Open AccessArticle

Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction

Yuan Gao

^1,2,3,4,

Qiuping Qin

^1,2,3,4,*,

Xiaobing Bian

^1,2,3,4,

Xiaoyang Wang

^1,2,3,4,

Wenjun Xu

⁵

and

Yanxin Zhao

⁵

Sinopec Key Laboratory of Drilling Completion and Fracturing of Shale Oil and Gas, Beijing 102206, China

State Center Research and Development of Oil Shale Exploitation, Beijing 102206, China

State Key Laboratory of Shale Oil and Gas Enrichment Mechanisms and Efficient Development, Beijing 102206, China

⁴

Sinopec Research Institute of Petroleum Engineering Co., Ltd., Beijing 102206, China

⁵

School of Petroleum Engineering, Yangtze University, Wuhan 430100, China

Author to whom correspondence should be addressed.

Processes 2024, 12(12), 2931; https://doi.org/10.3390/pr12122931

Submission received: 23 November 2024 / Revised: 17 December 2024 / Accepted: 19 December 2024 / Published: 21 December 2024

(This article belongs to the Special Issue Advanced Reservoir Simulation and Modelling, Thermal and Enhanced Oil Recovery Processes)

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Figure 1
The traction-separation law of cohesive elements [<a href="#B17-processes-12-02931" class="html-bibr">17</a>]. "> Figure 2
Schematic of fluid flow within a damaged unit [<a href="#B17-processes-12-02931" class="html-bibr">17</a>]. "> Figure 3
Comparison of indoor experiments and numerical simulation results. "> Figure 4
Numerical simulation diagram. "> Figure 5
Comparison of simulation results of different spacer thicknesses. "> Figure 6
Comparison of simulation results of stress difference between different layers. "> Figure 7
Comparison of simulation results of different tensile strength differences. "> Figure 8
Comparison of simulation results of different Young’s modulus differences. "> Figure 9
Comparison of simulation results of different injection rates. "> Figure 10
Comparison of simulation results of viscosity of different fracturing fluids. "> Figure 11
Calculation results of correlation degree of different influencing factors. ">

Versions Notes

Abstract

There are significant lithological and stress differences between continental shale layers, posing challenges for hydraulic fractures (HFs) to propagate through the formations, leading to weak fracture effects. To address this, this article adopts the finite element and cohesive force element methods to formulate a three-dimensional numerical model for hydraulic fracture (HF) propagation through layers, considering interlayer lithology and stress variations. The accuracy of the model was verified by physical experiments, and the one-factor analysis method was used to creatively reveal the complex mechanism of the effect of geological and engineering variables on the diffusion of HFs in continental shale reservoirs. The results show that high interlayer stress difference, high interlayer tensile strength difference, low interlayer Young’s modulus difference and large interlayer thickness are not conducive to the penetration of HFs, but increasing the injection rate and the viscosity of fracturing fluid can effectively improve the penetration of HFs. The influence ranking of each factor was determined using the grey relational degree analysis method: interlayer stress difference > interlayer Young’s modulus difference > interlayer tensile strength difference > interlayer thickness > injection rate > fracturing fluid viscosity.

Keywords:

continental shale; sandstone–shale interaction; hydraulic fracturing; penetration propagation; numerical simulation

1. Introduction

China has abundant reserves of continental shale gas with significant development potential. The shale gas reserves in the Fuxing block of the Sichuan Basin alone reach up to trillion cubic meters [1,2]. In contrast to marine shale reservoirs, continental shale reservoirs exhibit more complex geological characteristics, with interbedded sandstone and shale layers, large differences in vertical stress, and difficulties in the propagation of hydraulic fractures (HFs) through layers, causing poor volumetric fracturing effects. Accordingly, it is pressing to conduct thorough research on the propagation laws of HFs in continental shale reservoirs with sandstone–shale interaction.

In the last several years, national and foreign researchers have carried out a series of studies on the law of propagation of HFs in stratified layers. In terms of laboratory experiments, Warpinski [3], Teufel [4] and Liu [5] conducted true triaxial hydraulic fracturing experiments by studying natural rock samples to investigate the impact of geological factors on the vertical propagation of HFs. The findings indicated that the minimum horizontal stress difference and variation in Young’s modulus across different layers are important factors affecting vertical propagation of HFs. The larger the difference in minimum horizontal stress between successive layers (referred to as interlayer stress difference) and Young’s modulus difference, the fuller the vertical propagation of HFs. Hou [6,7] and Wang [8] carried out hydraulic fracturing laboratory experiments using marine shale outcrop rock samples to analyze the effects of engineering parameters on the spread of HF penetration. The experimental research showed that high-viscosity fluid and high injection rate can effectively enhance the penetration propagation ability of HFs and significantly increase the reservoir reconstruction volume. However, the laboratory experiments were limited by the capacity of the experimental equipment and the size of the sample, which could merely offer qualitative insights and had a confined impact on guiding the optimization of process parameters in design. Hence, numerous scholars conducted a range of numerical simulation studies. Gu [9], Wang [10] and Fu [11,12] constructed a numerical model by integrating the finite element method with the cohesive force element technique to reveal the law of penetration propagation of HFs under interlayer interface interference. However, the study only focused on the interference of the interlayer interface, ignoring the influence of interlayer structure on the penetration propagation of HFs. Mao [13] and Fu [14] established a numerical model of a three-layer structure (isolation, reservoir and isolation) by considering divergence in lithology and stress among various strata. They determined that the interlayer stress difference is the main controlling factor affecting the propagation of HFs through layers, and further verified that high-viscosity fluid and high injection rate can promote the full vertical propagation of HFs. Song [15] and Cui [16] established a multi-layer interaction model for sand-mud interaction reservoirs, but the analysis of influencing factors was not comprehensive enough, and only revealed the effects of interlayer stress difference, injection rate and fluid viscosity on the propagation of HFs through the layer.

Most of the above studies focus on the effect of interlayer interface and interlayer in terms of the dissemination behavior of HFs through the strata, but the principle governing HF propagation in continental shale reservoir with sandstone–shale interaction has not been fully clarified. In the context of this, in this article a three-dimensional numerical model of HF propagation in a continental shale reservoir with sandstone–shale interaction is established by leveraging the finite element and cohesive force element models, which fully reveals the influence of geological and engineering factors on HF propagation, and illustrates the primary and secondary relationships among the influencing factors, providing theoretical guidance for the optimal design of the volumetric fracturing of a continental shale reservoir.

2. Mathematical Model

The geological data of the Fuxing block show that the sandstone–shale interface in the continental shale reservoir in the target block has high cementation strength and is not easy to open during hydraulic fracturing. Therefore, the effect of the interlayer interface on the penetration propagation of HFs was not considered in the follow-up study. A three-dimensional fluid–structure coupling numerical model for HF propagation in a continental shale reservoir with sandstone–shale interaction was established by using the finite element and cohesive force element methods.

2.1. Fluid–Structure Interaction Governing Equation

During hydraulic fracturing, the fluid flow within pore spaces interacts with the deformation of the rock framework. The effective stress principle allows us to characterize the relationship between stress and fluid infiltration into the rock. Considering the control volume denoted as V and its bounding surface as S, the reference equation that couples the deformation of the solid skeleton and fluid flow can be expressed as [10]:

\int_{V} (\bar{σ} - p_{w} I) δ_{ε} d_{V} = \int_{S} t δ_{v} d_{S} + \int_{V} f δ_{v} d_{V}

(1)

Within the equation,

\bar{σ}

expresses the value of the effective stress tensor, Pa; p_w denote the pore pressure, Pa; I is the identity matrix, dimensionless; δ_ε refers to the virtual strain rate tensor, s⁻¹; δ_v is the virtual velocity vector, m/s; t stands for the surface force vector, N/m²; f represents the body force vector, N/m³.

The equation that governs the conservation of mass for fluid seepage through the pore spaces within the rock is:

\int_{V} \frac{1}{J} \frac{d}{d t} (J ρ_{w} φ_{w}) d_{V} + \int_{S} ρ_{w} φ_{w} n^{T} \cdot v_{w} d_{S} = 0

(2)

Darcy’s law governs the fluid flow velocity v_w through the rock:

v_{w} = - \frac{1}{φ_{w} g ρ_{w}} k (\frac{\partial p_{w}}{\partial x} - ρ_{w} g)

(3)

In the formula, J represents the rate of rock volume variation, dimensionless; ρ_w is the fluid density, kg/m³; φ_w is the porosity, dimensionless; n^T corresponds to the outward-facing normal vector on surface S, dimensionless; x corresponds to the vector in space, m; g corresponds to the vector of gravitational acceleration, m/s²; k corresponds to the permeability tensor associated with the rock skeleton, m/s.

2.2. Criteria for Crack Initiation and Propagation

This paper uses cohesive elements to model the initiation and propagation of HFs and weak bedding planes, with the secondary nominal stress criterion employed to determine crack initiation:

{\{\frac{〈σ_{n}〉}{σ_{n}^{o}}\}}^{2} + {\{\frac{τ_{s}}{τ_{s}^{o}}\}}^{2} = 1

(4)

where

〈σ_{n}〉 = \{\begin{cases} σ_{n} \begin{matrix} σ_{n} \geq 0 \end{matrix} \\ 0 \begin{matrix} σ_{n} < 0 \end{matrix} \end{cases}

(5)

where σ_n, τ_s and τ_t are the actual normal stress, the first tangential stress and the second tangential stress, Pa; σ_n^o, τ_s^o and τ_t^o represent the tensile strength, first tangential critical stress and second tangential critical stress belonging to the rock, Pa.

HF propagation is governed by the cohesive element’s stiffness degradation criterion, expressed as follows:

\{\begin{cases} σ_{n} = \{\begin{cases} (1 - D) \bar{σ_{n}} \bar{σ_{n}} \geq 0 \\ \bar{σ_{n}} \bar{σ_{n}} < 0 \end{cases} \\ τ_{s} = (1 - D) \bar{τ_{s}} \end{cases}

(6)

Within the equation,

\overset{—}{σ}

_n and

\overset{—}{τ}

_s are the stresses in the directions perpendicular and parallel to the cohesive element’s surface, respectively, assessed under the current strain conditions, according to the principle of undamaged linear elastic behavior. D is the damage factor, dimensionless, falling within the range of 0 to 1; when D = 0, the material is undamaged, and when D = 1, the material is fully damaged, indicating the initiation of hydraulic crack propagation (as shown in Figure 1). The formula for the calculation is given below:

D = \frac{δ_{f} (δ_{m} - δ_{o})}{δ_{m} (δ_{f} - δ_{o})}

(7)

In the formula, δ₀ and δ_f represent the displacements at initial damage and complete damage of the element, m; δ_m represents the greatest displacement that occurs during loading, m.

2.3. Fluid Flow Equation in Fractures

Once the cohesive element is fully damaged, fluid enters the damaged element. As illustrated in Figure 2, fluid movement within the damaged unit is divided into two types: tangential flow and normal flow. The specific form of the fluid flow equation can be seen in the study of Zhao (2022) [17].

3. Model Validation

To validate the ability of this numerical simulation method to model HF penetration and propagation in continental shale reservoirs, physical model experiments [14] were used to verify the model. A numerical model was constructed based on the above method. The scale of the model was 762 mm × 762 mm × 914 mm, and it was divided into 3 layers, in which the upper and lower layers were interlayers, and the middle layer was the reservoir. The thickness of the reservoir and interlayer were the same, and the stress difference between layers was set at 7 MPa. The specific parameters are shown in Table 1. The model adopted the same size mesh division, with the mesh size set as 15 mm × 15 mm × 15 mm, and the model boundary adopted the fixed displacement condition. According to Figure 3, the simulation results of HF propagation morphology are in substantial agreement with the morphology of the physics experiment, which affirms the feasibility and accuracy of the numerical simulation method. Based on this, the construction of subsequent numerical models and the method of grid division all applied the stated mesh size division guideline, which is to say, the side length of grid cells was less than or equal to 1/50 of the model scale. The smaller the mesh size, the more accurate the simulation results. However, reducing the mesh size too much will also make the mesh too many, which will cause the operation to be too slow. Therefore, the grid size setting standard of the subsequent numerical models adopted 1/50 of the model size.

4. Analysis of Influencing Factors

4.1. Model Establishment and Parameters

In Figure 4, referring to the geological features of the continental shale reservoir in Fuxing block and based on the above simulation modeling technique, a numerical model of HF propagation in continental shale with sandstone–shale interaction was established. The model size was 60 m × 30 m × 30 m, and it was divided into five layers. The sandstone layer was regarded as the interlayer, and the shale layer was regarded as the reservoir. The fundamental input parameters are outlined in Table 2. Based on this model, the single-factor analysis method was performed to examine the effects of six different parameters on the propagation of HFs through layers, such as the thickness of interlayer, the variation of minimum horizontal in situ stress difference among different layers, Young’s modulus difference between layers, the tensile strength difference between layers, injection rate and fluid viscosity. Three groups of horizontal values were investigated for each type of parameter, with a total of 13 groups of cases. In addition, to guarantee the comparability of the outcomes from state simulations, the same liquid injection volume scheme was adopted in each group of cases. The parameter design scheme of simulation cases is shown in Table 3.

4.2. Influence of Geological Parameters

4.2.1. Interval Thickness

As shown in Figure 5, with the augmentation of the thickness of the interlayer, the height of the HF progressively decreases, and length gradually increases. When the HF extended to the cementation interface between the reservoir and the interval, the expansion of the fracture height was limited and the pressure continued to rise. When the HF broke through the interface and entered the interval, the pressure decreased. When the fracture was confined within the interval and the fracture height could not break through the interval, the pressure continued to rise. When the HF broke through the interval, the injection pressure dropped abruptly and showed a decreasing trend. This is because HF propagation needs to overcome the minimum horizontal in situ stress and the tensile strength of rocks at the same time. Due to higher minimum horizontal in situ stress and tensile strength of the interlayer, the vertical propagation resistance of the HF after penetrating the reservoir into the interlayer increases, the height growth rate of the HF slows down and the length of the HF increases rapidly, creating a lengthy and broad fracture. Therefore, a high interval thickness was not conducive to the propagation of HFs through the layer. In the design of the fracturing construction scheme, the geological conditions of the region should be evaluated, and an appropriate location should be selected for well placement.

4.2.2. Interlayer Stress Difference

As shown in Figure 6, as the interlayer stress difference increases, the height of the HF gradually decreases, and its length gradually increases. The injection pressure curve shows that when the HF failed to penetrate the interlayer, the injection pressure presented an upward trend. The injection pressure curve dropped markedly upon the fracture penetrating the interlayer. This is because the minimum horizontal stress of the interlayer is larger, which implies that the propagation of the HF in the interlayer needs to overcome higher extension resistance [18]. As a result, the propagation of the HF in the interlayer is difficult, and the height of the HF is significantly inhibited, resulting in the formation of “short-width type” fractures, which leads to the propagation of HFs along the direction of the HF length in the reservoir.

4.2.3. Tensile Strength Difference

The definition of tensile strength difference is the disparity in tensile strength between the interlayer and the reservoir. As shown in Figure 7, the influence rules of tensile strength difference and interlayer stress difference on the penetration propagation of HFs are basically the same. As the tensile strength difference increases, the height of the HF is significantly reduced. The reason for this is that, along with the minimum horizontal in situ stress, the tensile strength of the rock needs to be surpassed during HF propagation. A smaller difference in tensile strength facilitates the penetration and propagation of HF. Specifically, when the HF initiates in a high tensile strength formation, it is more likely to penetrate the interlayer interface and spread into the lower tensile strength formation. As illustrated by the pressure curve, the expansion of the HF in the interlayer with high tensile strength requires a higher injection pressure. When the HF enters the low tensile strength formation from the high tensile strength formation, the injection pressure continues to decrease, indicating that it is easier for the HF to pass through the high strength formation into the low strength formation. Taking this into account, optimizations can be made to the wellbore’s traversal and perforation zones in fractured wells.

4.2.4. Young’s Modulus Difference

From Figure 8, with the increase in the Young’s modulus difference between layers, the HF height increases significantly, and length decreases significantly. The cause of this is that the high Young’s modulus of the interlayer limits the growth of the HF width in the interlayer. With the continuous injection of fluid, the net pressure continues to increase, resulting in the rapid expansion of the HF in the height direction, forming a “high-narrow” fracture. It should be noted that although the HF can penetrate the high Young’s modulus interlayer, the narrow width of the HF in the interlayer can make proppant migration and placement more difficult, which may not achieve effective support, and there is a risk of sand blocking. Therefore, the proppant particle size and concentration should be strictly optimized for volumetric fracturing of continental shale reservoirs with a large number of high Young’s modulus interlayers.

4.3. Influence of Engineering Parameters

4.3.1. Injection Rate

As shown in Figure 9, as the injection rate increases, the HF height gradually increases and the length decreases. The larger the injection rate is, the larger the injection pressure is, and the earlier the propagation time of the HF is. The reason for this is that fracture height expansion is restricted when the HF penetrates the reservoir and enters the interlayer. A high injection rate can significantly increase the hydraulic energy in the fracture, swiftly elevate the net pressure within the fracture [19], elevate the HF’s ability to propagate and significantly increase the fracture height. Therefore, the use of a high injection rate can significantly enhance the penetration propagation ability of HFs, but a high injection rate will also lead to high construction pressure. Reasonable optimization of the injection rate is essential in the design of fracturing construction schemes.

4.3.2. Fracturing Fluid Viscosity

Figure 10 shows that as the viscosity of the fracturing fluid increases, the HF height rises while the length decreases. This is because the high-viscosity fracturing fluid can effectively reduce the energy dissipation caused by fluid filtration and result in a quick surge in the fracture’s net pressure [20], resulting in an enhanced penetration propagation ability of HFs. Based on this, it is suggested that the fracturing fluid viscosity can be appropriately improved to promote HF longitudinal expansion in the field of fracturing construction, so as to communicate more high-quality reservoirs.

4.4. Primary and Secondary Relationship of Influencing Factors

To better understand the hierarchical relationships among these influencing factors, this paper adopts grey relational degree analysis [21] to analyze and compare the experimental results. The grey correlation method is a quantitative analysis tactic to judge the dynamic development trends of the system, which can elucidate the closeness of the relationship between each influencing factor and the experimental results, seeking to ascertain the primary variables that are shaping the experimental results. The calculation process can be seen below.

(1): Dimensionless processing. This paper selects the range transform method to achieve dimensionless data processing:

\begin{matrix} {x_{i}}^{'} (j) = \frac{x_{i} (j) - \min (X_{i})}{\max (X_{i}) - \min (X_{i})} \begin{matrix} i = 0, 1, 2, \dots, m \end{matrix} \begin{matrix} j = 1, 2, \dots, n \end{matrix} \\ {X_{i}}^{'} = ({x_{i}}^{'} (1), {x_{i}}^{'} (2), \dots, {x_{i}}^{'} (n)) \end{matrix}

(8)

In the formula, i is the serial number and 0 is the serial number (i.e., the experimental result); j is the quantity of data points during the experiment.

(2): Calculate the series difference and get the absolute value of the difference between the parent series and the sub-series of each data point:

\begin{matrix} Δ_{i} (j) = | {x_{0}}^{'} (j) - {x_{i}}^{'} (j) | \\ Δ_{i} = (Δ_{i} (1), Δ_{i} (2), \dots, Δ_{i} (n)) \begin{matrix} i = 1, 2, \dots, m \end{matrix} \end{matrix}

(9)

(3): Find the maximum and minimum values of the absolute difference of each series:

M = \max_{i} \max_{j} Δ_{i} (j) \begin{matrix} m = \min_{i} \min_{j} Δ_{i} (j) \end{matrix}

(10)

(4): Calculate the association coefficient of each data point:

\begin{matrix} γ_{0 i} (j) = \frac{m + ξ M}{Δ_{i} (j) + ξ M} \\ ξ \in (0, 1) \begin{matrix} i = 1, 2, \dots, m \end{matrix} \begin{matrix} j = 1, 2, \dots, n \end{matrix} \end{matrix}

(11)

(5): Calculate the correlation degree between each series, that is, the average value of the correlation coefficient of each series:

γ_{0 i} = \frac{1}{n} \sum_{j = 1}^{n} γ_{0 i} (j) \begin{matrix} i = 1, 2, \dots, m \end{matrix}

(12)

(6): Correlation degree ranking. The correlation degree obtained according to step (5) is arranged in order of magnitude. The higher the correlation degree, the greater the influence of this factor on the experimental results.

Based on the above steps, the degree of correlation between each factor and the height propagation of the HF is calculated, and the calculated results are drawn in a bar chart, as shown in Figure 11. The results show that the influence of the HF propagation morphology is most obvious in the interlayer horizontal stress difference, followed by the interlayer Young’s modulus difference, interlayer tensile strength difference, interlayer thickness, injection rate and fluid viscosity. By and large, the influence of engineering factors is significantly lower than that of geological factors.

5. Conclusions

(1): Incorporating both the finite element and the cohesive force element methods, a three-dimensional numerical model of the continental shale reservoir with sandstone–shale interaction was developed, and its feasibility and precision were confirmed through comparison with experimental data obtained in the laboratory. The single-factor and grey correlation methods served to investigate how geological and engineering factors influence the spread of HFs.
(2): High interlayer stress difference, high interlayer tensile strength difference, low interlayer Young’s modulus difference and large interlayer thickness were not conducive to the penetration of HF, but increasing the injection rate and the viscosity of fracturing fluid could effectively improve the penetration of HFs. The significance of each factor’s influence, both primary and secondary, was evaluated and ranked using the grey relational degree analysis method: interlayer stress difference > interlayer Young’s modulus difference > interlayer tensile strength difference > interlayer thickness > injection rate > fracturing fluid viscosity.
(3): It is suggested that high injection rate and high-viscosity fracturing fluid should be used in the design of the incoming fracturing scheme to promote the penetration of HFs and maximize the communication of high-quality reservoirs. However, high injection rate will also lead to high construction pressure, so it is necessary to optimize the injection rate reasonably when designing fracturing construction schemes.

Author Contributions

Conceptualization, Y.G., Q.Q., X.B., X.W., W.X. and Y.Z.; methodology, Q.Q., W.X. and Y.Z.; software, W.X. and Y.Z.; validation, W.X. and Y.Z.; formal analysis, Y.Z.; investigation, Y.G., Q.Q., X.B. and X.W.; data curation, W.X. and Y.Z.; writing—original draft preparation, Y.Z.; writing—review and editing, W.X. and Y.Z.; visualization, Q.Q., W.X. and Y.Z.; supervision, Q.Q. and W.X.; project administration, Y.G., Q.Q., X.B., X.W. and W.X.; funding acquisition, Y.G., Q.Q., X.B., X.W. and W.X. All authors have read and agreed to the published version of the manuscript.

Funding

This study was supported by the project team of “Research on the control theory of deep shale gas tight cutting and temporary plugging balanced fracturing” of the Sinopec Research Institute of Petroleum Engineering Co., Ltd. under Grant No. 35800000-22-ZC0613-0028.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors express their gratitude to other authors for their support in the experimental methods, experimental materials and amendments to the content of the article.

Conflicts of Interest

Authors Yuan Gao, Qiuping Qin, Xiaobing Bian and Xiaoyang Wang were employed by the company Sinopec Research Institute of Petroleum Engineering Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. The traction-separation law of cohesive elements [17].

Figure 2. Schematic of fluid flow within a damaged unit [17].

Figure 3. Comparison of indoor experiments and numerical simulation results.

Figure 4. Numerical simulation diagram.

Figure 5. Comparison of simulation results of different spacer thicknesses.

Figure 6. Comparison of simulation results of stress difference between different layers.

Figure 7. Comparison of simulation results of different tensile strength differences.

Figure 8. Comparison of simulation results of different Young’s modulus differences.

Figure 9. Comparison of simulation results of different injection rates.

Figure 10. Comparison of simulation results of viscosity of different fracturing fluids.

Figure 11. Calculation results of correlation degree of different influencing factors.

Table 1. Fracturing experiment parameters.

Parameter Type	Concrete Value
Young’s modulus/GPa	16
Poisson’s ratio	0.19
Fracture toughness/(MPa·m^0.5)	1
Tensile strength/MPa	3
Permeability/mD	5
Void ratio	0.2
Vertical in situ stress/MPa	20
Minimum horizontal in situ stress/MPa	15/8/15
Maximum horizontal in situ stress/MPa	12/5/12
Injection rate/(mL/min)	600
Viscosity/(mPa·s)	50

Table 2. Basic parameters of the model.

Parameter Type	Reservoir (Shale)	Interlayer (Sandstone)
Vertical in situ stress/MPa	68	68
Maximum horizontal in situ stress/MPa	56	56
Minimum horizontal in situ stress/MPa	48	52
Young’s modulus/GPa	25	25
Tensile strength/MPa	4	4
Poisson’s ratio	0.23	0.23
Permeability coefficient/(m/s)	1 × 10⁻⁷	1 × 10⁻⁷
Fluid loss coefficient/(m/Pa·s)	1 × 10⁻¹³	1 × 10⁻¹³
Void ratio	0.2	0.2

Table 3. Simulation scheme parameter design.

Case	Reservoir Thickness/m	Interval Thickness/m	Minimum Horizontal In Situ Stress in Interlayer/MPa	Tensile Strength of Interlayer/MPa	Young’s Modulus of Interlayer/GPa	Injection Rate/(m³/min)	Viscosity/(mPa·s)
1	8	3	4	4	25	4	20
2	8	2	4	4	25	4	20
3	8	4	4	4	25	4	20
4	8	3	2	4	25	4	20
5	8	3	6	4	25	4	20
6	8	3	4	2	25	4	20
7	8	3	4	6	25	4	20
8	8	3	4	4	15	4	20
9	8	3	4	4	35	4	20
10	8	3	4	4	25	2	20
11	8	3	4	4	25	8	20
12	8	3	4	4	25	4	10
13	8	3	4	4	25	4	40

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Gao, Y.; Qin, Q.; Bian, X.; Wang, X.; Xu, W.; Zhao, Y. Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction. Processes 2024, 12, 2931. https://doi.org/10.3390/pr12122931

AMA Style

Gao Y, Qin Q, Bian X, Wang X, Xu W, Zhao Y. Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction. Processes. 2024; 12(12):2931. https://doi.org/10.3390/pr12122931

Chicago/Turabian Style

Gao, Yuan, Qiuping Qin, Xiaobing Bian, Xiaoyang Wang, Wenjun Xu, and Yanxin Zhao. 2024. "Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction" Processes 12, no. 12: 2931. https://doi.org/10.3390/pr12122931

APA Style

Gao, Y., Qin, Q., Bian, X., Wang, X., Xu, W., & Zhao, Y. (2024). Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction. Processes, 12(12), 2931. https://doi.org/10.3390/pr12122931

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Propagation Law of Hydraulic Fractures in Continental Shale Reservoirs with Sandstone–Shale Interaction

Abstract

1. Introduction

2. Mathematical Model

2.1. Fluid–Structure Interaction Governing Equation

2.2. Criteria for Crack Initiation and Propagation

2.3. Fluid Flow Equation in Fractures

3. Model Validation

4. Analysis of Influencing Factors

4.1. Model Establishment and Parameters

4.2. Influence of Geological Parameters

4.2.1. Interval Thickness

4.2.2. Interlayer Stress Difference

4.2.3. Tensile Strength Difference

4.2.4. Young’s Modulus Difference

4.3. Influence of Engineering Parameters

4.3.1. Injection Rate

4.3.2. Fracturing Fluid Viscosity

4.4. Primary and Secondary Relationship of Influencing Factors

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Case	Reservoir Thickness/m	Interval Thickness/m	Minimum Horizontal In Situ Stress in Interlayer/MPa	Tensile Strength of Interlayer/MPa	Young’s Modulus of Interlayer/GPa	Injection Rate/(m³/min)	Viscosity/(mPa·s)
1	8	3	4	4	25	4	20
2	8	2	4	4	25	4	20
3	8	4	4	4	25	4	20
4	8	3	2	4	25	4	20
5	8	3	6	4	25	4	20
6	8	3	4	2	25	4	20
7	8	3	4	6	25	4	20
8	8	3	4	4	15	4	20
9	8	3	4	4	35	4	20
10	8	3	4	4	25	2	20
11	8	3	4	4	25	8	20
12	8	3	4	4	25	4	10
13	8	3	4	4	25	4	40

Case	Reservoir Thickness/m	Interval Thickness/m	Minimum Horizontal In Situ Stress in Interlayer/MPa	Tensile Strength of Interlayer/MPa	Young’s Modulus of Interlayer/GPa	Injection Rate/(m³/min)	Viscosity/(mPa·s)
1	8	3	4	4	25	4	20
2	8	2	4	4	25	4	20
3	8	4	4	4	25	4	20
4	8	3	2	4	25	4	20
5	8	3	6	4	25	4	20
6	8	3	4	2	25	4	20
7	8	3	4	6	25	4	20
8	8	3	4	4	15	4	20
9	8	3	4	4	35	4	20
10	8	3	4	4	25	2	20
11	8	3	4	4	25	8	20
12	8	3	4	4	25	4	10
13	8	3	4	4	25	4	40

Case	Reservoir Thickness/m	Interval Thickness/m	Minimum Horizontal In Situ Stress in Interlayer/MPa	Tensile Strength of Interlayer/MPa	Young’s Modulus of Interlayer/GPa	Injection Rate/(m³/min)	Viscosity/(mPa·s)
1	8	3	4	4	25	4	20
2	8	2	4	4	25	4	20
3	8	4	4	4	25	4	20
4	8	3	2	4	25	4	20
5	8	3	6	4	25	4	20
6	8	3	4	2	25	4	20
7	8	3	4	6	25	4	20
8	8	3	4	4	15	4	20
9	8	3	4	4	35	4	20
10	8	3	4	4	25	2	20
11	8	3	4	4	25	8	20
12	8	3	4	4	25	4	10
13	8	3	4	4	25	4	40