5.0×10^-4 N/m according to Fig. 11. Supposing the formation fluid viscosity is 30 mPa•s, the final maximum liquid production rate can be worked out at 100 m³/d based on Fig. 14. This production rate is the optimum rate which can avoid excessive plugging. According to this optimum production rate, the proper working scheme of the well is obtained by increasing the production in three steps.

The depth and gravel plugging permeability of sand-gravel mixed zone and the final gravel plugging permeability are in negative correlation with flow rate and viscosity of the sand-carrying fluid. The higher the flow rate and viscosity of the fluid, the lower the mixed zone permeability and the final gravel plugging permeability, and the smaller the mixed zone depth will be.

Flow rate and viscosity of fluid are the key parameters affecting the gravel plugging degree. The fluid viscosity-velocity index can characterize the fluid flowing properties well and make the analysis of gravel plugging mechanism easier. For different combinations of fluid viscosity and flow velocity, if the v-v index is the same or close, the final gravel plugging permeability would be the same or close. With the increase of v-v index, the gravel plugging permeability decreases rapidly at first, then slowly, till stabilizes finally.

The production optimization method for sand control well combines the optimum production rate and a multi-step production rate increase manner, which can reduce the damage to gravel permeability by sand invasion and increase well productivity.

Nomenclature

A—flow path area, m²;

K_s—gravel plugging permeability, m²;

L_s—thickness of gravel-packing zone, m;

Q—tested flow rate, m³/s;

v—flow velocity, m/s;

β—fluid viscosity-velocity index (v-v index), N/m;

Δp—differential pressure across gravel-packing zone, Pa;

μ—fluid viscosity, Pa•s.