Research on the Migration and Settlement Laws of Backflow Proppants after Fracturing Tight Sandstone

Abstract

This article studies the migration and settlement laws of backflow proppants after fracturing tight sandstone. This paper proposes a fitting method based on a multi-task learning network to address the issue of interference from multiple physical parameters during the transport and settlement processes of proppants. This method can effectively handle multi-dimensional interference factors and fit the mapping logic of multiple engineering parameters to transport patterns through the continuous correction of multi-layer networks. We first introduce the characteristics of tight sandstone reservoirs and their important value in mining, as well as the status of current research on the migration and settlement laws of proppants at home and abroad. Based on this, we then deeply analyze the sedimentation rate model of proppants in tight sandstone backflow and the equilibrium height of proppants under multiple factors of interference while considering the distribution characteristics of proppants. In order to more accurately simulate the transport and settlement laws of proppants, this paper introduces a multi-task learning network. This network can comprehensively consider multi-dimensional parameters, learn the inherent laws of data through training, and achieve accurate fitting of the transport and settlement laws of proppants. This study trained and tested the model using actual production data, and the results showed that the proposed model can fit the input–output relationship well, thus effectively supporting the study of proppant transport and settlement laws.

1. Introduction

Tight sandstone reservoirs are important fields of oil and gas exploration and development with special microstructures and genetic types [1]. These reservoirs typically have smaller pores and complex throat structures, with complex and diverse geneses, including primary genesis during sedimentation and secondary genesis during diagenesis [2]. The study of tight sandstone reservoirs involves multiple aspects such as sedimentation, diagenesis, and pore evolution, and is closely related to the process of oil and gas accumulation. With the expansion of oil and gas exploration and development, the proportion of unconventional oil and gas resources, especially tight sandstone gas reservoirs, is gradually increasing. However, due to the extremely low porosity and permeability of tight sandstone reservoirs, large-scale hydraulic fracturing is required to achieve industrial production capacity. Horizontal well segmented fracturing technology has become an important means of developing tight sandstone gas reservoirs, but the introduction of foreign fluids during the fracturing process may cause water lock and other damage to the reservoir, affecting the production capacity of the gas well. Therefore, developing a reasonable post-fracturing backflow system, reducing reservoir damage, and improving fracturing effectiveness are important issues facing the development of tight sandstone gas reservoirs at present. The densification process of tight sandstone reservoirs and their coupling relationship with oil and gas accumulation are currently popular research topics. Different regions and types of tight sandstone reservoirs may exhibit different densification processes and oil and gas accumulation mechanisms. Therefore, conducting in-depth research on tight sandstone reservoirs in specific regions is of great significance for improving the success rates of oil and gas exploration and development.

2. Literature Review

Stokes placed particles in an infinitely large fluid medium to observe their settling velocity, obtained the relationship between the Reynolds number and the drag coefficient, and established Stokes’ law for Newtonian fluids under different flow states to describe the free settling law of single particles in liquids [3]. Early research on single-particle settling has gradually evolved to consider the comprehensive effects of various factors such as fluid properties, particle shape, particle size, and crack wall [4]. For example, researchers have experimentally studied the settling behavior of proppant particles in viscoelastic fluids and established corresponding settling velocity models [5]. At the same time, through numerical simulation methods, researchers simulated the convection, settlement, and other forms of movement of proppants in cracks to better predict their distribution and transport patterns [6]. These research results not only provide a theoretical basis for understanding the settling process of proppants, but also provide guidance for practical engineering applications. In the study of particle transport laws, foreign research started earlier and achieved a series of important results [7]. Researchers studied the movement mode of proppants in cracks, including settlement, convection, and lateral transport, and established corresponding mathematical models and numerical calculation methods. These studies have revealed the migration mechanisms and the distribution patterns of proppants in fractures, thus providing a basis for optimizing fracturing design and improving fracturing effectiveness [8,9]. In contrast, domestic research started relatively late, but significant progress has been made in recent years. Drawing on foreign research achievements in combination with the actual situation in China, domestic scholars have carried out large amounts of experimental research and theoretical analysis work. Researchers conducted in-depth research on the settlement and transport laws of proppants through a combination of experiments and numerical simulations [10]. For example, by analyzing the force and motion mechanisms of particles in solid–liquid two-phase flows, the settlement formula of proppant in fracturing fluid was derived, and a corresponding mathematical model was established [11]. In addition, the influence of different parameters on the settling and migration patterns of proppants was simulated through experiments, providing an important theoretical basis for optimizing fracturing processes and improving oil and gas extraction efficiency [12]. At present, some achievements have been made in the study of the transport and settlement laws of proppants both domestically and internationally, but there are still some challenges and problems, including how to more accurately describe the movement process of proppant in cracks and how to consider the influence of factors such as crack morphology and rock properties on proppant transport [13]. Lupo, M et al. compared simulations to experiments to find the values of the interfacial adhesion surface energy and the rolling friction coefficient between particles used for simulation [14]. This article proposes a method based on a multi-task learning network to fit the migration and settling patterns of backflow proppants in tight sandstone fracturing, providing a better model for subsequent fracturing backflow. In the future, with the deepening of research and technological progress, we believe that these problems will be better solved.

3. Materials and Methods

3.1. Distribution Characteristics of Backflow Proppants in Tight Sandstone

As an important oil and gas reservoir, the fracturing effect of tight sandstone is directly related to the efficiency of oil and gas resource extraction [15]. The settling speeds and distribution characteristics of proppants play crucial roles in the formation, expansion, and oil and gas migration of fractures during the fracturing process [16,17]. Therefore, based on existing practical case data, we conducted in-depth research on the distribution characteristics of backflow proppants in tight sandstone. Firstly, in the experimental results of foreign researchers, the single-particle settling velocity under unconstrained conditions is as follows:

$$V_{ss}={\displaystyle \frac{\frac{{\rho }_s-{\rho }_l}{{\rho }_l}g{d}^2}{C_1\frac{{\mu }_s}{\rho }+\sqrt{0.75C_2\frac{{\rho }_s-{\rho }_l}{{\rho }_l}g{d}^3}}}$$

(1)

$V_{ss}$—the settlement velocity under unconstrained conditions, m/s;

$g$$—$the gravitational acceleration, $\mathrm{m}/{\mathrm{s}}^2$;

${\mu }_s$—the viscosity of the fluid in the fracturing fluid, Pa·s;

$d$—the diameter of the proppant particles, m;

${\rho }_s$—the densities of the proppant, kg/m³;

${\rho }_l$—the densities of fracturing fluid, kg/m³;

$C_1$—a constant related to the properties of the supporting agent particles, C₁ = 18;

$C_2$—the asymptotic value of the drag coefficient. When the surface of the proppant particles is smooth, C₁ = 0.4. When the proppant particles are natural sand particles with an uneven surface, C₂ = 1.

Liu Y and Sharma M.M. studied the relationship between crack width and proppant settlement velocity through experimental methods and obtained the calculation formula for the proppant settlement velocity model [18]:

$$\mathrm{When}{\displaystyle \frac{d}{W}}<0.9,{\displaystyle \frac{v_{s1}}{v_{ss}}}=1-0.16{\mu }_s^{0.28}{\displaystyle \frac{d}{W}}$$

(2)

$$\mathrm{When}{\displaystyle \frac{d}{W}}\ge 0.9,{\displaystyle \frac{v_{s1}}{v_{ss}}}=8.26e^{-0.0061{\mu }_s}(1-{\displaystyle \frac{d}{W}})$$

(3)

$W$—the width of the crack, m;

$v_{s1}$—proppant settling rate, m/s;

${\mu }_s$—the viscosity of the fluid in the fracturing fluid, mPa·s.

3.2. Research on the Influencing Factors of the Sedimentation Rates of Proppants

According to the derivation of Formulas (2) and (3), we believe that the main factors affecting the settling speed of proppants are as follows:

Firstly, the concentration of proppant has a significant impact on sedimentation rate. When the concentration of proppant increases, the interaction and interference between proppant particles are enhanced, which leads to a decrease in sedimentation rate [19,20]. This decreasing trend usually manifests as being fast first and then slow; that is, as the concentration further increases, the rate of decrease in sedimentation rate will gradually slow down [21]. In addition, when the flow state parameter α of the fluid is large, the effect of proppant concentration on settling velocity will be more sensitive, and the decrease in settling velocity will be more significant [22]. Secondly, the viscosity of the fluid in the fracturing fluid can also affect the settling speed of the proppant. As the viscosity of the fluid in the fracturing fluid increases, the resistance of the proppant particles during settlement also increases, which leads to a decrease in settlement velocity [23]. This is because the increase in viscosity increases the drag force of the fluid on the proppant particles, thereby slowing down the settling process [24]. The width of the cracks and the diameter of the proppants are also important factors affecting settlement speed. When the crack width is constant, an increase in the diameter of the proppant will lead to a smaller settlement space, thereby reducing the settlement speed [25]. This is because larger-diameter proppant particles are subjected to stronger wall blocking in cracks, making it difficult to settle quickly. Similarly, when the diameter of the proppant is constant, an increase in crack width will increase the settlement space, thereby increasing the settlement speed [26]. Taking into account the influence of parameters such as proppant concentration, diameter, crack width, and flow state, it can be found that proppant concentration has the greatest impact on settlement velocity, followed by the value of flow state parameter α and the ratio of proppant diameter to crack width. The combined effect of these factors determines the settling speed of the proppant under specific conditions.

3.3. Distribution Characteristics of Support Agents for Tight Sandstone Backflow

The diameter of proppant is one of the key factors affecting the distribution of proppant in cracks. The experimental data show that as the diameter of the proppant increases, the height and length of the proppant settling layer show an increasing trend [27]. This means that large-diameter proppants can better fill cracks and improve their conductivity. Therefore, in the actual fracturing process, the diameter of the proppant should be reasonably selected based on the reservoir characteristics and construction conditions in order to achieve the ideal fracturing transformation effect [28]. The fluid density in fracturing fluid also has a significant impact on the distribution of proppants. As the fluid density increases, the height and length of the settling layer of the proppant increase, and the distribution of the proppant in the crack becomes more uniform. This is because high-density fluids have a stronger ability to carry proppants, which is beneficial for the uniform distribution of proppants in fractures. However, excessive fluid density may lead to a reduction in crack length, so it is necessary to comprehensively consider the impact of fluid density on crack length and proppant distribution and choose an appropriate fluid density. The density of proppants also has a significant impact on the distribution of proppants in cracks. Experimental data show that increasing the density of proppants can improve the distribution uniformity of proppants in cracks, especially in the middle and later stages of cracks. This is because high-density proppants have better settling performance and can more effectively fill cracks. Therefore, during the fracturing process, different densities of proppants can be selected as needed to optimize the distribution of proppants in the fractures. The fluid viscosity in fracturing fluid is another important factor affecting the distribution of proppants. The experimental results show that as the fluid viscosity increases, the height and length of the settling layer of the proppant decrease, and the distribution of the proppant in the crack becomes uneven. This is because high-viscosity fluids have a strong carrying effect on proppants, which hinders the migration of proppants in fractures. In addition, high-viscosity fluids may also reduce crack length, which is not conducive to crack propagation. Therefore, during the fracturing process, low-viscosity fluids should be selected as much as possible to promote the uniform distribution of proppants in the fractures and increase the length of the fractures.

3.4. Mechanism of Free Settling of Proppants under Multi-Fluid Conditions

The distribution characteristics of proppants in tight sandstone fracturing are directly reflected in the fracturing quality and have an impact on the conductivity of the fractures [29]. It can be inferred that optimizing the distribution characteristics of proppants in fractures is of great value in improving fracturing quality. The physical properties of the support are closely related to changes in fluid conditions. When the external liquid is relatively stationary, the support agent mainly settles. If there is relative movement of the proppant, it is also accompanied by migration. Here, taking Newtonian fluids and power-law fluids as examples, we analyze their motion mechanisms [30].

Free settlement of proppant particles in Newtonian fluids:

Considering the acceptance of a single proppant in a relatively stationary Newtonian fluid, it can be concluded that in the vertical direction, the Earth’s gravity (downward), liquid buoyancy (upward), and fluid resistance (upward) should be considered. Assuming the particle is a sphere, the vertical downward direction is defined as the reference direction, and the resultant force can be described as follows:

$$F_1={\displaystyle \frac{\pi }{6}}{d}^3({\rho }_s-{\rho }_l)g-{\displaystyle \frac{\pi }{4}}{d}^2{C}_d{\displaystyle \frac{v^2}{2}}{\rho }_l$$

(4)

$v$—the settling rate of a single particle.

When the resistance, buoyancy, and self-weight of the proppant particles reach equilibrium, the velocity of the particles is shown in Formula (5):

$$v_s={\left[{\displaystyle \frac{4g({\rho }_s-{\rho }_l)d}{3C_d{\rho }_l}}\right]}^{0.5}$$

(5)

The formula for calculating the drag coefficient is as follows:

$$C_d={\displaystyle \frac{4g{d}_s}{3v_p^2}}{\displaystyle \frac{{\rho }_s-{\rho }_l}{{\rho }_l}}$$

(6)

According to the Reynolds number, settlement is divided into three flow regions (Stokes region, intermediate Reynolds number region, inertia region). In different fluid regions, there are different relationships between Cd and Re (

Table 1. Relationship table between Cd and Re.

Cd	Re
${\displaystyle \frac{24}{R_e}}$	≤1
${\displaystyle \frac{18.5}{R_e^{0.6}}}$	1~500
0.44	500~200,000

).

Sedimentation of proppant particles in non-Newtonian liquids:

Fracturing sand-carrying fluid is generally a power-law fluid. If the viscosity coefficient and rheological index of a power-law fluid are K and n, respectively, then the Reynolds number’s expression is as follows:

$$N_{\mathrm{Re}}={\displaystyle \frac{{\rho }_l{d}_s{v}_s}{{\mu }_a}}$$

(7)

Different shear rates result in differences in the velocity settlement equation under laminar flow conditions.

For a Daneshy shear rate $D={\displaystyle \frac{3v_s}{d_s}}$, ${\mu }_a=K{({\displaystyle \frac{3n}{2n+1}})}^n{D}^{n-1}$, the settlement velocity equation is as follows:

$$v_s={\displaystyle \frac{2n+1}{9n}}{\left[{\displaystyle \frac{({\rho }_s-{\rho }_l)g{d}_s^{n+1}}{6K}}\right]}^{{\displaystyle \frac{1}{n}}}$$

(8)

Figure 1 shows the relevant information on the force analysis of a single particle proppant. The settlement of proppants is also affected by viscosity, sand ratio, and wall efficiency. The viscous force will increase with an increase in fracturing fluid viscosity, hindering the settling movement of the proppant and weakening its speed [31]. When reaching a certain critical value, the proppant may be in a static suspension state. Therefore, low-density or small-particle-size proppants should be selected in actual construction to ensure that they can stay in the rock crevices for a long time. After fracturing tight sandstone, the proppant settles and migrates in the form of a group, and the mathematical relationship between particle concentration and settling speed can be described as follows:

$$v_{\varphi }=v_s{(1+6.88\varphi )}^{-1}$$

(9)

$\varnothing$ represents concentration, indicating that the sedimentation rate decreases with increasing concentration.

The fluid-containing proppant settles and transfers in a bounded system, so the influence of crack roughness and width cannot be ignored; that is, the wall effect should interfere with the proppant. Equation (9) provides the expression for the wall effect:

$$\overline{v_W}=v_s\left[1-{\displaystyle \frac{9}{16}}({\displaystyle \frac{a}{l-a}}+\ln {\displaystyle \frac{a}{2l-x}})\right]$$

(10)

Among them, l is the wall width, x is the particle position, and a is the diameter of the sphere.

4. Results and Analysis

4.1. Multi-Dimensional Transport and Settlement Laws of Proppants

The fracturing fluid studied in this article is slippery water, mainly composed of clean water, crosslinking agent, corrosion inhibitor, breaker, surfactant, and drag reducer in a certain proportion. In practical engineering, the transport and settlement of proppants are affected by multiple types of engineering parameters. The first is the construction operation parameters. During on-site construction, the construction displacement, time, sand ratio, and total operation time are often determined based on the technical personnels’ references to the fracturing pump injection program table, combined with construction experience. In terms of construction displacement, it is necessary to have bidirectional interference with the initial velocity of particle groups, phased changes in the length of sand embankments (initial sand dune/length unchanged → establishment stage/accumulation dominated → extension stage/height unchanged), dynamic accumulation in the direction of crack height (linear growth, slow growth, and stable maintenance), and changes in the accumulation angle at the entrance. The second is the interference between particles and fluid properties, mainly considering the particle size, viscosity, and density of fracturing fluid and proppant. The particle sizes of proppants are mostly dominated by 20/40 mesh or 30/60 mesh. The density of the proppant is directly related to the settling speed, and using a low-density proppant system can promote the increase in effective proppant seam length. Viscosity is an important characterization of the suspended sand performance of fracturing fluids and is particularly crucial for the placement of proppants. With the continuous exploration and production of unconventional oil and gas, low viscosity fracturing fluids are increasingly being valued. Figure 2 describes the multidimensional factors that affect the transport and settlement patterns of proppants.

4.2. Exploration of the Transport and Settlement Laws of Proppant Based on Artificial Intelligence

As mentioned above, the migration and settlement laws of backflow proppant after fracturing tight sandstone are affected by various factors such as construction factors and rock conditions. Currently, development mainly relies on the experience of engineering personnel. However, in the continuous construction of digital and green oilfields, improving quality and efficiency have become important goals for reservoir extraction. Therefore, it is particularly important to clarify the transport and settlement patterns of backflow proppants. In actual production, it is often difficult to establish an accurate mathematical model for backflow proppants to determine the optimal solution. This article considers introducing a multi-task learning network (MTLN), which has the capacity for multiple inputs and outputs and can adapt to the influence of multiple physical parameters on proppants. At the same time, through iterative learning of multiple hidden layers and continuous weight correction, it can effectively fit any mathematical mapping relationship, thus avoiding the problem of difficulty in establishing an accurate mathematical model for backflow proppants.

The proposed MTLN algorithm process includes the following: (1) network topology determination. This article intends to select construction operation parameters (construction displacement, time, sand ratio, and total operation time), particle size, viscosity, and density of proppant as input layers, with a total of seven items as input layers. The input matrix is represented as $[i_1,i_2,i_3,i_4,i_5,i_6,i_7]$, the output layer is the settling velocity and equilibrium height of proppant particles, the output matrix is represented as $[o_1,o_2]$, and the hidden layer can be determined by $c=\sqrt{a+b}+q$L. a and b represent the number of nodes in the output layer and the hidden layer, respectively, and q is an adjustable constant. (2) As mentioned earlier, the influencing factors of backflow proppant and settlement law indicators contain multiple physical parameter dimensions, and their numerical differences are significant. To prevent masking low-range data changes and resulting in poor training effectiveness, the maximum normalization method $X_{norm}={\displaystyle \frac{X}{\left|X_{max}\right|}}$ is used to map the data to the [−1,1] interval. (3) Taking the production data of a tight sandstone proppant as examples, model training was organized into 1000 sets of datasets, with 70 randomly selected as the training sets and the remaining 30% as the testing sets.

Table 2. Normalized data of a certain tight sandstone proppant.

Number of Groups	i1	i2	i3	i4	i5	i6	i7	o1	o2
1	0.5	−0.9	−0.7	0.3	−0.7	0.9	−0.1	0.7	−0.9
2	−0.3	0.1	0.4	−0.5	0.2	−0.4	0.8	−0.4	0.6
3	0.1	−0.3	−0.8	0.9	−0.9	0.6	−0.7	0.9	−0.3
4	−0.8	0.7	0.2	−0.8	0.5	−0.2	0.4	−0.6	0.7
5	0.1	−0.6	−0.6	0.6	−0.3	0.3	−0.5	0.2	−0.5
6	−0.9	0.8	0.9	−0.2	0.8	−0.8	0.2	−0.8	0.2
7	0.6	−0.5	−0.3	0.7	−0.6	0.7	−0.3	0.3	−0.8
8	−0.5	0.4	0.5	−0.1	0.1	−0.5	0.9	−0.1	0.1
9	0.2	−0.2	−0.1	0.4	−0.4	0.1	−0.6	0.5	−0.4
10	−0.7	0.6	0.7	−0.6	0.7	−0.6	0.1	−0.3	0.8
11	0.4	−0.7	0.6	0.8	−0.4	0.8	0.8	0.7	−0.9
12	−0.7	0.3	−0.8	−0.7	0.9	−0.6	−0.6	−0.4	0.6
13	0.9	−0.5	0.4	0.6	−0.7	0.4	0.4	0.8	−0.3
14	−0.2	0.8	−0.5	−0.4	0.3	−0.7	−0.7	−0.2	0.7
15	0.6	−0.2	0.2	−0.8	−0.5	0.2	0.2	0.9	−0.5
16	−0.3	0.9	−0.9	0.2	0.8	−0.9	−0.9	−0.5	0.2
17	0.8	−0.4	0.1	0.2	−0.2	0.1	0.1	0.6	−0.8
18	−0.5	0.6	−0.3	0.5	0.6	−0.3	−0.3	−0.3	0.1
19	0.1	−0.1	0.7	−0.6	−0.1	0.5	0.5	0.1	−0.4
20	−0.8	0.7	−0.6	0.1	0.7	−0.8	−0.8	−0.6	0.8

presents 20 sets of data, with the first 70% being the training sets and the rest being the testing sets.

Figure 3 shows the training fitting effect of the proposed MTLN algorithm. It can be seen from the figure that the proposed model can fit the input–output relationship well, effectively supporting the study of proppant transport and settlement laws.

The migration and settlement laws of backflow proppant after tight sandstone fracturing are important issues in unconventional oil and gas extraction, which is of great value for production planning and development plans. For a long time, the extraction of long-term gas reservoirs has mostly relied on the experience and judgment of engineering and technical personnel. The core of this method lies in the accumulation and effective judgment of technical personnel, which has significant engineering value in the development of conventional gas reservoirs. However, as oil and gas extraction gradually enters unconventional gas reservoirs, the engineering physical environment has undergone significant changes, and the drawbacks of this method, such as low accuracy in judgment and difficulty in replication and promotion, have gradually become apparent. However, due to the excessive interference factors affecting the transport and settlement of proppants, it is particularly difficult to establish an accurate mathematical model to simulate the effects. Therefore, this paper introduces a multi-task learning network that considers multi-dimensional interference factors.

5. Conclusions

This article focuses on the migration and settlement of backflow proppant after fracturing of dense sandstone, delves into the interference mechanisms of multiple physical parameters, introduces a multi-task learning network to achieve multi-parameter and nonlinear physical relationship fitting methods, and verifies its effectiveness through actual operating data. The paper first introduces the characteristics of tight sandstone reservoirs and the important value of their extraction. It extensively investigates the scientific and technological advances in the migration and settlement laws of proppants at home and abroad, explores the sedimentation rate model of proppants in tight sandstone backflow, and analyzes the equilibrium height of proppants under multi-factor interference. At the same time, its distribution characteristics are also considered. Finally, based on the analysis of the free settling mechanism of the proppant under multi-fluid conditions and multi-dimensional parameters, a multi-task learning network is introduced to achieve the fitting of multiple physical parameters to the law, and its feasibility is verified with actual production data.

In summary, this article proposes a multi task learning network-based method for fitting the migration and settlement patterns of backflow proppants in tight sandstone fracturing. This method has the ability to handle multi-dimensional interference factors, and its effectiveness has been verified through actual production data. The research results of this article are of great significance for improving the efficiency and accuracy of unconventional oil and gas extraction, and they provide useful reference and inspiration for future oil and gas extraction work.