Research on the Law of Crack Propagation in Oil Well Fracturing Process (2024)

1. Introduction

The Z oilfield is located in the eastern part of the Ordos Basin in Shaanxi Province, China. It belongs to a type of low-porosity and low-permeability reservoir. Hydraulic fracturing technology is one of the key methods to increase oil and gas production and recovery rates [1,2]. In recent years, there has been an increasing occurrence of water flooding in oil wells after hydraulic fracturing, which seriously affects oilfield production and has a significant impact on the economic benefits of oilfields and the environmental sustainability of oilfield areas.

Reasonably controlling the expansion of hydraulic fractures in oil wells is one of the key projects for well reformation measures. Yang et al. [3], Li et al. [4], and Varahanaresh Sesetty et al. [5] have conducted research on fracturing technology and studied the impact of fracturing technology on oil well production through field experiments. Mao et al. [6] conducted laboratory hydraulic fracturing experiments on pre-fractured granite samples, emphasizing the potential of hydraulic fracturing in controlling the expansion path of fractures. Shu et al. [7] introduced a method for qualitatively predicting the initiation direction of hydraulic fractures based on hyperbolic failure envelopes, aiming to improve the efficiency of fracturing in existing fractured rock masses. Zhao et al. [8] conducted hydraulic fracturing experiments on granite specimens with dual defects, highlighting the evolution of fractures during the fracturing process. Wang et al. [9] studied the extension of hydraulic fracturing cracks induced during supercritical CO₂ sequestration. Zhou et al. [10] optimized fracturing technology through laboratory experiments, improving development strategies and design methods, among others. Montgomery et al. [11] emphasized the spatial variability of tight oil well productivity and the influence of technology on the main oil well productivity levels. In shale gas and tight oil basins, B. Sobhaniaragh et al. [12] focused on the numerical simulation of non-planar hydraulic fracturing in multilateral horizontal wells, studying the influence of stress shadows on fracture spacing and the comparison of horizontal stress. Guo et al. [13] discussed the numerical simulation of single radial borehole guided hydraulic fracture propagation, providing a theoretical basis for directed hydraulic fracture propagation. Gunarathna et al. [14] analyzed the effect of effective vertical stress on the initiation pressure of hydraulic fractures in shale and engineering geothermal systems exploration. Li et al. [15] introduced pulsed plasma rock fracturing as a new technology to improve rock fracturing efficiency and recovery rates. Lv et al. [16] conducted a study which found that the proportion of shear damage is relatively low during the multi-cluster fracturing process, and it primarily occurs in the interlayer fractures activated by the outer fractures. Chang et al. [17] suggested that reducing the cluster spacing in horizontal fracturing can significantly increase the extension range of hydraulic fractures, but the phenomenon of uneven fracture propagation is severe. In recent years, research on fracturing has mainly focused on the optimization of fracturing design and fracturing technology. However, due to the complex development environment of low-permeability oilfields, studying the evolution of fractures during the fracturing process and aiming to improve the success rate of fracturing construction is a key focus of Z oilfield development.

The main reason for the failure of hydraulic fracturing in oilfields is often due to interlayer damage caused by the expansion of fractures, leading to interlayer water intrusion and ultimately resulting in post-fracturing water flooding [18]. Therefore, researching interlayer damage during the fracturing process is a key technical aspect to enhance the effectiveness of measures. Considering the distribution patterns of rock physical properties in oilfields, statistical damage constitutive models are frequently employed in rock damage research. Lee [19] developed a micromechanical damage constitutive model for discontinuous fiber-reinforced composite materials. Gong et al. [20] introduced a cohesive crack model combined with damage to describe fatigue crack propagation along interfaces. Hu et al. [21] studied factors such as gas-containing coal porosity/crack-induced damage and established equations for effective stress and damage deformation control. Li et al. [22] proposed a rock evaluation method based on the statistical constitutive relationship of rock damage for brittleness. He et al. [23], Fu et al. [24], and Xie et al. [25] investigated shear characteristics during the damage process and established shear constitutive models. Niu et al. [26] studied the damage constitutive model of rock with microcracks under tension. Liu et al. [27] delved into the damage and fracture evolution of hydraulic fracturing in shear fractures in rock under different stress conditions, proposing strength factor evolution equations for rocks with multiple fractures. Huang et al. [28] conducted experimental research on the shear dilation and fracturing behavior of soft rocks under unloading conditions, revealing rock behavior under specific circ*mstances. Shi et al. [29] found that during the fracturing process, fracture formation is primarily controlled by the major reverse faults of the reservoir, and under the influence of these faults, fractures propagate horizontally. Hui et al. [30] constructed an unconventional fracture model (UFM), taking into account the interaction between fluid flow, hydraulic fractures, and natural fractures, and concluded that the heterogeneous and complex fracture network provides a fluid diffusion path for hydraulic fracturing-induced earthquakes. Previous studies have mainly focused on the influence of damage on fracture propagation, and the established damage constitutive models did not take into account the compaction behavior of reservoir rocks under stress during the fracturing process, as well as the pressure drop characteristics after rock failure. As a result, the constitutive equations developed by previous researchers cannot accurately reflect the behavior of rocks under actual reservoir conditions, leading to inaccurate assessments of rock strength during the fracturing process and causing water flooding due to thin separating layers being ruptured between oil and water layers.

The purpose of this study was to delve into the evolution of damage during the fracturing process and analyze the primary influencing factors of wellbore water encroachment caused by damage. By establishing a new modified damage constitutive model, a more precise prediction of rock damage propagation during the fracturing process was achieved, ultimately optimizing hydraulic fracturing design through the model to reduce water encroachment risks and enhance well production efficiency. The study found that the modified constitutive model improves accuracy by more than 3% compared to previous research. Through the establishment of reservoir cross-section models with different perforation angles and wellbore cross-section models, the study investigated the patterns of damage evolution under different conditions during fracturing. It innovatively suggests that the primary factors influencing the direction of fracturing damage evolution are the perforation angle and the longitudinal and transverse pressure ratios at the wellbore interface. In conclusion, this study’s innovative contribution is to provide references for improving the construction effectiveness of hydraulic fracturing and enhancing oil recovery rates in the Z oilfield.

2. Rock Damage Constitutive Model

2.1. Basic Concepts of Rock Damage in Oil Reservoir

The damage to interlayer rock manifests as the generation, propagation, and interconnection of micro-cracks, as well as changes in porosity and permeability. Damage primarily results from the failure of microelements. Assuming the total number of microelements is Ni, the damage D can be expressed in terms of the damaged microelements Nt, as follows:

$D = \frac{N t}{N i}$

(1)

The damage variable represents the state of rock damage, with values ranging from 0 to 1. A higher damage value indicates more severe damage, and a damage variable of 1 signifies complete rock failure leading to crack formation.

The Mohr-Coulomb strength criterion holds significant applicability in studying the strength of Z zone reservoir rock. In Z zone reservoirs, lithology tends to be uniform, natural fractures are scarce, and rock porosity is low. Utilizing the Mohr-Coulomb strength criterion effectively characterizes the strength properties of reservoir rocks. When the stress exerted on the rock exceeds the yield stress, continuous damage occurs. The strength of rock elements is represented by F, which can be expressed as follows [31]:

$F = σ_{1}^{'} - \frac{(1 + s i n θ)}{1 - s i n θ} σ_{3}^{'} - \frac{2 c_{s} c o s θ}{1 - s i n θ}$

(2)

where c_s represents the cohesion of the rock, while θ denotes the friction angle.

Assuming the strength of the rock follows a Weibull distribution allows for describing the distribution characteristics of interlayer rock strength. By adjusting the shape and scale parameters of the Weibull distribution, one can simulate the non-hom*ogeneity and randomness of rock strength, thereby explaining variations in the mechanical properties of rocks observed in experiments. The internal particle distribution within the rock is heterogeneous and includes microfractures and cracks formed during the sedimentation-diagenesis process or later-stage exploitation activities. Hence, the actual variation in interlayer rock strength is also stochastic. To represent the non-hom*ogeneity of rocks, it is assumed that the microscopic mechanical properties of rock materials conform to the probability density function of the Weibull distribution, as follows:

$P (F) = \frac{m}{F_{0}} {(\frac{F}{F_{0}})}^{m - 1} e x p [- {(\frac{F}{F_{0}})}^{m}]$

(3)

In the equation, m and F₀ are statistical distribution parameters for composite rocks, their values being correlated with the mechanical properties of the rocks. F represents the stochastic distribution variable of microelement strength. Integrating Equation (3) yields the damage variable for the rock.

$D = \frac{N t}{N i} = \int_{0}^{F} P (F) d F = 1 - e x p [- {(\frac{F}{F_{0}})}^{m}]$

(4)

According to Lemaitre’s theory of strain equivalence [32], the constitutive relationship of interlayer rock can be expressed as follows:

$σ_{i j} = σ_{i j}^{'} (1 - D)$

(5)

In the equation, σ_ij represents the stress tensor of the rock, and σ_ij′ denotes the effective stress tensor. During the fracturing process, the rock is subjected to principal stress directions, and the stress-strain characteristics of the rock before compression fracture conform to the generalized Hooke’s law. Damage occurs when the reservoir reaches its breakdown pressure. During the fracturing process, it can be assumed that the lateral confining pressure of the interlayer rock is equal, and the load is only applied axially. In triaxial stress experiments, σ₁ represents the axial pressure, while σ₂ and σ₃ represent the lateral stresses. The corresponding effective stresses are given by σ₁′, σ₂′, and σ₃′.

${σ_{1}}^{'} = E ε_{1} + 2 μ σ_{3}$

(6)

Taking into account the influence of water pressure on rock stress, it has been adjusted accordingly. The formula now includes the effect of water pressure, and the expression for effective stress is represented as follows [33]:

${σ_{i j}}^{'} = σ_{i j} - η p_{u} δ_{i j}$

(7)

In the equation, p_u is the pore fluid pressure within the layer, $η$ is the Biot coefficient, and $δ_{i j}$ is the Kronecker delta.

For tight formations, where the compressibility of the skeleton particles is much smaller than the compressibility of the particles themselves, meaning the skeleton bulk modulus is much larger than the rock particle bulk modulus, $η$ can be approximated to be equal to 1. Therefore, the formula for effective stress can be simplified to:

${σ_{i j}}^{'} = σ_{i j} - p_{u} δ_{i j}$

(8)

The stress measured in experiments represents the effective stress of the undamaged portion of the rock, taking into account the pore water pressure. According to Equations (6) and (8), the axial effective stress under the influence of pore pressure can be obtained as follows:

$σ_{1} = E ε_{1} + 2 μ σ_{3} - (2 μ - 1) p_{u}$

(9)

In order to enhance the accuracy of the constitutive model under stress-permeability coupling, correction parameters ω and Ψ are introduced to modify the constitutive model. Here, Ψ represents the post-peak correction parameter which, based on the shape of the stress-strain curve, takes on an exponential form. Consequently, the post-peak parameter Ψ is introduced, and the established statistical damage correction formula is as follows:

$D = 1 - e x p [- {(\frac{F}{F_{0}})}^{ψ m}] \cdot e x p [- {(\frac{F}{F_{0}})}^{m}]$

(10)

The experimental stress σ_1p during the compaction stage of reservoir rock can be expressed as the difference between the calculated theoretical stress σ_1t and the softening flow stress Δσ_t:

$σ_{1_{p}} = σ_{1 t} - Δ σ_{t}$

(11)

$Δ σ_{t} = σ_{1 t} (1 - ω)$

(12)

Based on the difference between the experimental curve and the theoretical curve, the correction parameter $ω$ for the compaction stage can be expressed as follows:

$ω = b + c e x p (- Δ ε / a)$

(13)

2.2. Study on Seepage Characteristics in Fracturing Process

The seepage velocity of fracturing fluid in the reservoir is expressed by Darcy’s velocity. During the fracturing process, only the seepage condition of the fracturing fluid is considered, and in the case of Z oil field, water is used as the fracturing fluid. According to Darcy’s law, the seepage velocity of fracturing fluid can be represented as Equation (16):

$v = \frac{- k}{ϕ ϑ} (\nabla P_{u} + ρ_{w} g)$

(16)

where the permeability of the reservoir is denoted as k, the porosity is represented by $ϕ$ , viscosity of the fracturing fluid is denoted as ϑ, the fluid pressure is P_u, and the density of the fracturing fluid is represented by ρ_w.

The continuity equation of fracturing fluid in the reservoir can be expressed as Equation (17):

$\frac{\partial (ρ_{w} ϕ)}{\partial t} + \nabla (- \frac{ρ_{w}}{ϑ} \nabla p) + ρ_{w} α \frac{\partial ε_{v}}{\partial t} = Q_{m}$

(17)

$\frac{\partial (ρ_{w} ϕ)}{\partial t} = ρ_{w} S \frac{\partial P_{u}}{\partial t}$

(18)

where ε_v is the volumetric strain, the variable t represents time, $α$ is the Biot coefficient, Q_m represents the mass source term, and S represents the storage coefficient.

During the compaction stage of rock, it adheres to Hooke’s law, and the deformation equation of rock pore compaction stage is as follows [34]:

$ϕ = ϕ_{0} - \frac{(1 - 2 μ_{g}) (σ_{1} + 2 σ_{3})}{E_{g}}$

(19)

In the equation, $μ_{g}$ represents the Poisson’s ratio of the rock skeleton, and $E_{g}$ denotes the elastic modulus of the rock skeleton. Under high confining pressure conditions, the Poisson’s ratio and elastic modulus of the rock skeleton approximate those of the rock itself. Considering the influence of pore pressure, the above equation transforms to obtain the following:

$ϕ = ϕ_{0} - \frac{(1 - 2 μ) (σ_{1} + 2 σ_{3} - 3 Δ p)}{E}$

(20)

$\frac{k}{k_{0}} = {[1 - \frac{(1 - 2 μ) (σ_{1} + 2 σ_{3} - 3 Δ p)}{ϕ_{0} E}]}^{3}$

(21)

During the hydraulic fracturing process, taking into account the influence of the damage parameter of the rock, the relationship between permeability and damage is as follows:

$k = k_{m} {(\frac{ϕ}{ϕ_{m}})}^{3} e x p (χ D)$

(22)

where k_m represents the permeability of the rock after compaction, and $χ$ is the coefficient representing the influence of damage on permeability.

Therefore, the permeability of the rock can be divided into two parts with the variation of stress: the permeability before rock yielding and the permeability during the damage stage, the permeability equation can be represented as a piecewise function, as follows:

$k = \{\begin{array}{l} k_{0} \{{[1 - \frac{(1 - 2 μ) (σ_{1} + 2 σ_{3} - 3 Δ p)}{ϕ_{0} E}]}^{3}\}, ε < ε_{k m i n} \\ k_{m} {(\frac{1}{ϕ_{m}} - \frac{(1 - ϕ_{m}) (1 - Δ p / K_{m})}{(1 + ε_{v} - ε_{k m i n}) ϕ_{m}})}^{3} e x p (χ D), ε_{k m i n} \leq ε \end{array}$

(23)

where $ε_{k m i n}$ represents the strain corresponding to the onset of increased permeability under the influence of elastic damage.

3. Validation of the Damage Model

3.1. Core Testing under Different Confining Pressure with Fluid-Structure Coupling

In order to investigate the rock damage law of the reservoir in the Z area, laboratory core experiments were conducted using the TRC-100 high-temperature and high-pressure triaxial rheometer. The TRC-100 triaxial rheometer applies three mutually perpendicular stresses to simulate the in situ stress state under geological conditions, aiming to investigate the deformation and rheological properties of materials under different stress directions. This apparatus is capable of performing uniaxial compression, triaxial compression, shear compression tests, shear tests, and seepage tests. The maximum axial test force is 2000 kN, with a range of confining pressure from 0 MPa to 150 MPa, maximum pore pressure of 150 MPa, and temperature control ranging from −25 °C to 150 °C. The equipment is primarily comprised of a loading system, a confining pressure and temperature control system, a fluid injection system, a data acquisition system, and a control system. Rock specimens undergo testing within a pressure chamber, wherein the confining pressure system and fluid injection system apply the requisite confining pressure and pore pressure for the experiment, respectively. The control system and data acquisition system are responsible for configuring experimental parameters and recording stress, strain, displacement, and other experimental data, as illustrated in Figure 1.

To investigate the phenomenon of fracture propagation penetrating through separating layers during the fracturing process, leading to water channeling and flooding issues, this study selected interlayer rocks within the C61-3 reservoir of Zone Z. The rock type was mudstone sandstone, situated at a depth of 642 m–645 m. Fluid-structure coupling experiments were designed with confining pressures of 5 MPa, 10 MPa, and 15 MPa, while a pore pressure of 3 MPa was applied to the core samples. Mechanical parameters of the cores under different confining pressures were recorded. Experimental results indicated that with increasing confining pressure, both peak stress and peak strain of the cores increased. The experimental data of the fluid-structure coupling tests on the core samples are presented in Table 1.

3.2. Numerical Simulation Verification

The research of fracture propagation requires relevant software that can study fluid-structure coupling. Among the existing finite element analysis software, Comsol Multiphysics software is widely used in the study of fluid-structure coupling, especially in the study of damage evolution law. The software has fast analysis speed, accurate and scientific simulation results, and powerful secondary development function. In this paper, Comsol 6.0 software was used to simulate the law of artificial fracture propagation in the reservoir.

The new modified damage constitutive model was used to calculate the rock damage parameters, and the damage propagation and evolution characteristics were simulated by Comsol software. The established two-dimensional model had a size width of 50 mm, a height of 100 mm, a pore pressure of 3 MPa, a confining pressure of 5 MPa, and a grid number of 5949 units, as shown in Figure 2. The solid mechanics and seepage modules were added to simulate the damage characteristics under the fluid-structure coupling, and the simulated stress-strain data were compared with the experimental data to verify the accuracy of the modified constitutive model. The rock mechanical parameters necessary for the fluid-solid coupling numerical simulation were sourced from experimental data, with the elastic modulus being 5.02 GPa and the Poisson’s ratio being 0.2207, with other parameters presented in Table 2.

Figure 3 illustrates the damage pattern of rocks under the simulation of fluid-solid coupling using the newly developed damage constitutive model. To simulate the mechanical characteristics of the core under the coupling of fluid and solid, a two-dimensional rectangular core model domain with dimensions of 50 mm × 100 mm was established. The bottom of the model is fixedly constrained with a confining pressure of 8 MPa, while the top is subjected to a loading velocity of 0.05 MPa/s, with a pore pressure of 3 MPa. For further details regarding other parameters, please refer to Table 2. According to the simulation results, under uniaxial compression, cracks in the rock propagate along the diagonal direction, while under shear compression, cracks formed in the interlayer rock penetrate through the rock, eventually forming a shear fracture surface. A comparison with the experimental core fracture images indicated that the numerical simulation results were in good agreement with the observed fracture patterns.

Furthermore, utilizing the newly developed damage constitutive model with introduced correction parameters in this study, the theoretical values of peak stress of rocks under confining pressures of 5 MPa, 10 MPa, and 15 MPa were calculated to be 29.47 MPa, 41.31 MPa, and 51.38 MPa, respectively. The modified stress-strain curves better reflected the stress-strain relationship during the compaction stage and accurately captured the stress decay behavior in the post-peak stage. Compared to the uncorrected curves, the corrected curves exhibited higher precision in the post-peak stage, with errors in peak stress theoretical values compared to experimental data of 1.56%, 0.43%, and 0.14%, respectively. In contrast, the errors in peak stress for the uncorrected curves were 7.87%, 4.76%, and 2.95%, respectively, as shown in Table 3 and Figure 4. The trend of the theoretical curves calculated by the corrected constitutive model was consistent with the experimental curves, with peak errors within 3% under different confining pressures. This indicated that the corrected rock constitutive model had minimal errors and higher precision, enabling a scientific description of the damage evolution characteristics under fluid-solid coupling in the Z oilfield rocks.

4. Numerical Simulation Analysis of Hydraulic Pressure Damage

4.1. Numerical Simulation Parameters for Hydraulic Fracturing

Figure 5 depicts the hydraulic fracturing model established in this study. The model represents a rectangular domain with dimensions of 30 m in length and 15 m in width, resembling a cross-sectional view of the reservoir. At a vertical position of 3 m within the model, there is a 30 m long and 2 m wide barrier layer model. To simulate the geostatic stress, pressures of 15 MPa and 10 MPa are applied respectively at the longitudinal and transverse boundaries of the model. The mesh within the model consists of triangular elements, totaling 51,067 units, with a maximum mesh size of 1.11 m. According to the reservoir research program for Zone Z, the elastic modulus of the reservoir is determined to be 9.98 GPa, with a Poisson’s ratio of 0.2337. The parameters for interlayers were derived from laboratory experimentation. Detailed parameters required for numerical simulation are provided in Table 4.

Figure 6 presents the cross-sectional model of the wellbore. This model depicts a square domain with dimensions of 0.3 m on each side, featuring a wellbore model with a radius of 0.015 m at the center. The top and left boundaries are supported by rods, subjected to lateral loads of 10 MPa and longitudinal loads of 15 MPa. The domain is discretized into 58,480 triangular mesh elements, with a maximum element size of 0.00201 m. The model’s elastic modulus is 5.02 GPa, and the Poisson’s ratio is 0.2207. Other property parameters are detailed in Table 2.

4.2. Influence of Different Perforation-Formation Angle on Hydraulic Fracturing Effect

Due to the influence of factors such as formation dip angle, drilling techniques, and wellbore trajectory, there is a certain angle θ between the perforation hole and the formation during pre-fracturing perforation. In order to study the damage evolution patterns on the interlayer caused by the fracturing expansion process under different angles θ, we assume θ to be 0°, 10°, 30°, and 45°, respectively. The crack propagation patterns under different angles are shown in Figure 7, indicating that the total length of crack propagation decreases as the angle increases.

Figure 8 and Figure 9 depict the vertical and horizontal evolutionary curves of cracks under different angles, respectively. It can be observed that with larger angles, the cracks penetrate the interlayer earlier. In the fracturing process of oil wells in Zone Z, the vertical propagation of fractures is prone to penetrate interlayers and connect with adjacent water layers, leading to fracturing failure, water channeling in oil wells, and water flooding. Consequently, for reservoir intervals with larger angles, it is necessary to adjust the fracturing scale and reduce the injection volume of fracturing fluid to prevent interlayer fracturing during the fracturing process.

4.3. Effect of Natural Fractures on Hydraulic Fracturing

To investigate the influence of different types of natural fractures on hydraulic fracturing crack propagation, the boundary pressure σ_v was set to 15 MPa and the boundary pressure σ_h was set to 10 MPa. According to Figure 10, in the parallel double-fracture model, when the fracture propagates through the double fractures, it sequentially communicates with the fractures. The damage evolution rate within natural fractures is significantly higher than within the reservoir. When damage evolution occurs in intersecting fractures to the point of reaching natural intersecting fractures, the expansion mainly follows the natural fractures in the horizontal direction. The angle of horizontal natural fractures affects the path of damage evolution. In the “T” shaped fractures, after the fracture communicates with the T-shaped natural fracture, it mainly branches out along the bottom of the natural fracture, transforming the natural crack into a bridge for lateral expansion of artificial fractures. From the influence of different types of natural fractures on fracture propagation, it is evident that the distance of artificial fracture expansion is mainly related to the horizontal extension length of natural fractures. Natural fractures serve as pathways in the propagation path of artificial fractures. When damage evolution passes through natural fractures, it continues to expand horizontally. However, the lowest position of natural fractures in the direction of fracture propagation becomes the new starting point for horizontal expansion of damage.

From Figure 11, it is evident that under the influence of different natural fractures, the evolution characteristics of permeability are consistent with the damage evolution law. This indicates that during the hydraulic fracturing process, fluid primarily flows along the fractures, and a decrease in permeability occurs at the boundaries of damage.

4.4. Influence of Different Boundary Loads on Hydraulic Fracturing Effect

To investigate the influence of boundary pressure on the evolution pattern of damage around oil wellbores, longitudinal boundary pressures were set at 10 MPa, and lateral boundary pressure were respectively set at 4 MPa, 10 MPa, and 15 MPa. Different longitudinal and transverse pressure ratios were established at ratios of 2:5, 1:1, and 3:2, respectively. According to Figure 12, when the pressure ratio is 2:5, damage near the wellbore extends in a single direction, forming a single horizontal fissure during the pressure process. When the pressure ratio is 1:1, damage spreads around the wellbore, forming three distinct primary fractures during the process of damage evolution. When the pressure ratio is 3:2, damage evolves and extends in the vertical direction, forming a single fracture. It can be concluded that the evolution direction of fractures around the wellbore during hydraulic fracturing is primarily related to the longitudinal and transverse pressure ratios on the cross-section of the wellbore.

Figure 13 depicts the evolution of permeability under different pressure ratios. When damage occurs in the reservoir, permeability varies along the damage path, with the maximum permeability occurring within the fracture channels. Therefore, it can be inferred that post-fracturing wellbore flow primarily originates from the flow within the artificial fractures.

5. Discussion

In this study, utilizing the Z oilfield as a case study, a modified damage constitutive model was developed and a permeability evolution model was formulated. This model takes into account the compaction stage and post-peak stress changes, thereby improving the accuracy of the damage constitutive model. When compared with experimental results, the error is less than 2%. Through this damage model, the evolution law of damage during hydraulic fracturing was investigated to determine the fracture propagation mode. The newly developed damage constitutive model was validated for its rationality and stability using Comsol software.

Additionally, different perforation angle oil reservoir profile models and oil wellbore cross-section models were established. Numerical simulations yielded the following conclusions: during hydraulic fracturing, different perforation angles lead to differences in fracture length, and the presence of natural fractures facilitates damage communication with these natural fractures. Furthermore, the ratio of boundary load pressure directly influences the direction of fracture propagation. The research results indicate that the main reason for hydraulic fracturing failure is the change in fracture propagation direction caused by a high perforation angle and boundary load pressure ratio, thereby failing to achieve the expected production enhancement effect.

This paper applied the modified statistical damage model to the field of oil well hydraulic fracturing, considering the influence of damage evolution paths on the hydraulic fracturing process. This study fills a gap in research on damage evolution in the field of oil well hydraulic fracturing. Previous studies mainly focused on the expansion of fractures in reservoirs while neglecting the influence of factors such as fracture evolution direction, natural fracture morphology, and boundary pressure on hydraulic fracturing success rates.

Through theoretical research and numerical simulations, the characteristics of fracture propagation were analyzed, offering theoretical guidance to enhance the success rate of hydraulic fracturing operations in the Z oilfield. It is worth noting that in 2023, the success rate of hydraulic fracturing in the Z oilfield was only 63%, with the main reason for failure being the breakthrough of fractures near interlayer water barriers resulting in water influx. Our study clearly recognizes that perforation angles and boundary pressures are the primary reasons for this failure. The reservoir environment in the Z oilfield is complex, with interconnections among perforation angles, natural fractures, and boundary pressures. Process parameters during hydraulic fracturing operations also influence the success rate. Therefore, future research directions should focus on optimizing hydraulic fracturing techniques and improving success rates by altering the phase angles of perforation shots.

In summary, a scientific and rational model has been established, elucidating the evolution of damage during the hydraulic fracturing process, and fully revealing the reasons for fracture propagation leading to water influx in the Z oilfield. This study provides a theoretical basis for preventing water influx during hydraulic fracturing processes in the future.

6. Conclusions

Research was conducted on the expansion law of fracturing cracks in oil wells according to the hydraulic fracturing development process of the Z oilfield. Through numerical simulation and theoretical research methods, the evolution law of damage during the fracturing process was discussed, leading to the following conclusions.

A modified constitutive model considering the compaction stage and post-peak stage was established. Compared with traditional models, the numerical simulation accuracy improved by more than 3%. The modified model can more scientifically and reasonably explain the reservoir damage characteristics during the pressure process.

This study investigated the damage evolution patterns during the hydraulic fracturing process in Zone Z by incorporating a modified damage constitutive model and conducting numerical simulations. The research findings reveal that post-fracturing phenomena such as water seepage and inundation are induced by the propagation and connectivity of damage to adjacent aquifers. Additionally, numerical models with varying perforation angles and cross-sectional wellbore models were established to simulate the paths of damage propagation, thereby offering insights for optimizing fracturing designs.

This study revealed an increased risk of fracture propagation into adjacent layers when the angle between the perforation segment and the formation exceeds 30°. The presence of natural fractures within the reservoir accelerates the process of damage expansion. Furthermore, it was observed that changes in the direction of fracture propagation are primarily influenced by the ratio of perforation angles in the reservoir profile model and the boundary loads in the wellbore cross-sectional model. These findings hold significant implications for preventing water inundation and seepage phenomena during the hydraulic fracturing process.

In conclusion, by elucidating the evolution law of damage during the fracturing process using a scientifically reasonable model, the fundamental reasons for water infiltration caused by crack expansion were revealed. This provides an important theoretical basis and practical guidance for future fracturing operations, helping to improve the success rate of fracturing operations, reduce the risk of water infiltration, and promote the smooth progress of oilfield development work.

Author Contributions

Conceptualization, L.Z. and Q.L.; methodology, L.Z. and X.L.; investigation, Q.L. and X.L.; writing—original draft preparation, L.Z.; writing—review and editing, Q.L. supervision, Q.L.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Major Research Program for Science and Technology of China (Grant No. 2016ZX05050006).

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

Liang Zhao, Qi Li, Xiangrong Luo, hereby declare that we have no actual or potential conflict of interest with any individual, organization, or entity related to this research and the writing of this article.

References

Wang, X.; Wang, Z.; Chen, J.; Zeng, J. Genesis and hydrocarbon migration characteristics of low permeability reservoir in Yanchang Formation, Zhenbei Oilfield, Ordos Basin. Oil Gas Geol. Oil Recovery 2010, 17, 15–18+112. [Google Scholar]
Xi, S.; Li, M.; Zhao, W.; Fu, X.; Zu, K.; Zhao, X.; Liu, X.; Pei, W.; Gao, X.; Zhou, X.; et al. Reservoir formation conditions and key technologies for exploration and development of Hengshan large gas field, Ordos Basin. Acta Pet. Sin. 2023, 44, 1015–1028. [Google Scholar]
Yang, Z.; Tamhane, D.; Khurana, A.K.; Crosby, D.G.; Jones, M. Retrograde condensation or water imbibition? A case study of gas well productivity decline before and after hydraulic fracturing. APPEA J. 1996, 36, 562–574. [Google Scholar] [CrossRef]
Liu, Z.Y.; Zhang, P.P.; Wang, H.Z.; Zhu, X.R. Numerical Simulation of Deflection Fracturing with Finite Element Method Considering Crack Initiation Angle. Adv. Mater. Res. 2013, 712, 1027–1031. [Google Scholar] [CrossRef]
Sesetty, V.; Ghassemi, A. A Numerical Study of Sequential and Simultaneous Hydraulic Fracturing in Single and Multi-lateral Horizontal Wells. J. Pet. Sci. Eng. 2015, 132, 65–76. [Google Scholar] [CrossRef]
Mao, R.; Feng, Z.; Liu, Z.; Zhao, Y. Laboratory Hydraulic Fracturing Test on Large-scale Pre-cracked Granite Specimens. J. Nat. Gas Sci. Eng. 2017, 44, 278–286. [Google Scholar] [CrossRef]
Shu, B.; Zhu, R.; Zhang, S.; Dick, J. A Qualitative Prediction Method of New Crack-initiation Direction During Hydraulic Fracturing of Pre-cracks Based on Hyperbolic Failure Envelope. Appl. Energy 2019, 248, 185–195. [Google Scholar] [CrossRef]
Zhao, C.; Xing, J.; Zhou, Y.; Shi, Z.; Wang, G. Experimental Investigation on Hydraulic Fracturing of Granite Specimens with Double Flaws Based on DIC. Eng. Geol. 2020, 267, 105510. [Google Scholar] [CrossRef]
Wang, L.; Yao, B.; Xie, H.; Kneafsey, T.J.; Winterfeld, P.H.; Yin, X.; Wu, Y.-S. Experimental Investigation of Injection-induced Fracturing During Supercritical CO₂ Sequestration. Int. J. Greenh. Gas Control 2017, 63, 107–117. [Google Scholar] [CrossRef]
Zhou, Z.-L.; Zhang, G.Q.; Xing, Y.-K.; Fan, Z.-Y.; Zhang, X.; Kasperczyk, D. A Laboratory Study of Multiple Fracture Initiation from Perforation Clusters By Cyclic Pumping. Rock Mech. Rock Eng. 2018, 52, 827–840. [Google Scholar] [CrossRef]
Montgomery, J.B.; O’sullivan, F.M. Spatial Variability of Tight Oil Well Productivity and The Impact of Technology. Appl. Energy 2017, 195, 344–355. [Google Scholar] [CrossRef]
Sobhaniaragh, B.; Trevelyan, J.; Mansur, W.J.; Peters, F.C. Numerical Simulation of MZF Design with Non-planar Hydraulic Fracturing from Multi-lateral Horizontal Wells. J. Nat. Gas Sci. Eng. 2017, 46, 93–107. [Google Scholar] [CrossRef]
Guo, T.; Tang, S.; Liu, S.; Liu, X.; Xu, J.; Qi, N.; Rui, Z. Physical Simulation of Hydraulic Fracturing of Large-Sized Tight Sandstone Outcrops. SPE J. 2020, 26, 372–393. [Google Scholar] [CrossRef]
Gunarathna, G.; da Silva, B.G. Influence of The Effective Vertical Stresses on Hydraulic Fracture Initiation Pressures in Shale and Engineered Geothermal Systems Explorations. Rock Mech. Rock Eng. 2019, 52, 4835–4853. [Google Scholar] [CrossRef]
Li, Z.; Yan, T.; Hou, Z.; Sun, W.; Shao, Y. Visualization Experiment and Numerical Simulation of Cracks Caused By Pulsed- Plasma Rock Fracturing. J. Eng. Sci. Technol. Rev. 2020, 13, 232–239. [Google Scholar] [CrossRef]
Lv, J.-X.; Hou, B. Fractures interaction and propagation mechanism of multi-cluster fracturing on laminated shale oil reservoir. Pet. Sci. 2024. [Google Scholar] [CrossRef]
Chang, X.; Wang, X.; Yang, C.; Guo, Y.; Wei, K.; Hu, G.; Jaing, C.; Li, Q.; Dou, R. Simulation and optimization of fracture pattern in temporary plugging fracturing of horizontal shale gas wells. Fuel 2024, 359, 130378. [Google Scholar] [CrossRef]
Cheng, Y.; Zhang, G.; Chen, G.; Li, W.; He, Z. Treatment technology of interlayer water channeling after hydraulic fracturing of oil Wells. Pet. Drill. Tech. 2004, 32, 45–47. [Google Scholar]
Lee, H.K. A Computational Approach to The Investigation of Impact Damage Evolution in Discontinuously Reinforced Fiber Composites. Comput. Mech. 2001, 27, 504–512. [Google Scholar] [CrossRef]
Gong, B.; Paggi, M.; Carpinteri, A. A Cohesive Crack Model Coupled with Damage for Interface Fatigue Problems. Int. J. Fract. 2012, 173, 91–104. [Google Scholar] [CrossRef]
Hu, S.; Wang, E.; Kong, X. Damage and Deformation Control Equation for Gas-bearing Coal and Its Numerical Calculation Method. J. Nat. Gas Sci. Eng. 2015, 25, 166–179. [Google Scholar] [CrossRef]
Li, Y.; Jia, D.; Rui, Z.; Peng, J.; Fu, C.; Zhang, J. Evaluation Method of Rock Brittleness Based on Statistical Constitutive Relations for Rock Damage. J. Pet. Sci. Eng. 2017, 153, 123–132. [Google Scholar] [CrossRef]
He, Z.-L.; Zhu, Z.-D.; Ni, X.-H.; Li, Z.-J. Shear Creep Tests and Creep Constitutive Model of Marble with Structural Plane. Eur. J. Environ. Civ. Eng. 2019, 23, 1275–1293. [Google Scholar] [CrossRef]
Fu, T.; Zhu, Z.; Cao, C. Constitutive Model of Frozen-soil Dynamic Characteristics Under Impact Loading. Acta Mech. 2019, 230, 1869–1889. [Google Scholar] [CrossRef]
Xie, S.; Lin, H.; Chen, Y.; Yong, R.; Xiong, W.; Du, S. A Damage Constitutive Model for Shear Behavior of Joints Based on Determination of The Yield Point. Int. J. Rock Mech. Min. Sci. 2020, 128, 104269. [Google Scholar] [CrossRef]
Niu, H.; Zhang, X.; Tao, Z.; He, M. Damage Constitutive Model of Microcrack Rock Under Tension. Adv. Civ. Eng. 2020, 2020, 8835305. [Google Scholar] [CrossRef]
Liu, T.; Cao, P.; Lin, H. Damage and Fracture Evolution of Hydraulic Fracturing in Compression-shear Rock Cracks. Theor. Appl. Fract. Mech. 2014, 74, 55–63. [Google Scholar] [CrossRef]
Huang, X.; Liu, Q.; Liu, B.; Liu, X.; Pan, Y.; Liu, J. Experimental Study on The Dilatancy and Fracturing Behavior of Soft Rock Under Unloading Conditions. Int. J. Civ. Eng. 2017, 15, 921–948. [Google Scholar] [CrossRef]
Shi, C.; Lin, B.; Yu, H.; Shi, S.; Zhang, J. Characterization of hydraulic fracture configuration based on complex in situ stress field of a tight oil reservoir in Junggar Basin, Northwest China. Acta Geotech. 2023, 18, 757–775. [Google Scholar] [CrossRef]
Hui, G.; Chen, Z.; Schultz, R.; Chen, S.; Song, Z.; Zhang, Z.; Song, Y.; Wang, M.; Gu, F. Intricate unconventional fracture networks provide fluid diffusion pathways to reactivate pre-existing faults in unconventional reservoirs. Energy 2023, 282, 128803. [Google Scholar] [CrossRef]
Wu, S.; Zhang, M.; Zhang, S.; Jiang, R. Research on determination method of equivalent Mohr-Coulomb strength parameters based on modified Hoek-Brown criterion. J. Rock Soil Mech. 2019, 40, 4165–4177. [Google Scholar]
Li, H.; Zhang, S. Rock damage model based on modified Lemaitre strain equivalence hypothesis. Rock Soil Mech. 2017, 38, 1321–1326. [Google Scholar]
Hu, H.; Guan, Z.; Xu, Y.; Han, C.; Liu, Y.; Liang, D.; Lu, B. Analysis of bottom-hole stress field in deep Wells based on porous elasticity theory. J. China Univ. Pet. (Nat. Sci.) 2020, 44, 52–61. [Google Scholar]
Li, S.; Liu, X.; Li, Y.; Wang, W.; Zhou, Y. Study on deformation characteristics and damage evolution of carbonaceous mudstone during progressive failure process. China J. Highw. Transp. 2022, 35, 99–107. [Google Scholar]

Figure 1.TRC-100 high-temperature and high-pressure triaxial rheometer.

Figure 2.Cell division of reservoir rock specimen.

Figure 3.Damage diagram of rock in Z oilfield under fluid-structure coupling.

Figure 4.Comparison between theoretical stress-strain curves and test curves of water-soaked rocks under different confining pressures.

Figure 5.Hydraulic fracturing model schematic.

Figure 6.Schematic diagram of well hole plane model.

Figure 7.Numerical simulation of crack propagation patterns under different angles.

Figure 8.Evolutionary curve of crack propagation along the vertical direction under different angles.

Figure 9.Evolutionary curve of crack propagation along the horizontal direction under different angles.

Figure 10.Fracture damage evolution of rocks with natural fractures.

Figure 11.Evolution of permeability with damage induced by different natural fractures.

Figure 12.Damage evolution law under different boundary loads.

Figure 13.Evolution of permeability under different boundary loads.

Table 1.Rock mechanics test data under fluid-structure coupling.

Number	Confining Pressure (MPa)	Peak Strain (10⁻³)	Peak Stress (MPa)	Poisson’s Ratio	Young’s Modulus (GPa)
A1	5	6.15	29.94	0.2126	5.79
A2	10	7.1	41.13	0.2183	6.41
A3	15	7.88	51.31	0.2248	8.17

Table 2.Physical parameters and values of reservoir rock cylinder specimens.

Parameter Name	Symbol	Unit	Value
Internal Friction Angle	θ	Deg	22.7
Cohesion	c	MPa	5.85
Initial Permeability	k0	mD	1.00
Porosity	Φ	1	0.1
Viscosity	ϑ	mPa·s	1
Fluid Density	ρ	kg/m³	1000
Pore Pressure	p	MPa	3
Confining Pressure	Pc	MPa	8

Table 3.Comparison of numerical and experimental peak stress values under fluid-solid coupling.

ID	Peak Stress/MPa		Uncorrected Error/%	Corrected Error/%
ID	Corrected Value	Experimental Value	Uncorrected Error/%	Corrected Error/%
A1	29.47	29.94	7.87	1.56
A2	41.31	41.13	4.76	0.43
A3	51.38	51.31	2.95	0.14

Table 4.Physical parameters and values required for hydraulic fracturing simulation.

Parameter Name	Symbol	Unit	Value
Interlayer Elastic Modulus	E1	GPa	5.02
Reservoir Elastic Modulus	E2	GPa	9.98
Interlayer Poisson’s Ratio	μ1	1	0.2207
Reservoir Poisson’s Ratio	μ2	1	0.2337
Initial Permeability	k₀	m²	1.00 × 10⁻¹⁵
Viscosity	ϑ	mPa·s	1
Fluid Density	ρ	kg/m³	1000

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