Propagation Mechanism of Pressure Waves during Pulse Hydraulic Fracturing in Horizontal Wells

Abstract

Hydraulic fracturing, especially pulse hydraulic fracturing, is an important method for extracting oil and gas from low-permeability reservoirs, improving recovery rates significantly. Pulse hydraulic fracturing, which involves varying injection rates to create pressure waves, outperforms traditional constant-flow fracturing methods significantly. However, during pulse hydraulic fracturing operations, the flow properties of the fluid in the column change from moment to moment. Furthermore, current research on pulse hydraulic fracturing primarily focuses on vertical wells, while horizontal wells have become a common operational strategy. Therefore, a transient flow model of fluid within a horizontal well, considering variable-flow injection and unsteady friction conditions, is established in this paper. The model is solved using both the characteristic line method and the finite difference method. The hydrodynamic properties of the fracturing fluid were analyzed, and the propagation mechanisms of pressure waves within horizontal wells under various fluid injection schemes and well depths are analyzed to provide a reference for selecting appropriate fluid injection schemes in engineering practice. The study highlights the impact of fluid viscosity and injection flow amplitude on bottomhole pressure fluctuations, advancing the efficient development of low-permeability oilfields.

1. Introduction

With the continuous development of social economy, the demand for oil and gas resources has been increasing exponentially. However, the exploration of oil and gas resources has entered the middle and late stages, with increasingly high costs and difficulties. In particular, in recent years, low-permeability oilfields have accounted for more than half of the newly discovered reservoirs [1,2,3]. The reservoir environment of low-permeability reservoirs is complex, and traditional oil recovery methods have little effect; therefore, hydraulic fracturing technology is often used for reservoir transformation [4,5]. Hydraulic fracturing is a commonly used method for increasing production in low-permeability reservoirs. During the operation, high-pressure fracturing fluid is first injected into the wellbore, causing the rock in the reservoir to form cracks, making it easier for oil or natural gas to flow out of the reservoir [6,7,8].

In traditional hydraulic fracturing operations, multiple sets of fracturing pumps and pressure-resistant equipment are usually installed at the wellhead, allowing the fracturing fluid to be injected into the wellbore at a high pressure. However, due to the pressure-resistant equipment, the injection pressure at the wellhead cannot be too high, resulting in failure to achieve the expected fracturing effect. As a result, variable-flow hydraulic fracturing technology has emerged. Unlike traditional hydraulic fracturing technology, variable-flow hydraulic fracturing technology uses variable-flow circulation to inject fracturing fluid, generating fluctuating pressure at the bottom of the well, causing fatigue damage to the rock in the reservoir, and increasing the complexity of the cracks [9,10]. Thus, the wellhead injection pressure can be significantly reduced while ensuring the fracturing effect [11,12,13,14]. A comparison between constant-flow and variable-flow hydraulic fracturing is shown in Figure 1. Refs. [15,16,17,18,19,20] demonstrated in their 2019 study that the use of a variable-flow injection scheme could reduce the fracturing pressure by about 20%.

In variable-flow hydraulic fracturing operations, the length of the fracturing string typically extends several kilometers, and the key research questions revolve around the injection pressure of the fracturing fluid and the ability to generate fluctuating pressure at the well bottom. In traditional constant-flow hydraulic fracturing research, the pressure at the wellhead is essentially constant, and there are no significant fluctuations in the fluid flow velocity within the wellbore, which is considered steady-state flow [21]. The shear stress between the fracturing fluid and the pipe wall can be calculated using Darcy’s formula, and the pressure transmission characteristics within the wellbore are relatively simple. Liu et al. [22] revised the parameters in the formula of the drag reduction ratio method in 2010 and used the drag reduction ratio method to calculate the friction along the fracturing string. In variable-flow hydraulic fracturing, the velocity of the fracturing fluid within the wellbore is constantly changing, and it is not possible to calculate the shear stress based on steady-state flow; instead, the impact of non-constant friction must be considered. Among the existing research, non-constant friction models mainly include the Zielke weighted function model and the Brunone model [23,24]. Henclik [25] proposed a time-domain numerical solution algorithm based on the characteristic line method in 2018, which has been instrumental in solving transient fluid flow within the pipe column. In 2020, Tong and Gao [26] established a pressure fluctuation model in a vertical wellbore under variable-flow injection, considering the effects of fluid gravity and non-steady friction.

However, the research objects in these papers were all conventional vertical wells [27], which are generally perpendicular to the reservoir and have a relatively short length of reservoir penetration. Current common operational schemes mainly employ horizontal wells. A typical horizontal well is composed of three parts: a vertical or slightly inclined section drilled downward, known as the vertical section; after reaching the reservoir, the well inclination angle gradually changes to 90°, parallel to the reservoir, known as the build-up section; and a horizontal section that extends for a certain length, known as the horizontal section. Since conventional vertical wells only have a vertical section, there is a lack of research on the build-up and horizontal sections in horizontal wells.

Fluctuating pressure also exists in long-distance oil pipelines, and the MacCormack algorithm is commonly used for simulating transient flow within pipelines [28,29]. Moreover, research on pipeline leakage monitoring based on pressure fluctuations within the pipeline has matured [30,31], providing a reference for the study of pressure wave propagation in the stable section of horizontal wells.

In the second chapter of this paper, a transient flow model of fracturing fluids within horizontal well pipelines will be established. The third chapter will introduce two methods used to solve the model and verify the accuracy of these methods. The fourth chapter will analyze the variation characteristics of bottom hole pressure in horizontal wells under different fluid injection schemes and well depths. The fifth chapter will present some conclusions drawn from this study. This research is helpful for engineers to choose appropriate fluid injection schemes.

2. Fluid Transient Dynamics Model

In this study, a transient flow model for fracturing fluid in horizontal wells has been established, taking into account the influence of non-constant friction on pressure transmission. Figure 2 illustrates a variable-flow horizontal well.

The model is based on the following assumptions:

(I)

Axial flow only: it is assumed that the fracturing fluid exists only in the axial direction within the wellbore pipeline and does not have any radial flow.

(II)

Adiabatic flow: the flow of fracturing fluid in the wellbore pipeline does not involve any heat exchange with the external environment, indicating an adiabatic process.

(III)

Ignoring reservoir seepage effects: the influence of reservoir seepage on the fluid flow is neglected in this study.

(IV)

Homogeneous and compressible fluid: The fluid within the pipeline is assumed to be homogeneous and compressible, with the viscosity remaining constant. The non-Newtonian characteristics of the fluid are neglected.

(V)

Constant cross-sectional area and rigid wall: the cross-sectional area of the pipeline remains constant along its length, and the pipeline wall is considered to be rigid.

(VI)

Ignoring the impact of azimuth: the azimuth remains constant, and only the influence of wellbore inclination on pressure transmission is considered.

2.1. Continuity Equation

The fluid element shown in Figure 3 is taken at any position within the operating column; according to the principle of mass conservation, the difference in fluid mass flowing into the element from cross-section One and flowing out of the element from cross-section Two within a unit time dt is equal to the change in fluid mass within the element during the same time interval. Based on this, a continuity equation can be established.

The fluid mass flowing into the microelement from cross-section One can be calculated using the following formula:

$$m_1={\rho }_t{v}_t{A}_{t}dt,$$

(1)

The fluid mass flowing into the microelement from cross-section Two can be calculated using the following formula:

$$m_2=({\rho }_t+{\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }x}}dx)(v_t+{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}dx)A_{t}dt,$$

(2)

where ${\rho }_t$ is fluid density in element bodies, $v_t$ is velocity of axial fluid motion in element bodies, $A_t$ is cross-sectional area of working column, $t$ is time, $x$ is length of pipe.

The difference between the mass of fluid flowing into the element body from cross-section One and out of the element body from cross-section Two per unit time dt is

$$dm=m_1-m_2={\rho }_t{v}_t{A}_{t}dt=-{\displaystyle \frac{\mathsf{\partial }({\rho }_t{v}_t{A}_{t}dt)}{\mathsf{\partial }x}}dx,$$

(3)

The change in fluid mass within the element body per unit time dt is

$$dm={\displaystyle \frac{\mathsf{\partial }({\rho }_t{v}_t{A}_{t}dx)}{\mathsf{\partial }t}}dt,$$

(4)

This is easy to obtain according to the principle of conservation of mass:

$${\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}+{\displaystyle \frac{v_t}{{\rho }_t}}{\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }x}}+{\displaystyle \frac{1}{{\rho }_t}}{\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }t}}=0,$$

(5)

which is obtained according to the differential operator method:

$${\displaystyle \frac{d{\rho }_t}{dt}}={\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }x}}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{\mathsf{\partial }{\rho }_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{\rho }_t}{dx}},$$

(6)

$${\displaystyle \frac{d{p}_t}{dt}}={\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }x}}{\displaystyle \frac{dx}{dt}}={\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{p}_t}{dx}},$$

(7)

Substituting Equation (6) into Equation (5) gives

$${\displaystyle \frac{d{\rho }_t}{dt}}+\rho {\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}=0,$$

(8)

The fluid in the operational pipe column satisfies the linear elasticity eigen structure relationship, which is obtained as

$${\displaystyle \frac{d{\rho }_t}{{\rho }_t}}={\displaystyle \frac{d{p}_t}{K_t}},$$

(9)

where $K_t$ is the volumetric modulus of elasticity of the fluid in the operating column.

The functions that represent ${\rho }_t$ as $p_t$ are

$${\rho }_t={\rho }_0{e}^{{\displaystyle \frac{1}{K_t}}\left(p_t-p_0\right)},$$

(10)

where ${\rho }_0$ is fluid density at standard pressure $p_0$.

Taking the derivative of Equation (9) with respect to time t, and combining Equations (7) and (8), the continuity equation for the compressible fluid within the working pipe can be obtained:

$${\rho }_t{c}^2{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }x}}=0,$$

(11)

where $c$ is propagation velocity of pressure wave.

$$c=\sqrt{{\displaystyle \frac{K_t}{{\rho }_t}}},$$

(12)

2.2. Momentum Equation

In Figure 4, we show a situation of taking a fluid element within the working pipe at an arbitrary location, wherein the inclination angle of the wellbore is $\theta$. The forces acting on the fluid element include the pressure at both ends of cross-section One and cross-section Two, the self-weight of the fluid element, and the friction between the fluid element and the inner wall of the working pipe. When the force’s direction is the same as the fluid movement direction within the element, the force magnitude is positive; otherwise, the force magnitude is negative. Based on Newton’s second law, the motion equation for the compressible fluid within the working pipe could be established.

At cross-section One, the forces acting on the microelement are as follows:

$$F_1=p_t{A}_t,$$

(13)

At cross-section Two, the forces acting on the microelement are as follows:

$$F_2=(p_t+{\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }x}}dx)A_t,$$

(14)

The component of gravity force in the fluid flow direction for the microelement is

$$F_m={\rho }_{t}g{A}_{t}dx\sin {\theta }_x,$$

(15)

The friction force acting on the microelement due to the inner wall of the working pipe is

$$F_f={\tau }_t\pi D_{t}dx,$$

(16)

According to Newton’s second law, the motion equation for the compressible fluid within the working pipe can be obtained as

$${\rho }_t{A}_{t}dx({\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}})=F_1-F_2+F_m-F_f,$$

(17)

After simplification, the motion equation for the compressible fluid within the working pipe becomes

$${\rho }_t{A}_{t}dx({\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}})=p_t{A}_t-[p_t{A}_t+{\displaystyle \frac{\mathsf{\partial }(p_t{A}_t)}{\mathsf{\partial }x}}dx]-{\tau }_t\pi D_{t}dx+{\rho }_{t}g{A}_{t}dx\sin {\theta }_x,$$

(18)

where $p_t$ is the injection pressure within the microelement, ${\tau }_t$ is the shear stress between the fluid inside the microelement and the inner wall of the working pipe, $D_t$ is the inner diameter of the working pipe, $g$ is the gravitational acceleration, and ${\theta }_x$ is the angle between the axis of the pipe and the horizontal plane.

In the study of constant-flow hydraulic fracturing, the change rate of the fracturing fluid velocity inside the working pipe is not significant. Therefore, the shear stress between the fluid and the inner wall of the working pipe can be investigated under the assumption of steady-state flow. At this point, the shear stress between the fluid and the inner wall of the working pipe can be calculated using the Darcy formula.

The Darcy formula is as follows:

$${\tau }_t={\displaystyle \frac{{\rho }_t{v}_t\left|v_t\right|f_d}{8}},$$

(19)

where $f_d$ is the constant friction coefficient.

In constant-flow hydraulic fracturing, the rate of change in the actual fracturing fluid velocity is relatively small, and the error caused by simplifying the shear stress using the Darcy formula is acceptable. However, in variable-flow hydraulic fracturing, the injection rate of the fracturing fluid varies frequently, leading to rapid changes in the velocity. Continuing to use the Darcy formula to calculate the shear stress in this case would result in significant errors. Therefore, the influence of non-steady friction should be considered in variable-flow hydraulic fracturing operations.

At present, the main non-steady friction models include the Zielke weighted function model, the Brunone model, and modified models based on these two models. In Brunone’s modified model, the friction coefficient consists of two parts: the constant friction coefficient and the non-steady friction coefficient. This formulation facilitates subsequent programming. The non-steady friction coefficient considers the influence of instantaneous local acceleration and instantaneous convection acceleration on friction loss. According to the Brunone model, the corrected friction coefficient can be expressed as

$$f_c=f_d+{\displaystyle \frac{2k_t{D}_t}{v_t\left|v_t\right|}}\left({\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }t}}+c\cdot sign\left(v_t\right){\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}\right),$$

(20)

where $f_c$ is the corrected friction coefficient, and $k_t$ is the Brunone friction coefficient.

$$k_t={\displaystyle \frac{\sqrt{C^{\ast }}}{2}},$$

(21)

The flow state of the fluid is determined by the Reynolds number (Re):

$$\mathrm{Re}={\displaystyle \frac{{\rho }_t{v}_t{D}_t}{\mu }},$$

(22)

where $\mu$ is the dynamic viscosity of the fracturing fluid.

When the fluid is in a laminar flow state,

$$C^{\ast }=0.00476,$$

(23)

$$f_d={\displaystyle \frac{\mathrm{Re}}{64}},$$

(24)

When the fluid is in a turbulent flow state [32],

$$C^{\ast }={\displaystyle \frac{7.41}{{\mathrm{Re}}^{{\log }_{10}\left(\frac{14.3}{{\mathrm{Re}}^{0.05}}\right)}}},$$

(25)

$${\displaystyle \frac{1}{\sqrt{f_d}}}=-2\log \left[{\displaystyle \frac{{\delta }_1/D_t}{3.7}}-{\displaystyle \frac{4.518}{\mathrm{Re}}}\log \left[{\displaystyle \frac{6.9}{\mathrm{Re}}}+{\left({\displaystyle \frac{{\delta }_1/D_t}{3.7}}\right)}^{1.11}\right]\right],$$

(26)

where $C^{\ast }$ is an empirical parameter, and ${\delta }_1$ is the pipe wall roughness.

By substituting the corrected friction coefficient of $f_c$ into Equation (18), the motion equation for compressible fluid in the working pipe column considering non-uniform friction becomes

$${\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }t}}+v_t{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}+{\displaystyle \frac{1}{{\rho }_t}}{\displaystyle \frac{\mathsf{\partial }{p}_t}{\mathsf{\partial }x}}-g\sin {\theta }_x+{\displaystyle \frac{f_d{v}_t\left|v_t\right|}{2D_t}}+k_t\left[{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }t}}+c\cdot sign\left(v_t\right)\left|{\displaystyle \frac{\mathsf{\partial }{v}_t}{\mathsf{\partial }x}}\right|\right]=0,$$

(27)

2.3. Boundary Condition

a.

Entrance boundary condition

Variable-flow hydraulic fracturing necessitates precise control over the injection flow rate of fracturing fluid. Currently, electric hydraulic fracturing pumps and variable frequency control technology are capable of satisfying engineering requirements. The entrance boundary conditions of the working pipe column can be ascertained by the injection flow rate at the wellhead:

$${\left.v\right|}_{x=0}={\displaystyle \frac{4Q_t}{\pi D^2}},$$

(28)

where $Q_t$ is the injection flow rate at the wellhead.

b.

Exit boundary condition

In the present study, the influence of reservoir seepage at the bottom of the well is not considered. Therefore, the exit boundary condition can be expressed as

$${\left.v\right|}_{x=L_t}=0,$$

(29)

where $L_t$ is the total length of the working pipe column.

2.4. Initial Condition

Prior to the initiation of hydraulic fracturing, it is assumed that the pipe column is already filled with fracturing fluid, and there is no flow of the fluid within the pipe column.

$$\left\{\begin{array}{l}{\left.v_t\right|}_{t=0}=0\\ {\left.p_t\right|}_{t=0}=p_0+{\rho }_{t}g{z}_x\end{array}\right.,$$

(30)

where $p_0$ is the initial entrance pressure, and $z_x$ is the longitudinal depth of the pipe column.

When in the vertical section, $z_x$ is numerically the same as $x$. However, when in the build-up and horizontal sections, $z_x$ needs to be represented using a recursive algorithm.

$$z_x=z_{\left(x-1\right)}+\sin {\theta }_x,$$

(31)

3. Simulation Solution

In order to enhance the accuracy of the simulation results and verify the precision of the simulations, this chapter employs two numerical methods, the characteristic line method and the finite difference method, to solve the model.

First, we determine the space step: based on the length of the wellbore, we divide the space into grids with a spacing of 1m, such that Dx = 1.

Next, we determine the time step: to ensure simulation accuracy, we divide the time into grids based on the time it takes for a pressure wave to pass through one space grid, such that Dt = Dx/c.

3.1. Method of Characteristics

In the numerical simulation of transient fluid flow, the method of characteristics (MOCs) is frequently employed, as illustrated in Figure 5. In this representation, C₁ denotes the propagation of pressure waves from the upstream to the downstream direction, while C₂ indicates the opposite direction, from the downstream to the upstream. Further integration over space and time yields the following equation [33,34]:

Case 1:

$$v_P+{\displaystyle \frac{1}{\alpha \rho }}{p}_P=v_M+{\displaystyle \frac{1}{\alpha \rho }}{p}_M+g\sin {\theta }_x\Delta t-{\displaystyle \frac{v_M\left|v_M\right|f}{2D}}\Delta t-k_t\Delta t\left({\displaystyle \frac{v_M-v_{M_0}}{\Delta t}}-\alpha {\displaystyle \frac{v_{P_0}-v_M}{\Delta x}}\right),$$

(32)

Case 2:

$$v_P+{\displaystyle \frac{1}{\alpha \rho }}{p}_P=v_N+{\displaystyle \frac{1}{\alpha \rho }}{p}_N+g\sin {\theta }_x\Delta t-{\displaystyle \frac{v_N\left|v_N\right|f}{2D}}\Delta t-k_t\Delta t\left({\displaystyle \frac{v_N-v_{N_0}}{\Delta t}}-\alpha {\displaystyle \frac{v_N-v_{P_0}}{\Delta x}}\right),$$

(33)

3.2. Finite Difference Method

The partial derivative with respect to time can be expressed in the following form:

$${\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }t}}={\displaystyle \frac{S_j^i-\left[{\alpha }_1{S}_j^{i-1}+\frac{1-{\alpha }_1}{2}\left(S_{j+1}^{i-1}+S_{j-1}^{i-1}\right)\right]}{\Delta t}},$$

(34)

In the equation, S represents the fluid pressure or velocity within the pipeline, $S_j^i$ denotes the function value at time i and point j, and ${\alpha }_1$ is the weighting coefficient. Through multiple simulation calculations, it is found that the best result is obtained when ${\alpha }_1$ is set to 0.1 in the model established in this study, which exhibits good stability and convergence.

The partial derivative with respect to distance can be expressed in the following form:

$${\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }x}}={\displaystyle \frac{S_{j+1}^{i-1}-S_{j-1}^{i-1}}{2\Delta x}},$$

(35)

By organizing the finite difference format and incorporating it into Equations (5), (11) and (27), along with the initial and boundary conditions of the pipe column, the flow state of the proppant within the column can be computed.

4. Results and Discussion

This section primarily focuses on validating the simulation accuracy of two numerical methods, the characteristic line method and the finite difference method, and analyzes the influence of various injection schemes on the pressure fluctuations at the bottom of the wellbore. The injection schemes considered include the following aspects: injection flow rate amplitude, injection flow rate waveform, injection flow rate mean value, and the dynamic viscosity of the fracturing fluid. In the simulation model employed within this study, the primary parameters selected for the oil well and construction are presented in

Table 1. Main oil well and construction parameters.

Parameter	Value
$\rho$/(kg/m3)	1050
$\mu$/(Pa·s)	0.01
$D$/mm	89
$L_v$/m	1500
$L_h$/m	1500
${\beta }_v$/rad	$\pi /2$
${\beta }_h$/rad	0
$P_0$/Mpa	2
$Q$/(m3/min)	2.8

.

Where $\rho$ is the fracturing fluid density, $\mu$ is the fracturing fluid dynamic viscosity, $D$ is the inside diameter of the pipe column, $L_v$ is the length of the pipe column’s vertical section, $L_h$ is the length of the pipe column’s horizontal section, ${\beta }_v$ is the average deviation angle of the pipe column’s vertical section, ${\beta }_h$ is the average deviation angle of the pipe column’s horizontal section, $P_0$ is the initial pressure, and $Q$ is the mean injection flow rate.

4.1. Accuracy Verification

For the sake of the computation, it is assumed that the fracture fluid flow rate at the wellbore is injected in a sinusoidal manner. The horizontal well consists of a 1500 m vertical section and a 1500 m horizontal section. At time zero, the fracture fluid has already filled the entire pipe column. The problem is solved using the characteristic line method and the finite difference method, and the results of the two solution approaches are compared. The comparison results are shown in Figure 6.

The two methods yield nearly identical solutions for pressure transmission, with differences remaining within 1%. In terms of solution speed, the characteristic line method is faster than the finite difference method. However, due to the omission of the convection term in the solution process of the characteristic line method, its accuracy is inferior to the finite difference method when there is significant convection in the fracturing fluid within the pipe column. In the subsequent calculations, the finite difference method will continue to be used to study the problem.

As illustrated in Figure 7, when the sinusoidal waveform method is employed for fluctuating injections of the fracturing fluid, the wellhead pressure exhibits an immediate approximate sinusoidal variation. Due to the time required for pressure wave transmission, pressure fluctuations throughout the pipe column exhibit a certain degree of lag, with the lag time being directly proportional to the distance. As the distance increases, the amplitude of the pressure fluctuations decreases gradually due to the influence of friction losses along the pipe.

4.2. Analysis of Fracturing Fluid Pressure and Velocity

In this section, it is assumed that the fracturing fluid flow rate at the wellhead varies in a sinusoidal pattern. Simulation calculations are performed using the finite difference method, resulting in the pressure variation with time and well depth presented in Figure 8, and the variation in velocity with time and well depth shown in Figure 9.

As illustrated in Figure 8, at t = 0, the pressure in the vertical section increases with increasing well depth, while the pressure in the horizontal section remains constant due to the constant wellbore depth. Ignoring the influence of reservoir seepage at the bottom of the wellbore, the pressure throughout the wellbore increases with time. Owing to the time required for pressure wave transmission, the pressure fluctuations at the bottom of the well exhibit different degrees of lag compared to the wellhead, with the lag time being proportional to the distance from the wellhead. During the transmission process, the fracturing fluid is influenced by friction with the inner wall of the pipe column, resulting in a gradual decrease in pressure along the transmission path.

As illustrated in Figure 9, the variation in fracturing fluid velocity at the wellbore mouth is identical to the change in injection flow rate. Ignoring the impact of leakage at the bottom of the pipe column, the velocity at the downhole is zero. Although there are fluctuations in the velocity with increasing well depth, the overall trend is downward.

4.3. Influence of Wellbore Section Length

4.3.1. Influence of Horizontal Section Length

In order to investigate the influence of horizontal section length on pressure fluctuations at the downhole, a constant liquid injection scheme and vertical section length of the well tubing are maintained, while the horizontal section lengths are varied to 1000 m, 1500 m, and 2000 m.

As evident from the simulation results in Figure 10, the downhole pressure and amplitude of pressure fluctuations gradually decrease with the increase in the horizontal section length of the well tubing. For every 500 m increase in the length of the horizontal section, the downhole pressure decreases by approximately 5 MPa, and the amplitude of pressure fluctuations reduces by 15% to 25%.

4.3.2. Influence of Vertical Section Length

In order to investigate the influence of vertical section length on pressure fluctuations at the downhole, a constant liquid injection scheme and horizontal section length of the well tubing are maintained, while the vertical section lengths are varied to 1000 m, 1500 m, and 2000 m.

As depicted from the simulation results in Figure 11, with the increase in the vertical section length of the well tubing, the initial value of downhole pressure gradually increases, the amplitude of pressure fluctuations at the downhole gradually decreases, and the rate of increase in downhole pressure slows down. With the passage of time, shorter vertical sections are more susceptible to generating higher downhole pressures. For every additional 500 m in the vertical segment length, the initial value of the downhole pressure increases by approximately 5.145 MPa, and the amplitude of pressure fluctuations decreases by 10–20%.

4.3.3. Analysis of Purely Vertical and Purely Horizontal Conditions

In this section, we assume the wellbore to be purely horizontal and purely vertical, directly comparing the pressure transmission patterns between the two. Assuming both wellbores are 3000 m long and other conditions remain unchanged, the simulation results are shown in Figure 12. As is evident from Figure 12, under the influence of gravity, pressure attenuates faster in a purely horizontal wellbore, with the average fluctuation amplitude of pressure reduced by about 13% compared to a purely vertical wellbore and the maximum fluctuation amplitude reduced by about 17%.

4.4. Influence of Fluid-Injection Scheme

4.4.1. Influence of Injection Flow Rate Amplitude

When injecting fracturing fluid using a sinusoidal waveform, the mean injection flow rate of the fracturing fluid is kept constant, while the amplitude varies at 30%, 50%, and 80% of the mean value. The calculation results are presented in Figure 13. The simulation demonstrates that when the mean injection flow rate remains constant, the greater the amplitude variation in the injection flow rate, the more significant the change in downhole pressure, and the better the actual fracturing effect.

4.4.2. Influence of Injection Flow Rate Waveform

In variable-flow hydraulic fracturing operations, there are differences in the manner of injecting fracturing fluid. This sub-section will discuss three representative injection waveforms, namely rectangular wave injection, sinusoidal wave injection, and sawtooth wave injection. Figure 14 depicts the entrance flow rate waveforms of the three different injection schemes, while Figure 15 shows the simulation results of the three distinct injection waveforms.

It is noteworthy that when fracturing fluid is injected in the form of rectangular waves, the abrupt changes in injection lead to corresponding pressure transient responses at the downhole. The rectangular waves, compared to sinusoidal and sawtooth waves, are capable of generating greater pressure fluctuations at the downhole, resulting in more effective fracturing.

4.4.3. Influence of Injected Flow Rate Average Value

Under other constant parameters, the impact of varying injected flow rate mean values on the simulation results was examined. The mean values of injected flow rate were set at 2 m³/min, 3 m³/min, 4 m³/min, and 6 m³/min. The simulation results from Figure 16 reveal that the downhole pressure increases with the increase in the mean value of the injected flow rate. As the mean value of injected flow rate increases, the velocity of fracturing fluid in the pipe column also increases, resulting in increased pressure losses along the pipeline. Consequently, although the downhole pressure increases with the injected flow rate mean value, the fluctuation in downhole pressure decreases.

A further analysis was conducted on the average injection flow rate, with the average injection flow rate as the variable, simulating the amplitude of downhole pressure fluctuations at depths of 1000 m and 3000 m, respectively. The results indicate that a higher average flow rate is not always better, but is related to the well depth, and there exists an optimal solution. In Figure 17a,b, taking wells at depths of 1000 m and 3000 m as examples, it is evident that in the 1000 m deep well, with 10 m³/min as the threshold, the fluctuation amplitude decreases as the average injection flow rate increases. Therefore, we can conclude that the optimal average injection flow rate for a 1000 m deep well is approximately 10 m³/min; similarly, in the 3000 m deep well, this optimal solution occurs earlier, at 3 m³/min.

4.4.4. Influence of Fracturing Fluid Dynamic Viscosity

In this sub-section, the dynamic viscosity of the fracturing fluid was varied and analyzed. The selected dynamic viscosity values were 0.01 Pa·s, 0.05 Pa·s, and 0.10 Pa·s. The simulation results from Figure 18 demonstrate that as the dynamic viscosity of the fracturing fluid increases, the amplitude of downhole pressure fluctuations decreases, indicating a decline in fracturing effectiveness.

4.4.5. Influence of Variation Frequency

In this section, the impact of injection frequency on the amplitude of downhole pressure fluctuations will be analyzed. With the injection frequency as the variable, simulations were conducted on the amplitude of downhole pressure fluctuations at depths of 1000 m and 3000 m. The simulation results are shown in Figure 19a,b. From the simulation results, it can be observed that when the well depth is smaller, the amplitude of downhole pressure fluctuations exhibits a fluctuating state, which is particularly evident in Figure 19a. The analysis suggests that the reason for this is the proximity of the injection frequency to the natural frequency of the wellbore. Therefore, it can be concluded that, similar to the average injection flow rate, there is an optimal solution for the injection frequency, and this optimal solution is related to the natural frequency of the wellbore.

In conclusion, the friction loss between the fracturing fluid and the inner wall of the pipe plays a dominant role in affecting the downhole pressure fluctuations in the variable-flow-rate hydraulic fracturing of horizontal wells. Therefore, exploring methods to reduce the friction loss between the fracturing fluid and the inner wall of the pipe should be one of the important research directions for future variable-flow-rate hydraulic fracturing.

5. Conclusions

In this paper, a transient flow model for variable-flow-rate injection in horizontal wells was developed while considering the impact of unsteady friction. Based on this model, the effects of wellbore section lengths and liquid injection schemes on downhole pressure fluctuations were analyzed. The results of the analyses are as follows:

The vertical section length of the wellbore affects the magnitude of the downhole pressure and also has an impact on the amplitude of downhole pressure fluctuations. The horizontal section length, however, only affects the amplitude of downhole pressure fluctuations. Furthermore, the attenuation rate of pressure waves is faster in pure horizontal wells, with the average fluctuation amplitude of pressure reduced by about 13% compared to pure vertical wells.

The viscosity of the fracturing fluid and the amplitude of the injection flow rate have a significant impact on the amplitude of downhole pressure fluctuations. To improve the fracturing effectiveness, it is recommended to minimize the viscosity of the fracturing fluid and maximize the amplitude of the injection flow rate in engineering practice. Utilizing rectangular wave injection is more prone to induce fatigue failure in reservoir rocks.

The influence of the mean injection flow rate and injection frequency on downhole pressure fluctuations is not linear, but rather, there exists an optimal solution that is related to the wellbore parameters. This optimal solution can be calculated in advance before operation, thereby improving the operational efficiency.