Effects of Geological and Fluid Characteristics on the Injection Filtration of Hydraulic Fracturing Fluid in the Wellbores of Shale Reservoirs: Numerical Analysis and Mechanism Determination

Abstract

To mitigate the influence of wellbore heat transfer on the physicochemical properties of water-based fracturing fluids in the high-temperature environments of low-permeability shale reservoirs, this study investigates the fluid filtration behavior of water-based fracturing fluids within the wellbore under such reservoir conditions. A wellbore heat-transfer model based on solid–liquid coupling was constructed in order to analyse the effects of different reservoir and wellbore factors on fluid properties (viscosity and filtration volume) in the water-based fracturing fluids. Concurrently, boundary conditions and control equations were established for the numerical model, thereby delineating the heat-transfer conditions extant between the water-based fracturing fluid and the wellbore. Furthermore, molecular dynamics theory and microgrid theory were utilised to elucidate the mechanisms of the alterations of the fluid properties of the water-based fracturing fluids due to wellbore heat transfer in low-permeability shale reservoirs. The findings demonstrated that wellbore heat transfer in low-permeability shale reservoirs exerts a pronounced influence on the fluid viscosity and filtration volume of the water-based fracturing fluids. Parameters such as wellbore wall thickness, heat-transfer coefficient, radius, and pressure differential introduce distinct variation trends in these fluid properties. At the microscopic scale, the disruption of intermolecular hydrogen bonds and the consequent increase in free molecular content induced by thermal effects are the fundamental mechanisms driving the observed changes in viscosity and fluid filtration. These findings may offer theoretical guidance for improving the thermal stability of water-based fracturing fluids under wellbore heat-transfer conditions.

1. Introduction

The geological reserves of conventional energy sources, including coal and oil, have a direct and profound impact on economic development and human civilisation [1,2]. This reliance has led to the substantial utilisation of fossil fuels in production processes. The over-exploitation of traditional energy sources gives rise to a number of significant challenges, including global energy security and economic development issues [3,4]. Moreover, environmental pollution and the greenhouse effect represent further catastrophic problems that demand urgent resolution [5]. The exploration of new energy sources and the enhancement of the efficiency of traditional energy extraction have emerged as pivotal research domains aimed at mitigating these challenges [6,7]. These domains have garnered significant attention from energy and materials scientists worldwide [8,9]. New energy sources, such as hydrogen, solar energy, and wind energy, are increasingly favored due to their advantages, including environmental friendliness, low cost, and reduced carbon emissions. However, limitations such as the low efficiency of energy storage systems and energy conversion remain significant barriers to their complete replacement of traditional fossil fuels [10,11]. These challenges have driven a substantial number of energy scientists to shift their research focus toward developing innovative approaches for the extraction of conventional energy [12,13,14]. This shift has also accelerated the advancement of reservoir-transformation techniques and the promotion and application of unconventional energy sources [15,16]. Unconventional energy sources, such as shale gas and methane hydrates, serve as effective supplements to traditional energy resources [17,18]. However, their unique geological structures and extreme storage conditions hinder effective extraction through conventional reservoir-transformation techniques [19]. By adapting conventional reservoir-transformation methods to account for the specific characteristics of unconventional reservoirs, it is possible to achieve efficient extraction of these energy sources [20,21]. Therefore, it is crucial to investigate the extraction efficiency and the governing mechanisms of conventional reservoir transformation under unconventional geological conditions.

As the most widely used reservoir-transformation technique, water-based fracturing has achieved high recovery efficiency in tertiary oil recovery within conventional reservoirs [22,23]. Recent studies have demonstrated that water-based fracturing technology can also be applied to unconventional shale reservoirs, offering potential improvements in recovery efficiency in low-permeability shale formations [24,25]. However, the inherent disadvantages of water-based fracturing fluids—such as weak shear resistance and significant fluid seepage—pose substantial challenges to the effective extraction of unconventional energy [26,27]. Various strategies, including optimizing the structure of cross-linking agents and adjusting the fracturing fluid formulation [28,29], can mitigate the physical damage caused by low-permeability shale reservoirs to the water-based fracturing fluids [30,31]. Li et al. have devised a nanoscale 3D cross-linker that has the capacity to achieve a greater number of hydroxyl groups in the molecule than conventional commercial cross-linkers. This, in turn, facilitates enhanced intermolecular hydroxyl cross-linking capacity at a constant cross-linker content. Concurrently, other rheological parameters of water-based fracturing fluids, including fluid viscosity and filtration coefficient, have been demonstrated as exhibiting enhanced performance [32]. Additionally, the hydroxyl groups within montmorillonite—which constitutes a major component of shale—can form intramolecular hydrogen bonds with water and guar gum in the water-based fracturing fluid, thereby inducing a water-sensitive effect that exacerbates reservoir damage in low-permeability formations [33,34]. Zhang et al. investigated the geological characteristics of shale reservoirs and determined that contact with the water-based fracturing fluids can result in a reduction in stiffness, while the viscoelasticity of brittle shale materials is augmented. Furthermore, the water sensitivity exhibited anisotropy and a substantial clay-swelling capacity [35]. Adjustments to the cross-linking agent structure and fracturing fluid composition can also alleviate the reduction in recovery efficiency caused by this water-sensitive effect [36,37]. Nevertheless, the influence of reservoir geological structures and multiphase heat transfer on the properties of water-based fracturing fluids remains an underexplored area [38,39]. During the injection process [40], water-based fracturing fluids undergo significant property changes—such as variations in rheology [41] and seepage capacity [42]—due to heat transfer in high-temperature reservoirs. These property changes directly affect key mechanical parameters and fracture propagation behaviors once the fluid enters reservoir fractures. Yang et al. [43] investigated the flow behavior of water-based fracturing fluids in thermally active wellbores; however, their study was primarily confined to the interaction of multiple fluids (e.g., fracturing fluid and drilling fluid). Moreover, prior research has largely neglected the seepage and filtration behaviors of fracturing fluids under wellbore heat-transfer conditions. Analyses of flow dynamics and filtration mechanisms have also lacked systematic and mechanistic interpretation. These research gaps form the core motivation of the present study, which aims to elucidate the fluid flow and microscale transport mechanisms of fracturing fluids under the influence of thermal gradients in the wellbore environment.

This study establishes a multi-field coupled numerical simulation model to investigate the behavior of water-based fracturing fluid in low-permeability shale reservoirs and verifies the applicability of the proposed mathematical model. Additionally, the influence of deep reservoir heat transfer on the filtration behavior of the water-based fracturing fluid during the injection process is analyzed. From a molecular dynamics perspective, the study further explores the underlying mechanisms by which wellbore heat transfer affects the filtration of water-based fracturing fluid.

2. Numerical Model

2.1. Temperature Field of Wellbore Heat Transfer in Low-Permeability Shale Reservoirs

2.1.1. Heat-Transfer Equation for the Wellbore and Formation, Considering Seepage

The temperature fields of wellbore heat transfer in low-permeability reservoirs involve the thermal interactions between the wellbore fluid, formation rock, and wellbore structure (including the casing, cement sheath, etc.) [44], which can be described by the transient heat conduction equation, accounting for both heat diffusion and convection effects (Equation (1)).

$$\rho C_p{\displaystyle \frac{\partial T}{\partial t}}=\nabla ·\left(k\nabla T\right)+Q$$

(1)

where ρ represents the effective density of the porous media; C_p is the specific heat capacity of the wellbore material; T is the reservoir temperature; t is the heat-transfer time; and k and Q are the thermal conductivity and heat source terms (fluid friction heat generation or formation heat flow).

When water-based fracturing fluid is injected into the production well, a small amount of seepage occurs through the wellbore wall [45,46]. The heat convection induced by this seepage must also be considered in the temperature field equation of the numerical model (Equation (2)) [47]. After incorporating fluid seepage, the governing equation for the wellbore temperature field in low-permeability reservoirs can be expressed by Equation (3).

$$q_{conv}={\rho }_f{C}_{pf}v·\nabla T$$

(2)

$$\rho C_p{\displaystyle \frac{\partial T}{\partial t}}+{\rho }_f{C}_{pf}v·\nabla T=\nabla ·\left(k\nabla T\right)+Q$$

(3)

where ρ_f represents the density of the water-based fracturing fluid; C_pf is the specific heat capacity of the water-based fracturing fluid; q_conv presents the thermal convection caused by a small amount of seepage of the water-based fracturing fluid; v is the seepage velocity of the water-based fracturing fluid; and ρ is the rock density of the wellbore.

2.1.2. Heat-Transfer Equation in a Wellbore, Considering Seepage

The heat-transfer equation for water-based fracturing fluid within a wellbore comprises three principal components: net convection input (Equation (2)) [48], net conduction input $q_{cond}$ (Equation (4)), and external heat input $q_w$ (Equation (5)).

$$q_{cond}=k_f{\displaystyle \frac{{\partial }^2{T}_f}{\partial z^2}}·\pi r_w^2\mathrm{\Delta }z$$

(4)

$$q_w=h(T_w-T_f)·2\pi r_w\mathrm{\Delta }z$$

(5)

where $k_f$ is the thermal conductivity of the water-based fracturing fluid in the wellbore; $T_f$ is the real-time temperature of the water-based fracturing fluid in the wellbore; $r_w$ and $\mathrm{\Delta }z$ are the well inner radius and wellbore length; h is the convective heat-transfer coefficient; and $T_w$ is the well-wall temperature.

The heat-transfer equation for the water-based fracturing fluid in the wellbore is derived from the law of conservation of energy and the correlation analysis of the convection [49], conduction, and boundary heat-transfer terms, as previously mentioned (Equation (6)):

$${\rho }_f{C}_{pf}({\displaystyle \frac{\partial T}{\partial t}}+vz{\displaystyle \frac{\partial T}{\partial t}})=\nabla ·(k_f\nabla T_f)+{\displaystyle \raisebox{1ex}{$2h$}\!\left/ \!\raisebox{-1ex}{$r_w$}\right.}(T_w-T_f)$$

(6)

2.2. Continuity Equation for the Water-Based Fracturing Fluid in a Wellbore, Considering Seepage

The continuity equation (mass conservation equation) (Equation (7)) for water-based fracturing fluid in the wellbore is a complex equation that takes into account many factors [49,50]. These include the flow characteristics of the fluid, the wellbore geometry, and the physical properties of the fracturing fluid.

$${\displaystyle \frac{\partial {\rho }_f}{\partial t}}+{\displaystyle \frac{\partial ({\rho }_f{v}_z)}{\partial z}}=-{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$r_w$}\right.}{\rho }_f{v}_r{|}_{r=r_w}$$

(7)

where $v_r$ is the radial seepage velocity of the water-based fracturing fluid in the wellbore; $v_z$ is the axial (depth direction) flow velocity of the water-based fracturing fluid in the wellbore; and $r_w$ is the well inner radius.

The net axial mass influx of the water-based fracturing fluid into the wellbore along the depth direction can be described by Equation (8).

$$m={\displaystyle \frac{\partial ({\rho }_f{v}_z)}{\partial z}}·\pi r_w^2\mathrm{\Delta }z$$

(8)

In addition, the liquid seepage of the water-based fracturing fluid at the wellbore wall (Equation (9)) is also considered in the continuity equation.

$$Q_s=q_s·\mathrm{\Delta }z={\rho }_f{v}_r·2\pi r_w\mathrm{\Delta }z$$

(9)

where m is the net axial mass; Q_s is the fluid seepage volume; q_s is the injection volume; and v_r is the seepage velocity of the water-based fracturing fluid

The liquid seepage velocity of the water-based fracturing fluid at the wellbore wall can be solved by utilizing Equation (10).

$$v_r=-{\displaystyle \frac{k}{\mu }}{\displaystyle \frac{\partial P}{\partial r}}$$

(10)

where k is the wellbore rock permeability, and $\mu$ is the apparent viscosity of the water-based fracturing fluid. Equation (10) demonstrates the relationship between the seepage velocity of the water-based fracturing fluid inside the wellbore, due to various factors, including fluid viscosity, permeability, and pressure difference [51,52]. This equation can be used to analyse the influence of fluid viscosity on seepage capacity at different wellbore depths and to perform mechanism analysis.

2.3. Fluid Flow Field of Water-Based Fracturing Fluid in a Wellbore of a Low-Permeability Reservoir

The injection of water-based fracturing fluid along the wellbore induces fluid flow in both the axial (z direction) and radial (r direction) directions. The axial flow propagates downward along the wellbore (Equation (11)) [53,54], while the radial flow results in fluid seepage through the wellbore wall (Equation (12)).

$${\rho }_f({\displaystyle \frac{\partial v_z}{\partial t}}+v_r{\displaystyle \frac{\partial v_z}{\partial r}}+v_z{\displaystyle \frac{\partial v_z}{\partial z}})=-{\displaystyle \frac{\partial P}{\partial r}}+\mu \left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}(r{\displaystyle \frac{\partial v_z}{\partial r}})+{\displaystyle \frac{{\partial }^2{v}_z}{\partial z^2}}\right]+{\rho }_{f}g\cos \theta$$

(11)

$${\rho }_f({\displaystyle \frac{\partial v_r}{\partial t}}+v_r{\displaystyle \frac{\partial v_r}{\partial r}}+v_z{\displaystyle \frac{\partial v_r}{\partial z}})=-{\displaystyle \frac{\partial P}{\partial r}}+\mu \left[{\displaystyle \frac{\partial }{\partial r}}({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial (r{v}_r)}{\partial r}})+{\displaystyle \frac{{\partial }^2{v}_r}{\partial z^2}}\right]$$

(12)

where g is the gravity acceleration; θ is considered to be the angle between the wellbore and the vertical direction; and P is the wellbore pressure. The horizontal well orientation, with an angle of 90° (θ = 90° (cosθ = 0)), is frequently observed in low-permeability reservoirs. This observation indicates that the gravitational effect of the water-based fracturing fluid in these reservoirs can be disregarded.

Furthermore, the flow of water-based fracturing fluid in a low-permeability reservoir wellbore must also take into account the wellbore friction, which is approximated by the Darcy–Weisbach formula (see Equation (13)) [55,56].

$$F_{\mathrm{friction}}=-{\displaystyle \frac{f{\rho }_f{v}_z\left|v_r\right|}{4r_w}}$$

(13)

where F_friction is the wellbore friction of the water-based fracturing fluid, and f is the friction factor.

2.4. Determination of the Physical Property Parameters of the Coupling Model

The relevant parameters of the coupling model of the water-based fracturing fluid in the wellbore in a low-permeability reservoir can be determined according to the following equations [57,58].

$${\rho }_f={\rho }_0\left[1+\beta (P-P_0)-\alpha (T-T_0)\right]$$

(14)

where ${\rho }_0$ is the water density; α and $\beta$ are the thermal expansion coefficient and compression coefficient of the water-based fracturing fluid; P and T are the wellbore pressure and wellbore temperature; and P₀ and T₀ are the initial pressure and initial temperature.

$$\mu ={\mu }_0{e}^{-a(T-T_0)}$$

(15)

where ${\mu }_0$ is the water viscosity, and a is the temperature correlation coefficient of the water-based fracturing fluid.

$$k_f=k_0(1+b(T-T_0))$$

(16)

where $k_f$ and $k_0$ are the thermal conductivities of the water-based fracturing fluid at different temperatures, and b is the temperature coefficient of the water-based fracturing fluid.

$$h={\displaystyle \frac{k_f}{2r_w}}{N}_u, N_u=0.023R{e}^{0.8}P{r}^{0.4}$$

(17)

$$Pr={\displaystyle \frac{\mu C_{p}f}{k_f}}$$

(18)

where N_u is the Nusselt number (Equation (17)); Pr is the Prandtl number, and Re is the Reynolds number. Equation (18) shows the detailed calculation method and influencing factors for the Prandtl number [59,60].

Furthermore, the friction factor of the water-based fracturing fluid flowing in the wellbore must also be reasonably analysed, as demonstrated in Equation (19) (laminar flow).

$$f={\displaystyle \frac{64}{Re}} , Re={\displaystyle \frac{{\rho }_f{v}_{z}2r_w}{\mu }}$$

(19)

However, a water-based fracturing fluid characterised by a relatively high flow rate has the capacity to generate a friction factor in a turbulent flow that is entirely distinct from a laminar flow (Equation (20)):

$${\displaystyle \frac{1}{\sqrt{f}}}=-2{\log }_{10}({\displaystyle \frac{\zeta /2r_w}{3.7}} +{\displaystyle \frac{2.51}{Re\sqrt{f}}})$$

(20)

where $\zeta$ is the wellbore roughness, which directly determines the friction resistance of the wellbore wall and affects the axial (depth direction) flow velocity $v_z$.

2.5. Model Geometry and Boundary Conditions

The reservoir depth in the water-based fracturing fluid geometric model for heat transfer in a low-permeability shale reservoir is set to 3500 m, and the wellbore length and diameter are set as 10 m and 0.25 m, respectively. Furthermore, the lateral area of the geometric model encompasses a shale reservoir with a diameter of 200 m, and the reservoir rock is regarded as an isotropic brittle material (Figure 1).

In the geometric model shown in Figure 1, the appropriate selections of boundary conditions and initial conditions are crucial for accurately simulating the behavior of the water-based fracturing fluid within the wellbore under the influence of the heat-transfer properties.

2.5.1. Temperature Boundary Conditions

Temperature differences and heat-transfer processes occur between the reservoir shale and the wellbore, as well as between the wellbore and the fracturing fluid [61,62]. To accurately characterize these thermal interactions, it is essential to define temperature boundaries while considering all the aforementioned materials as an integrated model. (Equation (21)).

$$h(T_f-T_w)=-k_g{\displaystyle \frac{\partial T}{\partial r}}{|}_{r=r_w}$$

(21)

where $T_w$ is the wall temperature; $T_f$ is the fluid temperature; and $k_g$ is the thermal conductivity of the reservoir shale.

2.5.2. Pressure Boundary Conditions

The pressure boundary condition at the wellhead is defined as the injection pressure, while the bottom-hole pressure is approximately equal to the pore pressure of the shale reservoir [63,64]. Additionally, the pressure within the wellbore is assumed to be equivalent to the pressure in the reservoir near the wellbore (Equation (21)).

$$P(r=r_w,z,t)=P_w(z,t)=P_0+{\rho }_{f}gz$$

(22)

where P is the wellbore pressure; P_w is the matrix pressure in shale reservoirs; P₀ is the initial pressure of shale reservoir at 3500 m; and ρ_f is the fluid density.

2.6. Model Decoupling, Solution, and Adaptability Validation of Investigation Methodology

2.6.1. Model Decoupling and Solution

The equivalent integral form of the continuity equation (mass conservation equation) (Equation (7)) for the water-based fracturing fluid in the wellbore can be written as follows [65]:

$${\displaystyle \frac{\partial }{\partial t}}{\displaystyle {\int }_{z_1}^{z_2}{\rho }_{f}dz+({\rho }_f{v}_z){|}_{z=z_2}-({\rho }_f{v}_z){|}_{z=z_1}+\frac{2}{r_w}}{\displaystyle {\int }_{z_1}^{z_2}({\rho }_f{v}_r){|}_{r=r_w}dz}=0$$

(23)

where z is the wellbore axial coordinate, and z₁ and z₂ are the wellbore axial critical values.

Subsequent derivation reveals that the weak integral form of the quality control equation is as presented in Equation (24).

$${\displaystyle {\int }_0^{H}w\frac{\partial {\rho }_f}{\partial t}dz-}{\displaystyle {\int }_0^H({\rho }_f{v}_z)\frac{\partial w}{\partial z}dz+}{\displaystyle {\int }_0^{H}w(\frac{2}{r_w}{\rho }_f{v}_r{|}_{r=r_w})dz}=0$$

(24)

In addition, the temperature control equation (Equation (6)) demonstrates temporal correlation. It is imperative to decouple the weak integral form of the temperature control equation in the time domain (Equation (25)) [66,67].

$$\begin{array}{l}{\displaystyle {\int }_0^{H}w{\rho }_f{C}_{pf}\frac{\partial T_f}{\partial t}dz+}{\displaystyle {\int }_0^H{\rho }_f{C}_{pf}{v}_z\frac{\partial T_f}{\partial z}wdz+}{\displaystyle {\int }_0^H{k}_f\frac{\partial T_f}{\partial z}\frac{\partial w}{\partial z}dz}\\ +{\displaystyle \frac{2h}{r_w}}{\displaystyle {\int }_0^{H}w{T}_{f}dz}={\displaystyle \frac{2h}{r_w}}{\displaystyle {\int }_0^{H}w{T}_{w}dz}\end{array}$$

(25)

where T_w is the well-wall temperature; r_w is the wellbore radius; and C_pf is the specific heat capacity of the water-based fracturing fluid.

The weak integral form of the seepage-field governing equation is shown in Equation (26):

$$\begin{array}{l}{\displaystyle {\int }_0^H{w}_v{\rho }_f\frac{\partial v_z}{\partial t}dz+}{\displaystyle {\int }_0^H{\rho }_f{w}_v{v}_z\frac{\partial w_v}{\partial z}dz-}{\displaystyle {\int }_0^{H}P\frac{\partial w_v}{\partial z}dz}+{\displaystyle {\int }_0^H{w}_v(\frac{f{\rho }_f{v}_z\left|v_z\right|}{4r_w})dz}\\ -{\displaystyle {\int }_0^H{w}_v({\rho }_{f}g\cos \theta )dz}=0\end{array}$$

(26)

where w_v is the test function for the momentum equation.

The aforementioned numerical framework establishes a filtration model capable of analyzing the behavior of water-based fracturing fluids within reservoir wellbores, enabling in-depth investigation and mechanistic interpretation of the effects of wellbore heat transfer on fluid mobility and filtration characteristics under various influencing factors. As illustrated in Figure 1, the numerical model considers a deep wellbore at a depth of 3500 m as the study object, with both the wellbore casing material and reservoir rock assumed to be homogeneous. Furthermore, the radial filtration domain of the water-based fracturing fluid is set at 200 m, which is sufficient to capture the full extent of fluid filtration during injection. The validity and reliability of the model depicted in Figure 1 are substantiated by the governing equations presented in Section 2 and the previous research data shown in Figure 2.

All of the aforementioned models necessitate the development of corresponding theoretical frameworks and assumptions based on the premisses of a homogeneous reservoir and a constant injection flow rate. However, actual reservoir formations often exhibit more complex and heterogeneous behaviors, which may introduce deviations or uncertainties into the numerical simulation results.

2.6.2. Adaptability Validation of Investigation Methodology

Figure 2 presents a comparative analysis of the filtration behavior in the numerical model developed in this study and that of a previous study. As shown in

Table 1. Sensitivity of different factors to reservoir wellbore heat transfer.

Factor Type	Factor Range	Fluid Viscosity (mPa·s)	Proportional Relationship	Sensitivity
Crack diameter	8 mm–18 mm	−32	Inverse	Strength
Reservoir pressure	10 MPa–15 MPa	−16	Inverse	Medium
Wellbore thickness	10 mm–15 mm	+11	Proportional	Medium

, the filtration behavior of the water-based fracturing fluid in shale reservoirs exhibits a similar trend in both studies. However, slight discrepancies are also observed between the two sets of results. The consistent trends indicate that the proposed numerical model is applicable for analyzing the filtration behavior of the water-based fracturing fluid within the wellbores of shale reservoirs [68,69]. The observed discrepancies are primarily attributed to the modification of the seepage-field control equation, which accounts for the influence of wellbore heat transfer on the temperature field—an aspect not considered in the previous study [70,71]. However, the study is lacking in experimental verification of the filtration behaviour of the water-based fracturing fluid in the reservoir wellbore under varying conditions, which necessitates further in-depth exploration following the construction of the experimental device.

3. Results and Discussion

The heat-transfer coefficient is an evaluation parameter which is critical in assessing the rheological behavior and filtration loss characteristics of water-based fracturing fluids in the context of wellbore and reservoir conditions. It has been extensively applied across various operational stages of reservoir stimulation. This study investigates the variation in heat-transfer coefficients under different wellbore and reservoir conditions and systematically analyzes the influences of multiple geological and engineering factors on the properties of the fluid.

3.1. Effects of the Wellbore Heat-Transfer Coefficient on the Filtration Behavior of Fracturing Fluid

As the water-based fracturing fluid is injected into the low-permeability shale reservoir, it undergoes a gradient temperature rise along the wellbore with increasing reservoir depth, which may cause significant changes in its physical and chemical properties. The wellbore heat-transfer coefficient h, determined by the material properties of the wellbore, is a critical parameter influencing both the reservoir temperature and the behavior of the water-based fracturing fluid [72]. This coefficient directly affects the fluid characteristics of the water-based fracturing fluid. The relationship between the wellbore heat-transfer coefficient h and various influencing factors related to the water-based fracturing fluid is described by Equation (6) [73,74].

The heat-transfer term in Equation (6) indicates that the temperature difference between the water-based fracturing fluid in the wellbore and the wellbore wall directly influences the seepage capacity of the fracturing fluid, with the heat-transfer coefficient being a key factor in determining the efficiency of heat exchange between these two temperatures [75,76]. Furthermore, variations in the wellbore heat-transfer coefficient alter the fluid temperature, which in turn affects the viscosity of the water-based fracturing fluid—a critical parameter governing its filtration into the reservoir rock. Figure 3 presents the influence curves for different wellbore thermal conductivities and the fluid viscosity and filtration behavior of the water-based fracturing fluid, directly validating the relationships among thermal conductivity, fluid viscosity, fluid temperature, and filtration rate, as described in the different forms of Equation (6). In Figure 3, the wellbore thermal conductivity and the viscosity of the water-based fracturing fluid exhibit a clear, inversely proportional relationship, in which the viscosity of the water-based fracturing fluid gradually decreases with increasing thermal conductivity [77,78]. Conversely, the filtration volume of the water-based fracturing fluid increases as the thermal conductivity rises, also demonstrating an inverse relationship with fluid viscosity. When the wellbore thermal conductivity is relatively low, the reduction in fluid viscosity occurs at a slower rate. However, a higher thermal conductivity accelerates the decrease in viscosity. Specifically, when the wellbore thermal conductivity reaches 1000 W/(m²·K), it marks a critical threshold for the change in viscosity. Beyond this point, an increase in thermal conductivity to values greater than 1000 W/(m²·K) results in a viscosity reduction of approximately 25 mPa·s, while thermal conductivity below 1000 W/(m²·K) only causes a viscosity decrease of about 10 mPa·s. Moreover, the wellbore thermal conductivity significantly influences the filtration volume of the water-based fracturing fluid, which can be divided into two distinct regions: In the low thermal conductivity region (below 1000 W/(m²·K)), the filtration volume increases slowly. For instance, raising the thermal conductivity from 800 W/(m²·K) to 1000 W/(m²·K) leads to only a 2 mL increase in fluid filtration. In the high thermal conductivity region (above 1000 W/(m²·K)), the filtration volume increases more rapidly. When the thermal conductivity rises from 1000 W/(m²·K) to 1200 W/(m²·K), the filtration volume increases by approximately 5 mL.

The effect of the wellbore heat-transfer coefficient on the viscosity and filtration behavior of the water-based fracturing fluid primarily depends on changes in the microscopic hydrogen bonds and molecular dynamics parameters under different heat-transfer conditions. This relationship can be quantitatively described by the seepage velocity equation for the water-based fracturing fluid (Equation (10)) [79]. A lower wellbore heat-transfer coefficient slows the transfer of heat from the deep low-permeability reservoir to the water-based fracturing fluid, resulting in a gradual increase in fluid temperature [80,81]. This slow temperature rise reduces the kinetic energy of fluid molecules, preventing large-scale disruption or destruction of intermolecular hydrogen bonds. As a result, the water-based fracturing fluid retains a higher viscosity on a macroscopic scale. Furthermore, the gradual temperature increase preserves the microscopic grid density of the fluid, contributing to a slower rate of viscosity reduction. Conversely, when the wellbore heat-transfer coefficient reaches 1000 W/(m²·K), the reservoir rock temperature is rapidly transferred to the water-based fracturing fluid, accelerating molecular motion. This increased molecular activity facilitates the breaking and stretching of intermolecular hydrogen bonds, leading to a rapid decrease in fluid viscosity. Additionally, under conditions associated with higher heat-transfer values, weakened hydrogen bonds are more likely to undergo shear deformation during molecular escape, further accelerating the reduction in fluid viscosity on a macroscopic scale.

Similarly, the wellbore heat-transfer coefficient influences the fluid filtration behavior by affecting the viscosity of the water-based fracturing fluid, a process primarily explained by the micro-grid theory. A lower wellbore heat-transfer coefficient transfers less heat to the water-based fracturing fluid, resulting in limited disruption of intermolecular hydrogen bonds and the formation of fewer free molecules. The reduced number of free molecules leads to lower fluid filtration into the reservoir rock through the pores around the wellbore. As the wellbore heat-transfer coefficient increases, the fluid filtration gradually rises due to enhanced infiltration into the reservoir rock. In contrast, a higher wellbore heat-transfer coefficient significantly increases the molecular activity within the water-based fracturing fluid, accelerating the breakage of intermolecular hydrogen bonds and the generation of free molecules. The greater number of free molecules not only reduces the fluid viscosity but also leads to a continuous increase in fluid filtration on a macroscopic scale. Therefore, selecting wellbore materials with a lower heat-transfer coefficient during oilfield fracturing operations can effectively mitigate the impact of reservoir temperature on the performance of water-based fracturing fluids, thereby improving fluid retention and reducing the fluid seepage.

To compare the sensitivity of various factors influencing heat transfer in the reservoir wellbore, Table 1 presents the effects of different fracture diameters, reservoir pressures, and wellbore wall thicknesses. It is important to note that the sensitivity of heat transfer, as indicated in Table 1, is evaluated using the change in fluid viscosity as a proxy parameter—in which a greater variation in viscosity corresponds to a higher heat-transfer capacity.

3.2. Effects of Wellbore Radius on Filtration Behavior of the Fracturing Fluid and Heat Conversion Efficiency

The wellbore geometry, including the wellbore radius, exerts a substantial influence on the heat exchange between the reservoir rock and the water-based fracturing fluid, the fluid parameters of the water-based fracturing fluid, and the fluid friction. The aforementioned parameters are illustrated in Figure 4.

Figure 4 illustrates the effects of different wellbore radii on the viscosity of the water-based fracturing fluid, which is closely related to the heat transfer between the reservoir rock and the fracturing fluid. A smaller wellbore radius leads to a rapid decrease in fluid viscosity. Specifically, increasing the wellbore radius from 0.08 m to 0.1 m results in a sharp reduction in fluid viscosity by 30 mPa·s, while further increasing the radius from 0.1 m to 0.15 m only causes a 10 mPa·s decrease. The relationship between the wellbore radius and heat transfer is a critical factor affecting the viscosity of the water-based fracturing fluid, as described by the heat exchange term in Equation (6). This equation demonstrates a clear inverse relationship between the amount of heat transfer and the wellbore radius (r_w), indicating that an increase in the wellbore radius inevitably reduces the efficiency of heat transfer from the reservoir rock to the fracturing fluid. Under a smaller wellbore radius, a higher heat-transfer rate is observed, allowing the fracturing fluid to absorb more heat and experience a rapid temperature rise. Conversely, a larger wellbore radius reduces the heat-transfer rate, thereby slowing the increase in fluid temperature and mitigating the reduction in fluid viscosity caused by heat transfer. Moreover, the wellbore radius also significantly influences the fluid filtration capacity, as described by Equation (10). This equation indicates a pronounced inverse relationship between the wellbore radius and the fluid filtration capacity. A smaller wellbore radius facilitates greater heat transfer from the reservoir rock to the fracturing fluid, increasing fluid temperature and enhancing molecular mobility, which in turn leads to higher fluid filtration. In contrast, a larger wellbore radius reduces the heat-transfer efficiency, resulting in a slower decline in fluid viscosity and a decrease in seepage velocity. Therefore, the heat-transfer mechanism driven by the wellbore radius is a key factor governing both the viscosity and the fluid filtration behavior of water-based fracturing fluids during reservoir stimulation.

The relationship between the wellbore radius and the filtration behavior of the water-based fracturing fluid is primarily governed by wellbore heat transfer and microscopic molecular interactions. According to the heat-transfer term in Equation (6), the wellbore radius is inversely proportional to the wellbore heat-transfer coefficient. This implies that a smaller wellbore radius facilitates greater heat transfer from the reservoir rock to the water-based fracturing fluid. Moreover, as the wellbore radius decreases, the temperature and internal energy of the fracturing fluid increase more rapidly, which significantly affects molecular interactions and kinetic parameters. Previous studies have indicated that the rheological properties and fracture propagation behavior of water-based fracturing fluid are primarily determined by intermolecular hydrogen bonds, with reservoir temperature being a critical factor influencing these rheological parameters [82]. According to the Arrhenius equation, minimal temperature changes at near-surface conditions result in negligible alterations to the fluid’s properties due to reduced molecular activity [83,84]. However, Equation (6) suggests that a smaller wellbore radius leads to enhanced heat transfer, thereby increasing the temperature of the fracturing fluid. As the fracturing fluid absorbs thermal energy from the reservoir rock, molecular motion intensifies, causing irregular molecular movement [85,86]. Simultaneously, the intermolecular repulsive forces increase as molecules gain thermal energy. This enhanced repulsion extends the length of the hydrogen bonds, weakening their bond energy. Furthermore, weaker hydrogen bonds may undergo breakage, resulting in a reduction in microscopic grid density, which manifests as a significant decrease in fluid viscosity on a macroscopic scale. Additionally, the disruption of intermolecular hydrogen bonds generates more free molecules, which can more easily penetrate the microscopic pores around the wellbore, thereby increasing the filtration volume of the fracturing fluid. Conversely, a larger wellbore radius results in reduced heat-transfer efficiency, limiting the temperature increase of the fracturing fluid. This prevents significant stretching or breaking of hydrogen bonds, reduces the formation of free molecules, and ultimately suppresses the filtration behavior of the water-based fracturing fluid [87]. Thus, a larger wellbore radius facilitates the stable flow of fracturing fluid along the wellbore by mitigating thermal-induced molecular disruptions.

Additionally, the frictional resistance of water-based fracturing fluid during wellbore flow, as described by Equation (13), further elucidates the relationship between the wellbore radius and the flow and filtration capacity of the fluid. A reduction in the wellbore radius inevitably increases fluid resistance, which significantly decelerates the flow rate of the water-based fracturing fluid within the wellbore [88,89]. As a result, the prolonged residence time of the fluid allows for greater seepage into the reservoir rock along the wellbore wall. Conversely, an increase in the wellbore radius reduces flow resistance and facilitates higher flow velocities. Under these conditions, the faster-moving fracturing fluid spends less time in the wellbore, thereby decreasing the extent of fluid infiltration into the reservoir rock [90]. Therefore, it can be posited that a moderate increase in the wellbore radius will help to achieve a reduced flow resistance in the water-based fracturing fluid. Furthermore, a diminished wellbore heat transfer will also reduce the influence of heat transfer on the filtration capacity and rheological parameters of the water-based fracturing fluid.

3.3. Effects of Wellbore Pressure Gradient on Filtration Behavior of the Fracturing Fluid and Heat Conversion Efficiency

In addition to the wellbore parameters that transfer heat to the fracturing fluid, various environmental conditions in the wellbore exert a significant influence on the filtration behavior and fluid properties of the water-based fracturing fluid. Among these, the wellbore pressure gradient, as a key physical factor, plays a crucial role in altering the characteristics of water-based fracturing fluid and directly impacts the fracturing efficiency of low-permeability reservoirs. Figure 5 illustrates the effects of different wellbore pressure gradients on the fluid viscosity, filtration capacity, and density of the water-based fracturing fluid. These variations also induce notable changes in the expansion behavior of reservoir fractures during fracturing operations.

The pressure gradient in the wellbore does not have a direct correlation with wellbore heat transfer; however, it significantly influences the viscosity of the water-based fracturing fluid. An inverse relationship is observed between the pressure gradient and fluid viscosity, whereby an increase in the pressure gradient leads to a more rapid decline in viscosity. Although a higher wellbore pressure gradient results in a notable reduction in fluid viscosity, the rate of decline exhibits two distinct stages. Specifically, when the pressure gradient reaches 12 MPa, the viscosity decreases by 4 mPa·s, whereas a pressure gradient of 14 MPa leads to a more substantial reduction of 12 mPa·s. In contrast, the wellbore pressure gradient demonstrates a direct, positive correlation with the filtration behavior of the water-based fracturing fluid, where an increase in pressure gradient significantly enhances filtration volume. A 12 MPa pressure gradient results in an increase in filtration volume of only 1.5 mL, whereas a 14 MPa pressure gradient leads to a rise in filtration volume of 5 mL. Figure 5 not only illustrates the relationship between pressure gradient and fluid properties but also highlights the pronounced inverse correlation between fluid viscosity and filtration volume under the influence of pressure gradient. Additionally, the wellbore pressure gradient impacts the fluid density of the water-based fracturing fluid, potentially establishing an inverse relationship between these two parameters. As the wellbore pressure gradient increases, a distinct downward trend in fluid density is observed, which can be further analyzed through intermolecular forces and molecular dynamics theory.

Previous studies have demonstrated that reservoir pressure plays a crucial role in enhancing the viscosity of water-based fracturing fluids. Lower reservoir pressures result in a lower viscosity of the fracturing fluid, whereas higher reservoir pressures significantly increase its viscosity [90,91]. The pressure gradient within the wellbore leads to continuous fluctuations in the pressure of the water-based fracturing fluid, which in turn causes dynamic changes in its viscosity [92]. As the wellbore pressure difference transitions from high to low, the viscosity of the fracturing fluid exhibits varying declining degrees. At the microscopic scale, a decrease in wellbore pressure alters the hydrogen bond parameters of individual fluid molecules, leading to significant changes in hydrogen bond length and bond energy. Higher wellbore pressure compresses intermolecular distances, resulting in stronger intermolecular hydrogen bonds with higher bond energy. Conversely, under lower wellbore pressure, the weaker intermolecular interactions lead to an increase in hydrogen bond length and a decrease in bond energy [93]. When the wellbore pressure gradient remains relatively small, the increase in hydrogen bond length is minimal, and the intermolecular hydrogen bonds remain largely intact. However, molecular spacing expands considerably (Figure 6 and

Table 2. Effect of reservoir pressure on the microscopic chemical bond parameters of the water-based fracturing fluid in the wellbore (n = 30).

Reservoir pressure/MPa	10	12	14	15
Bond length/nm	74	72	67	61
Bond energy/×10−3 J/mol	3.8	4.0	4.4	5.1

). A slight reduction in pressure may cause the cleavage of a small number of intermolecular hydrogen bonds, leading to a reduction in microscopic grid density [94]. The breakdown of a limited number of hydrogen bonds generates a small quantity of free molecules, which are subsequently forced into the wellbore wall pores under the influence of wellbore pressure, resulting in minimal fluid filtration. In contrast, a larger pressure gradient induces significant pressure variations within the wellbore, leading to a marked decrease in molecular spacing and substantial changes in the hydrogen bond parameters at the microscopic level. A considerable number of intermolecular hydrogen bonds break, forming a large number of free molecules, which, under elevated fluid pressure, further enhance fluid filtration. Moreover, the wellbore pressure gradient continuously reduces the external pressure acting on the water-based fracturing fluid, leading to a decline in fluid density. As the wellbore pressure gradient increases, the reduction in external pressure intensifies, amplifying the decrease in fluid density.

3.4. Effects of Wellbore Thickness on Filtration Behavior of the Fracturing Fluid and Crack Extension

The wellbore thickness is a pivotal factor in ensuring the stability of reservoir and production wells. It also has a crucial impact on the physical properties and heat-transfer performance of the water-based fracturing fluid in the wellbore. As illustrated in Figure 7, the impacts of varying the wellbore thickness on the fluid viscosity, fluid filtration, and fracture propagation performance of the water-based fracturing fluid are significant for reservoir protection and fracture propagation. There exists a clear positive correlation between wellbore wall thickness and the viscosity of the water-based fracturing fluids, indicating that increased wellbore thickness can effectively mitigate the influence of reservoir heat on the fluid and help maintain higher fluid viscosity. Notably, this relationship can be divided into two distinct regimes: when the wellbore thickness is less than 13 mm, the viscosity of the fracturing fluid increases slowly; whereas, once the thickness exceeds 13 mm, the rate of viscosity increase becomes more pronounced, indicating a steeper viscosity growth trend. The effect of wellbore wall thickness on fluid viscosity is primarily governed by changes in fluid temperature due to reservoir heat transfer and variations in the density of microscopic fluid grids. A thinner wellbore wall allows greater heat transfer from the reservoir to the fracturing fluid, resulting in elevated molecular kinetic energy. This energy increase weakens intermolecular hydrogen bonds, potentially stretching or disrupting them, which leads to a reduction in viscosity at the macroscopic scale. In contrast, a thicker wellbore wall substantially impedes heat transfer from the reservoir, thereby minimizing the increase in molecular kinetic energy within the fracturing fluid. As a result, hydrogen bonds remain largely intact and unaltered [95,96], preserving the microstructural integrity of the fluid grid. Consequently, the fracturing fluid retains a significantly higher viscosity under conditions of increased wellbore thickness. Therefore, the thickness of the wellbore mainly changes the fluid viscosity of the water-based fracturing fluid by changing the transfer of reservoir heat to the water-based fracturing fluid [97,98]. Appropriate wellbore thickness can not only effectively avoid the many changes in the properties of the water-based fracturing fluid caused by reservoir heat transfer, but also have a crucial impact on the filtration behavior of the water-based fracturing fluid.

As demonstrated in Figure 7, the impact of varying the wellbore thickness on the filtration volume of the water-based fracturing fluid is also illustrated. It is evident that there exists a highly significant inverse relationship between the wellbore thickness and the filtration volume of the water-based fracturing fluid [99,100]. As the wellbore thickness increases, the filtration volume of the water-based fracturing fluid gradually decreases, thereby facilitating the achievement of a weaker filtration capacity under a high mirror-wall thickness. Furthermore, it has been observed that a wellbore thickness of less than 13 mm results in a weaker downward trend in the filtration volume of the water-based fracturing fluid, while a wellbore thickness greater than 13 mm leads to a rapid decrease in fluid filtration volume. The inversely proportional relationship between wellbore thickness and fluid filtration volume is attributable to two primary factors: Firstly, an increase in wellbore thickness can prevent excessive reservoir heat from being transferred to the water-based fracturing fluid in the wellbore, which results in the microscopic grid density of the water-based fracturing fluid not being stretched or sheared. Consequently, the number of free molecules that can be released is reduced, and only a limited number of the free molecules that were not originally bonded seep into the deep reservoir. The reduction in well-wall thickness leads to an increase in reservoir heat exposure to the water-based fracturing fluid, thereby directly inducing the stretching or breaking of intermolecular hydrogen bonds [101,102,103]. The increased number of free molecules resulting from the fracturing process is known to enhance the filtration capacity of the water-based fracturing fluid. Furthermore, the presence of a thicker well-wall serves to impede the entry of free molecules into the reservoir. Conversely, the thinner well-wall reduces the difficulty involved in free molecules entering the reservoir, thereby promoting the seepage of more free molecules. It is therefore vital to select the optimum wellbore thickness, as this has a significant effect on the physical properties and filtration capacity of the water-based fracturing fluid. This, in turn, is directly related to the fracturing efficiency and reservoir damage capacity of low-permeability reservoirs.

Variations in fluid properties and heat-transfer efficiency induced by wellbore thickness exert a significant influence on the performance of water-based fracturing operations. In reservoirs with low levels of wellbore thickness, sealing techniques such as cement sheaths are essential to mitigate excessive fluid filtration due to filtration. Conversely, in reservoirs with high levels of wellbore thickness, adjustments to the formulation of the water-based fracturing fluid are required to minimize excessive frictional resistance, which may otherwise hinder fracturing effectiveness.

To verify the universality of the constructed numerical model,

Table 3. Filtration behavior of various fracturing fluids in a reservoir wellbore under different thermal stress conditions (×10⁻² m/min^1/2).

Fracturing fluid type	150 °C	160 °C	170 °C	180 °C
Water-based fracturing fluid	0.4	0.405	0.415	0.43
CO2-based fracturing fluid	0.63	0.65	0.68	0.74
Foam-based fracturing fluid	0.48	0.49	0.505	0.53

presents a comparative analysis of the filtration behaviors of various fracturing fluids within a reservoir wellbore under identical conditions. The data in Table 3 indicate that other fracturing fluids, such as CO₂-based and foam-based systems, can also be analyzed using this modeling approach to investigate fluid filtration behavior under thermal stress. These findings provide a theoretical foundation for optimizing fracturing fluid performance and improving operational efficiency in thermally stressed reservoir environments.

Furthermore, the results of the numerical simulations indicate that future research should prioritise the development of advanced wellbore materials to mitigate the parameter variations in water-based fracturing fluids induced by wellbore heat transfer. Concurrently, the design and experimental investigation of fluid filtration loss within the reservoir wellbore have also emerged as pivotal domains for future exploration.

4. Conclusions

This study established a wellbore heat-transfer model for water-based fracturing fluids, based on the geological characteristics and fluid parameters of low-permeability shale reservoirs. The model enables the investigation of variations in fluid properties induced by heat transfer from the reservoir through the wellbore. Furthermore, the combination of microgrid modeling and molecular dynamics theory was employed to elucidate the mechanisms underlying viscosity reduction and changes in the filtration behavior of the water-based fracturing fluid under thermal influences. The transfer of heat to the fracturing fluid is identified as a critical factor affecting viscosity and filtration capacity, wherein the disruption of intermolecular hydrogen bonds and the generation of free molecules serve as the fundamental microscopic drivers. Any factor that enhances the transfer of heat into the fluid promotes hydrogen bond breakage and free molecular formation, consequently leading to increased fluid leakage into the shale formation. In contrast, limited heat transfer minimizes these molecular changes, thereby reducing filtration efficacy. Moreover, the filtration behaviors of different fracturing fluids under wellbore heat transfer also verify the universality of the numerical model. Thus, investigating wellbore heat transfer is of considerable importance in addressing wellbore stability issues associated with excessive fluid filtration during hydraulic fracturing in low-permeability shale reservoirs.