A Fractional Step Method to Solve Productivity Model of Horizontal Wells Based on Heterogeneous Structure of Fracture Network

Abstract

The existing productivity models of staged fractured horizontal wells in tight oil reservoir are mainly linear flow models based on the idealized dual-medium fracture network structure, which have a certain limitation when applied to the production prediction. Aiming at the difficulty in describing the shape of the complex fractal fracture network, a two-dimensional heterogeneous structure model of the fracture network is proposed in this paper. Considering the deformation characteristics of porous media and the characteristic of non-Darcy fluid flow, a three-zone steady-state productivity model with the combination of radial and linear flow is established. To eliminate strong nonlinear characteristics of the mathematical model, a fractional step method is employed to deduce the production formulas of staged fractured horizontal wells under infinite and finite conductivity fractures. The established productivity model is verified with the actual data of three horizontal wells in different blocks of S oilfield, and the error between the model calculation results and the actual production data is less than 4%. The analysis results of productivity sensitive factors show that production of horizontal wells is primarily influenced by the reservoir physical properties and fracturing parameters. The steady-state productivity model established in this study can be applied to effectively predict the average production of a horizontal well in stable stage of production, and it has theoretical and practical application value for improving the development effect of tight oil reservoir.

1. Introduction

With the exploitation of conventional oil resources entering the middle and late stage, numerous scholars have begun to pay attention to unconventional oil and gas resources. As a member of unconventional petroleum resources, tight oil has the characteristics of low permeability, poor reservoir properties, and strong heterogeneity, of which the development issues have been a hot topic discussed in the global petroleum industry in recent years.

The successful exploitation of tight oil in North America by applying horizontal well hydraulic fracturing technology has attracted extensive attention from global scholars. Horizontal wells volume fracturing technology is considered the most effective method to develop tight oil reservoirs. Deliverability evaluation research on staged multi-cluster fractured horizontal wells is an important prerequisite for optimizing fracturing parameters and guiding the effective exploitation of tight oil reservoirs. In addition to the conventional numerical simulation method [1,2,3,4,5] and the analytical and semi-analytical methods [6,7,8,9,10], there are also some special analysis methods [11,12,13,14,15] for productivity evaluation.

In the study on deliverability evaluation of staged fractured horizontal wells, many scholars [16,17,18,19,20,21,22,23] primarily established simple productivity prediction models targeting hydraulic fractures and surrounding matrix areas without considering the complex fracture network reconstruction area around hydraulic fractures. With the widespread application of volumetric fracturing technology in horizontal wells, to accurately predict the production performance of fractured horizontal wells, numerous experts [24,25,26,27,28] have conducted plenty of studies on the productivity model for staged fractured horizontal wells aiming at the complex fracture network. Considering the impact of threshold pressure gradient and pressure-sensitive effect on reservoir matrix permeability, Wang Zhiyuan et al. [29] employed the material balance dynamic zoning method and proposed a deliverability evaluation model of fractured horizontal wells in tight reservoirs with matrix–natural fracture–artificial fracture coupling seepage. This model adopted homogeneous grids to divide the matrix and fracture system. Given the effects of imbibition on oil extraction, Su Yuliang et al. [30] established a production prediction model for staged fractured horizontal wells coupled with imbibition based on the seven-zone composite seepage model. The imbibition and fracture network reconstruction area were both idealized dual-porous medium structures. Considering the coupling of micro-fractures and matrix in the fractured stimulated area, Yuan Hongfei [31] respectively established unsteady productivity models of fractured horizontal wells under the conditions of equal-length and unequal-length primary fractures. Liu Yunfeng [32] proposed a multi-regional steady-state productivity model of horizontal wells by applying conformal transformation. The primary fracture in this model was surrounded by an elliptical seepage region with evenly distributed fracture network. In the above research, although the fracture network formed around the primary fracture of fractured horizontal wells was considered, some models lacked the morphological description of the complex fracture network, and the other models regarded the complex fracture network area as an idealized homogeneous dual-porosity medium structure. In fact, it is impossible to form a homogeneous fracture network after the primary fracture connecting with the natural fractures in the reservoir. Therefore, the production capacity prediction models above based on the idealized double porosity medium structure have certain limitations when applied to productivity prediction of staged fractured horizontal wells in tight oil reservoirs.

For the sake of breaking the limitations of idealized fracture network structure applied to productivity prediction, some scholars have conducted further research on complex fracture networks. Wang Wendong et al. [33] described the fracture network as fractal fracture in the established trilinear flow model, which indicates the heterogeneity of the fracture network, but the specific heterogeneous structure of the fracture network is still unclear. Jiang Ruizhong et al. [34] assumed the fracture network as a fractal reservoir and established a nonlinear seepage flow model for a dual-porosity fractal reservoir. Although the model introduced the fractal principle to describe the heterogeneity of the reservoir, it lacked the division of the reservoir seepage area and the description of the complex fracture network. Xie Bin et al. [35] established a three-zone seepage mathematical model of tight oil reservoirs based on the fractal dual media model and obtained the analytical solution of the model. The fractal medium was adopted in the model to describe the fracture network reconstruction area and reservoir boundary regions for characterizing the heterogeneity of the complex fracture network. However, the specific distribution of the complex fracture network was not demonstrated in the model. Li Xianwen et al. [36] established a five-zone seepage model for staged fractured horizontal wells in tight oil reservoirs based on the fracture network morphology obtained by inversion of microseismic data. Liu Siyu et al. [37] established a trilinear flow model with a non-uniform fracture network structure. This model is a step further than previous studies in describing the shape of a complex fracture network, but the specific non-uniform structure of fracture network should be further discussed. Overall, the existing studies have limitations in understanding and describing the complex fracture network of fractured horizontal wells. Meanwhile, the solution of mathematic model is complicated to a certain extent. Therefore, the deliverability evaluation method of staged fractured horizontal wells still requires further study.

In this study, we proposed a two-dimensional heterogeneous fracture network structure model, in which the secondary fractures are unevenly distributed along both directions of the horizontal wellbore and primary fractures. The fractal theory was adopted to characterize the two-dimensional heterogeneous fracture network. Different from the conventional linear and elliptical flow model, in this study, the hydraulic fracture was considered to be surrounded by an approximately elliptical seepage region composed of cuboid and elliptical reservoirs. Therefore, a three-zone seepage physical model with the combination of plane radial and bilinear flow based on the heterogeneous fracture network structure was proposed. The corresponding mathematical model, considering the deformation characteristics of porous medium and the characteristic of non-Darcy fluid flow, was established. For eliminating the strong nonlinear characteristics of the equations caused by the medium deformation characteristics, we applied a fractional step calculation method to simplify the fracture permeability, which greatly reduces the difficulty of solving the mathematical equations and is innovative in the algorithm. The productivity formulas of staged fractured horizontal wells under the conditions of infinite and finite conductivity fractures were obtained, respectively, through the calculation method. Finally, we adopted the basic parameters and actual production data of three oil wells in different blocks of the S oilfield to verify the reliability of the established model and analyzed the influences of reservoir and fracturing sensitive factors on the production of horizontal wells.

2. Methodology

2.1. Physical Model

2.1.1. Heterogeneous Fracture Network Structure Model

The primary fractures produced by the fracturing of horizontal wells are connected with the natural fractures in the oil reservoir to form secondary fractures, thus generating a complex fracture network around the hydraulic fractures. In previous productivity evaluation models, the complex fracture network was mostly assumed to be an idealized homogeneous double porosity structure with the matrix rock block uniformly divided by secondary fractures, which is inconsistent with the actual heterogeneous fracture network structure. Based on the research of Liu Siyu et al. [37] proposing a non-uniform distribution structure of fracture network and Sheng Guanglong et al. [38] obtaining the fracture network morphology by means of micro-seismic data inversion, a two-dimensional heterogeneous structure model of the fracture network was proposed in this study. The plane figure of this fracture network model was depicted in Figure 1.

The matrix rock block is unevenly divided by secondary fractures in the heterogeneous fracture network structure model. The closer the secondary fractures are to the position of the primary fracture and the horizontal wellbore, the denser the distribution of the fracture network. The farther away from the primary fracture and the horizontal wellbore, the smaller the number of secondary fractures. In the actual fracturing process of horizontal wells, the formation near the horizontal wellbore is first stimulated to produce hydraulic fractures, and the hydraulic fractures tend to connect with the nearby natural fractures to generate secondary fractures and then extend to the distance. Therefore, the two-dimensional heterogeneous structure model of the fracture network with secondary fractures unevenly distributed along both directions of the horizontal wellbore and primary fractures can reflect the actual fracture network shape, to a certain extent.

2.1.2. The Steady-State Physical Model of Fractured Horizontal Well

Multiple clusters of fractures are generated after the volume fracturing of horizontal wells. The seepage physical model of staged fractured horizontal wells was established in Figure 2. We divided the physical model into three areas, including the primary fracture seepage area, the cuboid seepage area, and the elliptical seepage zone.

Each hydraulic fracture constitutes the same seepage environment, and there is no interference between these fractures. The two-dimensional plan of fluid flow in a single fracture and its surrounding reservoir is shown in Figure 3. Viewing a quarter of the oil drainage area controlled by a single primary fracture as the research object, the reservoir was divided into three areas, namely region I, region II, and region III. Region I is a seepage area of the primary fracture. Region II is a reconstruction volume area of the fracture network, which is a cuboid reservoir composed of secondary fractures and matrix rock blocks. The fracture network in this area is heterogeneous and non-uniformly distributed along the $xoy$ plane. Region III is an unstimulated volume area at the reservoir boundary composed of matrix rock blocks, which is a 1/4 elliptical reservoir. Fluid in this area behaves as non-Darcy flow, which is characterized by the threshold pressure gradient. The fluid seepage process in our model can be divided into two stages. The first stage is the linear flow of the fluid in region II perpendicular to the direction of the primary fracture, and the pseudo radial flow of the fluid in region III into region I through the tip of the primary fracture. The second stage is the linear flow of the fluid in region I into the horizontal wellbore along the direction of the primary fracture.

The assumptions of the steady-state productivity seepage physical model for fractured horizontal wells can be listed as follows: (1) The upper and lower boundaries of the reservoir are closed. The half-width and the thickness of the reservoir are, respectively, $y_e$ and $h$. The horizontal wellbore passes through the center of the reservoir. (2) Primary fractures are symmetrically distributed perpendicular to the horizontal wellbore, of which the height equals to the reservoir thickness, the width is $w$, and the half-length is $y_f$. (3) The half-distance between adjacent primary fractures is $x_e$. (4) The fluid in the reservoir and fractures is single-phase isothermal stable seepage, without considering the influences of gravity and capillary forces. (5) Porous media and fluids are both incompressible. (6) Region II is a fractal reservoir with fractal dimension $D$, and the fracture network is embedded in a two-dimensional Euclidean rock block. (7) The pressure loss in the horizontal well section is not considered. (8) The model considers the deformation characteristics of porous media. (9) The effect of the threshold pressure gradient is considered in the reservoir. (10) The bottom hole pressure of the horizontal well is constant.

2.2. Mathematical Model and the Solution

2.2.1. Consideration of the Threshold Pressure Gradient

2.2.1.1. Primary Fractures of Infinite Conductivity

There is no pressure drop in the primary fracture because of the infinite conductivity. Therefore, the pressure in Region I is uniformly distributed, and its value equals to the bottom hole pressure $p_w$.

(1)

Fracture network stimulated volume area (Region II)

Region II is a rectangular reservoir composed of a dual-porosity medium with the matrix and fracture network. Fluid flows vertically into the primary fracture from the fracture system along the opposite direction of the $x$-axis. The fluid seepage law in this area is very complicated. The fracture network in this area is unevenly distributed along the horizontal wellbore and the primary fracture. The closer to the horizontal wellbore and the hydraulic fracture, the greater the density of secondary fractures, thus leading to the non-uniform change of fluid flow capacity in this region. To describe the non-uniform variation of the fluid flow capacity in the region, the fractal theory was employed to characterize the permeability of the fracture network. Next, we established and solved the seepage mathematical model in the volume area of fracture network stimulation.

It is assumed that the fluid storage space in the fractal dual-medium is at the node with the unit volume of $V_s$, and each node has the same volume. The radial distance between the node and the horizontal wellbore is $r$, and the node density is $N\left(r\right)$. The relationship between $r$ and $N\left(r\right)$ is as follows [39].

$$N(r)=\alpha \cdot r^{D-1}$$

(1)

where $\alpha$—A parameter related to fracture porosity, $D$—The fractal dimension, an important symbol of the complexity of the fractal body, which reflects the geometric characteristics of the fractal.

Fracture fractal porosity was defined as ${\varphi }_2$ and M was applied to represent the corresponding symmetry relationship. We could obtain the equation at $\left(r+dr\right)$ as follows [40]:

$${\varphi }_2=\frac{V_{\varphi }}{V}=\frac{V_s\cdot N(r)dr}{M{r}^{d-1}dr}$$

(2)

where $V_{\varphi }$ is pore volume, and V is the volume of the whole medium.

Equation (1) was substituted into Equation (2), then the equation was obtained as follows:

$${\varphi }_2=\frac{V_s\cdot \alpha \cdot r^{D-1}dr}{M{r}^{d-1}dr}$$

(3)

Rearranging the above formula, we could get:

$${\varphi }_2=\frac{V_s\cdot \alpha }{M}{r}^{D-d}$$

(4)

where $d$—The dimension of the Euclidean space embedded by the fractal, and the value of $d$ in two-dimensional space is 2.

The fracture fractal permeability $k_2$ and $r$ obey the relationship of a power-law as follows [41]:

$$k_2\propto r^{-\theta }$$

(5)

In above equation, $\theta$ is abnormal diffusion coefficient, indicating the connectivity of the fractal network (the smaller value of $\theta$, the better connectivity of the network).

In Euclidean geometry, the relationship between the fracture permeability and the fracture porosity is as follows [41]:

$$k_2=m{\varphi }_2$$

(6)

where $m$—A parameter related to fracture model.

Substituting Equation (4) into Equation (6), the equation was obtained as follows:

$$k_2=\frac{m{V}_s\cdot \alpha }{M}{r}^{D-d}$$

(7)

Extending the above formula to the general fractal system, the expression of the fracture network fractal permeability was given by

$$k_2=\frac{m{V}_s\cdot \alpha }{M}{r}^{D-d-\theta }$$

(8)

when $r=r_w$, we assumed $k_2=k_{2w}$, then we got the following equation:

$$k_{2w}=\frac{m{V}_s\cdot \alpha }{M}{r}_w{}^{D-d-\theta }$$

(9)

Combining Equations (8) and (9), the fracture fractal permeability was obtained:

$$k_2=k_{2w}{\left(\frac{r}{r_w}\right)}^{D-2-\theta }$$

(10)

when the above equation was extended to region II, the expression of fracture fractal permeability should be written as

$$k_2=k_{20}{\left(\frac{y}{r_w}\right)}^{D-2-\theta }{\left(\frac{x}{w/2}\right)}^{D-2-\theta }$$

(11)

In the above formula, $r_w$ is the horizontal wellbore radius, and $k_{20}$ is the permeability at the position near the primary fracture and horizontal wellbore in zone II (physical unit: ${\mathrm{m}}^2$). D is the fractal dimension of the fracture network, which can be obtained by the box-counting dimension method [42] after inverting the shape of fracture network based on micro-seismic data of horizontal wells [38].

Considering the stability of fluid flow and the incompressibility of the fracture network and fluid, the governing equation of the fracture system was obtained as follows

$$\frac{{\partial }^2{p}_2}{\partial x^2}+\frac{D-2-\theta }{x}\frac{\partial p_2}{\partial x}=0$$

(12)

The pressure in the primary fracture of finite conductivity is evenly distributed, so the pressure at inner boundary of region II is equal to $p_w$. It was assumed that the constant pressure at the outer boundary of the fracture network reconstruction area was the reservoir pressure $p_i$.

Combining the governing equation of the fracture system with the boundary conditions, the seepage mathematical model of region II was obtained:

$$\{\begin{array}{l}\frac{{\partial }^2{p}_2}{\partial x^2}+\frac{D-2-\theta }{x}\frac{\partial p_2}{\partial x}=0\\ p_2|{}_{x=w/2}=p_w\\ p_2|{}_{x=x_e}=p_i\end{array}$$

(13)

We obtained the expression of the pressure in region II after solving the Equation (13).

$$p_2=p_i+\frac{p_i-p_w}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left(x^{3+\theta -D}-x_e{}^{3+\theta -D}\right)$$

(14)

Assuming the productivity in zone II was $q_2$, we could obtain the equation as follows according to the flow formula.

$$q_2=\frac{k_{20}{y}_{f}h\left(3+\theta -D\right)\left(p_i-p_w\right)}{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}$$

(15)

(2)

Unstimulated volume area at the reservoir boundary (Region III)

This area is a 1/4 volume of elliptical reservoir in which pseudo plane radial flow occurs, flowing from the tip of the primary fracture into region I. The coupled flow model between the whole elliptical reservoir and the primary fracture is shown in Figure 4. Considering the influence of threshold pressure gradient on fluid flow in the reservoir, the pressure distribution law and production expression in this area were obtained.

Region III is viewed as an oval with the minor axis semidiameter of $x_e$ and the major axis semidiameter of $\left(y_e-y_f\right)$ (The minor axis semidiameter and major axis semidiameter is determined by the specific value of corresponding parameters). We obtained the half-focal length of the oval after calculating, and the expression is $\sqrt{\left|{\left(y_e-y_f\right)}^2-x_e^2\right|}$. Since the elliptical seepage field increases the difficulty in solving the mathematical model, the flow in an elliptical plane can be equivalent to the flow in a circular region with the seepage radius of $r$ through conformal transformation. The expression of equivalent seepage radius $r_e$ in the circular seepage area can be written as follows.

$$r_e=\frac{y_e-y_f+x_e}{\sqrt{\left|{\left(y_e-y_f\right)}^2-x_e^2\right|}}$$

(16)

For the plane radial flow, the governing equation in polar coordinates is

$$\frac{d^2{p}_3}{d{r}^2}+\frac{1}{r}\frac{d{p}_3}{dr}=0$$

(17)

The outer boundary of region III is a supply boundary with the pressure of $p_i$ and the inner boundary pressure is equal to the pressure of the primary fracture. Therefore, the seepage mathematical model of this area was obtained as follows:

$$\{\begin{array}{l}\frac{d^2{p}_3}{d{r}^2}+\frac{1}{r}\frac{d{p}_3}{dr}=0\\ p_3|{}_{r=w/2}=p_w\\ p_3|{}_{r=r_e}=p_i\end{array}$$

(18)

The mathematical model was solved to obtain the pressure expression at any point in region III

$$p_3=p_w+\frac{p_i-p_w}{\ln \frac{2r_e}{w}}\ln \frac{2r}{w}$$

(19)

Assuming the production in region III was $q_3$ (productivity in a quarter elliptical reservoir), we obtained the production formula in this area.

$$q_3=\frac{\pi k_{3}h\left[p_i-p_w-G\left(r_e-\frac{w}{2}\right)\right]}{2\mu B\ln \frac{2r_e}{w}}$$

(20)

(3)

Production capacity of horizontal wells

The fluid in regions II and III flows into the horizontal wellbore through the primary fracture. Assuming that the productivity of the infinite conductivity fracture was $q_0$, the productivity formula of a single fracture was obtained as follows.

$$q_0=4\left(q_2+q_3\right)=\frac{4k_{20}{y}_{f}h\left(3+\theta -D\right)\left(p_i-p_w\right)}{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}+\frac{2\pi k_{3}h\left[p_i-p_w-G\left(r_e-\frac{w}{2}\right)\right]}{\mu B\ln \frac{2r_e}{w}}$$

(21)

For staged multi-cluster fractured horizontal wells, the distribution of primary fractures along the horizontal wellbore is shown in Figure 5. Based on the locations of primary fractures, these fractures in the following figure can be divided into the fractures at both ends of the reservoir, the fractures at both ends of the fracturing section, and the fractures inside the fracturing segment. $x_a$ with the unit of $\mathrm{m}$ represents the distance between the fracture at the end of the reservoir and the reservoir boundary, $x_b$ with the unit of $\mathrm{m}$ characterizes the distance between the last and first fractures of two adjacent fracturing sections, and $x_c$ with the unit of $\mathrm{m}$ represents the distance between any two adjacent fractures inside fracturing sections.

It was assumed that the number of fracturing sections in the horizontal well was $n$, and the number of fractures in each section was $z$, so the total number of primary fractures is $N_F=z\times n$. The length of the horizontal well was defined as $l$, then the relationship between $x_a$, $x_b$, $x_c$, and $l$ follows the equation $l=n(z-1)x_c+(n-1)x_b+x_a$. $x_e$ represents the half-distance between two adjacent primary fractures in the Formula (21) of single-fracture productivity, so the production capacity of a single horizontal well can be obtained by substituting the value of $x_e$ when calculating the production of a single well. Referring to the principle of multiple fractures production superposition proposed by Meyer et al. [43], the productivity formula of a single horizontal well was given by

$$Q=n\left(z-1\right)q_0\left(\frac{x_c}{2}\right)+\left(n-1\right)q_0\left(\frac{x_b}{2}\right)+q_0\left(\frac{x_a}{2}\right)$$

(22)

2.2.1.2. Primary Fractures of Finite Conductivity

(1)

The primary fracture area (region I)

The primary fracture of finite conductivity indicates the pressure drop in the fracture. Suppose the pressure at the tip of the primary fracture was $p_0$, and the bottom hole pressure was still $p_w$. The fluid in the primary fracture flows into the horizontal wellbore along the opposite direction of the $y$-axis, and the direction of pressure drop is consistent with the direction of the value of $y$ decreases.

The fluid seepage is stable and the porous medium is incompressible, so the governing equation of the fluid seepage can be written as:

$$\frac{{\partial }^2{p}_1}{\partial y^2}=0$$

(23)

Combining the inner and outer boundary conditions, the seepage mathematical model in region I was obtained:

$$\{\begin{array}{l}\frac{{\partial }^2{p}_1}{\partial y^2}=0\\ p_1|{}_{y=y_f}=p_0\\ p_1|{}_{y=0}=p_w\end{array}$$

(24)

Solving the Equation (24), we got the expression of the pressure in the primary fracture.

$$p_1=p_w+\frac{p_0-p_w}{y_f}y$$

(25)

Then, we obtained the expression of the average pressure of the primary fracture as follows

$$\overline{p_1}=\frac{{\displaystyle {\int }_0^{y_f}\left(p_w+\frac{p_0-p_w}{y_f}y\right)dy}}{{\displaystyle {\int }_0^{y_f}dy}}=\frac{p_w+p_0}{2}$$

(26)

The productivity of region I was supposed to be $q_1$, and the expression of the production capacity in this area was obtained according to the production formula.

$$q_1=\frac{p_0-p_w}{R_1}$$

(27)

where $R_1=\frac{2\mu B{y}_f}{w{k}_{10}h}$.

(2)

Fracture network stimulated volume area (Region II)

For finite conductivity fractures, the boundary conditions in Zone II have changed, and the internal boundary pressure in this area becomes the average pressure of the primary fracture.

The seepage mathematical model in region II was written as:

$$\{\begin{array}{l}\frac{{\partial }^2{p}_2}{\partial x^2}+\frac{D-2-\theta }{x}\frac{\partial p_2}{\partial x}=0\\ p_2|{}_{x=w/2}=\frac{p_w+p_0}{2}\\ p_2|{}_{x=x_e}=p_i\end{array}$$

(28)

Solving the model, the pressure at any point in zone II is:

$$p_2=p_i+\frac{p_i-\frac{p_w+p_0}{2}}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left(x^{3+\theta -D}-x_e{}^{3+\theta -D}\right)$$

(29)

The fracture network is non-uniformly distributed in region II, and the matrix rock block is unevenly divided by secondary fractures. The schematic diagram of fluid flow in this area on the two-dimensional plane is shown in Figure 6.

It can be seen from the figure that the fluid in the fracture network reconstruction area flows into region I from the matrix rock block through the fracture network along the direction perpendicular to the primary fracture. The pressure at the outer boundary in region II is the reservoir pressure of $p_i$, and the pressure at the inner boundary is the pressure of the primary fracture of $p_1\left(y\right)$. The productivity expression of zone II was obtained by the seepage law of single-phase flow.

$$q_2={\displaystyle \sum _{i=1}^n\frac{k_{20}h}{\mu B}}{\left(\frac{x}{w/2}\right)}^{D-2-\theta }\frac{p_i-p_1(y_i)}{\Delta x}\Delta y$$

(30)

Separating the variables and integrating the above formula, we obtained the following equation.

$$\frac{q_2\mu B}{k_{20}h}{\displaystyle {\int }_{\frac{w}{2}}^{x_e}{\left(\frac{x}{w/2}\right)}^{2+\theta -D}dx}={\displaystyle {\int }_0^{y_f}\left[p_i-p_1(y)\right]dy}$$

(31)

Substituting Equation (25) into the above formula, the expression of the productivity in zone II was obtained as:

$$q_2=\frac{\left(p_i-\frac{p_w+p_0}{2}\right)}{R_2}$$

(32)

where $R_2=\frac{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}{k_{20}{y}_{f}h\left(3+\theta -D\right)}$.

(3)

Unstimulated volume area at the reservoir boundary (Region III)

For finite conductivity fracture, the seepage-flow mathematical model in zone III is:

$$\{\begin{array}{l}\frac{d^2{p}_3}{d{r}^2}+\frac{1}{r}\frac{d{p}_3}{dr}=0\\ p_3|{}_{r=w/2}=p_0\\ p_3|{}_{r=r_e}=p_i\end{array}$$

(33)

Solving the mathematical model of seepage flow, the pressure at any point in the area was obtained as:

$$p_3=p_0+\frac{p_i-p_0}{\ln \frac{2r_e}{w}}\ln \frac{2r}{w}$$

(34)

The productivity expression of this area was further obtained as follows:

$$q_3=\frac{\left[p_i-p_0-G\left(r_e-\frac{w}{2}\right)\right]}{R_3}$$

(35)

where $R_3=\frac{2\mu B\ln \frac{2r_e}{w}}{\pi k_{3}h}$.

(4)

Production capacity of horizontal wells

The productivity of the primary fracture is composed of the production of the unstimulated area at the reservoir boundary (region III) and the complex fracture network reconstruction volume area (region II). We had acquired the productivity expressions of the three regions in previous derivations, so the following relationship could be listed:

$$q_1=q_2+q_3$$

(36)

Substituting Equations (27), (32) and (35) into the above formula, the pressure expression at the tip of the primary fracture was obtained:

$$p_0=\frac{\left(R_1{R}_2+R_1{R}_3\right)p_i+\left(R_2{R}_3-\frac{R_1{R}_3}{2}\right)p_w-R_1{R}_{2}G\left(r_e-\frac{w}{2}\right)}{R_1{R}_2+\frac{R_1{R}_3}{2}+R_2{R}_3}$$

(37)

We could obtain the productivity formula of region II as follows by substituting the Equation (37) into the Equation (27).

$$q_1=\frac{p_0-p_w}{R_1}=\frac{p_i-p_w-\frac{R_2}{R_2+R_3}G\left(r_e-\frac{w}{2}\right)}{\frac{R_1{R}_2}{R_2+R_3}+\frac{1}{2}\frac{R_1{R}_3}{R_2+R_3}+\frac{R_2{R}_3}{R_2+R_3}}$$

(38)

Since the fluid in region I finally flows into the horizontal wellbore, the production formula of a single-fracture horizontal well could be obtained:

$$q_0=4q_1=4\frac{p_i-p_w-\frac{R_2}{R_2+R_3}G\left(r_e-\frac{w}{2}\right)}{\frac{R_1{R}_2}{R_2+R_3}+\frac{1}{2}\frac{R_1{R}_3}{R_2+R_3}+\frac{R_2{R}_3}{R_2+R_3}}$$

(39)

For staged fractured horizontal wells, the distribution diagram of primary fractures was depicted as Figure 5 in Section 2.2.1.1, then the production expression of a single horizontal well was given by:

$$Q=n\left(z-1\right)q_0\left(\frac{x_c}{2}\right)+\left(n-1\right)q_0\left(\frac{x_b}{2}\right)+q_0\left(\frac{x_a}{2}\right)$$

(40)

In Section 2.2.1, the physical symbol $p_i$ represents the reservoir pressure, $\mathrm{Pa}$; $p_w$ expresses the bottom hole pressure, $\mathrm{Pa}$; $p$ represents the pressure at any point of the seepage flow area, $\mathrm{Pa}$; $p_0$ is the pressure at the tip of the primary fracture, $\mathrm{Pa}$; $k$ represents the permeability, ${\mathrm{m}}^2$; $G$ expresses the threshold pressure gradient, $\mathrm{Pa}/\mathrm{m}$; $\mu$ is the viscosity of crude oil, $\mathrm{Pa}·\mathrm{s}$; $B$ represents the crude oil volume factor; $x_e$ is the half-distance between two adjacent primary fractures, $\mathrm{m}$; $y_f$ is the half-length of the primary fracture, $\mathrm{m}$; $h$ expresses the reservoir thickness, $\mathrm{m}$; $w$ represents the width of primary fractures, $\mathrm{m}$; $q_0$ is the production of the single-fracture horizontal well, ${\mathrm{m}}^3/\mathrm{s}$; $Q$ is the productivity of a single horizontal well, ${\mathrm{m}}^3/\mathrm{s}$; $q$ represents the flow of the seepage flow area, ${\mathrm{m}}^3/\mathrm{s}$; the subscripts 1, 2, and 3 in the symbols represent regions I, II, and III, respectively; the subscript 0 in the symbols indicates the initial value. It should be noted that the superscript “′” of the physical symbols in Section 2.2.2 indicates the consideration of the deformation characteristics of porous media in the production capacity model. The physical meaning and units of other symbols in Section 2.2.2 are the same as those in the Section 2.2.1 except for individual symbols, so the explanation of symbols will not be repeated.

2.2.2. Consideration of the Deformation Characteristics of Porous Medium

In the process of reservoir exploitation, with the continuous recovery of underground fluid, the formation pressure gradually decreases, and the effective overburden pressure on the rock grain increases, then the porous media undergoes elastic–plastic deformation, thus resulting in changes of permeability and porosity. According to the deformation theory of pore and throat, the throat is closed but the pores nearly remain unchanged when the tight sandstone is compressed. Therefore, in tight reservoirs, compared with porosity, the effective overburden pressure has a greater impact on permeability. Meanwhile, for porous media, the deformation degree of the fracture is more obvious than that of matrix rock block under the influence of effective confining pressure, so only the change of fracture system permeability was considered in our deliverability model.

Primary Fractures of Infinite Conductivity

(1)

Fracture network stimulated volume area (Region II)

The permeability of the fracture network changes under the impact of the effective peripheral pressure. According to the empirical formula, the expression is

$$k_2=k_{20}{e}^{-\alpha \left(p_i-p_2\right)}$$

(41)

where $\alpha$—medium deformation coefficient,${ \mathrm{Pa}}^{-1}$.

While considering the deformation characteristics of the fracture network, the fractal theory was introduced to characterize the non-uniform distribution of the fracture network in region II. Combining the above formula and Formula (11), the permeability expression of the fracture network in zone II could be obtained:

$$k_2=k_{20}{\left(\frac{y}{r_w}\right)}^{D-2-\theta }{\left(\frac{x}{w/2}\right)}^{D-2-\theta }{e}^{-\alpha \left(p_i-p_2\right)}$$

(42)

Since the above formula contains an exponential function, which greatly increases the difficulty of solving the pressure distribution law and the production formula, we simplified the actual permeability of the fracture network to the permeability under the average pressure, namely:

$$\overline{k_2}=k_{20}{\left(\frac{y}{r_w}\right)}^{D-2-\theta }{\left(\frac{x}{w/2}\right)}^{D-2-\theta }{e}^{-\alpha \left(p_i-\overline{p_2}\right)}$$

(43)

where $\overline{p_2}$ represents the average pressure in region II under the condition of infinite conductivity fracture when only the influence of threshold pressure gradient is considered in the deliverability model.

The average pressure expression of zone II in Section 2.2.1.1 was obtained as follows based on Equation (14).

$$\overline{p_2}=\frac{{\displaystyle {\int }_{\frac{w}{2}}^{x_e}{p}_{2}dx}}{{\displaystyle {\int }_{\frac{w}{2}}^{x_e}dx}}=p_i+\frac{p_i-p_w}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left\{\frac{\left[x_e{}^{4+\theta -D}-{\left(\frac{w}{2}\right)}^{4+\theta -D}\right]}{\left(4+\theta -D\right)\left(x_e-\frac{w}{2}\right)}-x_e{}^{3+\theta -D}\right\}$$

(44)

Substituting Equation (44) into Equation (43), the expression of fracture network permeability under the average pressure could be obtained:

$$\overline{k_2}=\overline{k_{20}}{\left(\frac{y}{r_w}\right)}^{D-2-\theta }{\left(\frac{x}{w/2}\right)}^{D-2-\theta }$$

(45)

In the above equation, $\overline{k_{20}}=k_{20}{e}^{\alpha \left\{\frac{p_i-p_w}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left\{\frac{\left[x_e{}^{4+\theta -D}-{\left(\frac{w}{2}\right)}^{4+\theta -D}\right]}{\left(4+\theta -D\right)\left(x_e-\frac{w}{2}\right)}-x_e{}^{3+\theta -D}\right\}\right\}}$.

The pressure distribution of region II was obtained with the same derivation process in Section 2.2.1.1:

$$P_2=p_i+\frac{p_i-p_w}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left(x^{3+\theta -D}-x_e{}^{3+\theta -D}\right)$$

(46)

Similarly, the flow expression of zone II was obtained:

$$q_2^{\prime }=\frac{\overline{k_{20}}{y}_{f}h\left(3+\theta -D\right)\left(p_i-p_w\right)}{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}$$

(47)

(2)

Unstimulated volume area at the reservoir boundary (Region III)

Region III is a quarter elliptical volume area, which is an unstimulated matrix block at the reservoir boundary. The deformation characteristics of the medium in this area were not considered in the model, so the pressure distribution law and production formula in zone III could be written directly according to the derivation results in Section 2.2.1.1.

The pressure formula in region III is

$$P_3=p_w+\frac{p_i-p_w}{\ln \frac{2r_e}{w}}\ln \frac{2r}{w}$$

(48)

The production formula in region III is

$$q_3^{\prime }=\frac{\pi k_{3}h\left[p_i-p_w-G\left(r_e-\frac{w}{2}\right)\right]}{2\mu B\ln \frac{2r_e}{w}}$$

(49)

(3)

Production capacity of horizontal wells

The fluid in regions II and III flows into the horizontal wellbore through the primary fracture. Assuming the productivity of infinite conductivity fracture was $q_0^{\prime }$, the productivity formula of a single fracture was obtained as follows.

$$q_0^{\prime }=4\left(q_2^{\prime }+q_3^{\prime }\right)=\frac{4\overline{k_{20}}{y}_{f}h\left(3+\theta -D\right)\left(p_i-p_w\right)}{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}+\frac{2\pi k_{3}h\left[p_i-p_w-G\left(r_e-\frac{w}{2}\right)\right]}{\mu B\ln \frac{2r_e}{w}}$$

(50)

For staged multi-cluster fractured horizontal wells, the production expression of a single horizontal well is:

$$Q^{\prime }=n\left(z-1\right)q_0^{\prime }\left(\frac{x_c}{2}\right)+\left(n-1\right)q_0^{\prime }\left(\frac{x_b}{2}\right)+q_0^{\prime }\left(\frac{x_a}{2}\right)$$

(51)

Primary Fractures of Finite Conductivity

(1)

The primary fracture area (region I)

Considering the change of the primary fracture permeability with the effective overburden pressure, the permeability expression was obtained as:

$$k_1=k_{10}{e}^{-\alpha \left(p_i-p_1\right)}$$

(52)

Simplifying the actual permeability of the primary fracture to the permeability under the average pressure, we got:

$$\overline{k_1}=k_{10}{e}^{-\alpha \left(p_i-\overline{p_1}\right)}$$

(53)

In the above equation, $\overline{p_1}$ represents the average pressure of region I under the condition of finite conductivity fracture when only the effect of the threshold pressure gradient is considered in the production capacity model, which was shown as Equation (26).

Substituting Equation (26) into Equation (53), the permeability of the primary fracture was obtained as follows.

$$\overline{k_1}=k_{10}{e}^{-\alpha \left(p_i-\frac{p_0+p_w}{2}\right)}$$

(54)

where, the expression of $p_0$ was shown as the Equation (37).

Supposing the pressure at the tip of the finite conductivity fracture was $p_0^{\prime }$, and the derivation process about the expressions of pressure and production was the same as in Section 2.2.1.2, then we obtained the pressure formula of zone I.

$$P_1=p_w+\frac{p_0^{\prime }-p_w}{y_f}y$$

(55)

The average pressure of the primary fracture was further obtained as:

$$\overline{P_1}=\frac{{\displaystyle {\int }_0^{y_f}\left(p_w+\frac{p_0^{\prime }-p_w}{y_f}y\right)dy}}{{\displaystyle {\int }_0^{y_f}dy}}=\frac{p_w+p_0^{\prime }}{2}$$

(56)

Similarly, we could obtain the production formula of the primary fracture.

$$q_1^{\prime }=\frac{p_0^{\prime }-p_w}{R_1^{\prime }}$$

(57)

In the above equation, $R_1^{\prime }=\frac{2\mu B{y}_f}{w\overline{k_1}h}$.

(2)

Fracture network stimulated volume area (Region II)

The expression of the fracture system permeability in this area is:

$$\overline{k_2}=\overline{k_{20}}{\left(\frac{y}{r_w}\right)}^{D-2-\theta }{\left(\frac{x}{w/2}\right)}^{D-2-\theta }$$

(58)

where $\overline{k_{20}}=k_{20}{e}^{\alpha \left\{\frac{p_i-\frac{p_w+p_0}{2}}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left\{\frac{\left[x_e{}^{4+\theta -D}-{\left(\frac{w}{2}\right)}^{4+\theta -D}\right]}{\left(4+\theta -D\right)\left(x_e-\frac{w}{2}\right)}-x_e{}^{3+\theta -D}\right\}\right\}}$.

The seepage mathematical model in region II was written as:

$$\{\begin{array}{l}\frac{{\partial }^2{P}_2}{\partial x^2}+\frac{D-2-\theta }{x}\frac{\partial P_2}{\partial x}=0\\ P_2|{}_{x=w/2}=\frac{p_w+p_0^{\prime }}{2}\\ P_2|{}_{x=x_e}=p_i\end{array}$$

(59)

Solving the model, the pressure at any point in zone II is:

$$P_2=p_i+\frac{p_i-\frac{p_w+p_0^{\prime }}{2}}{x_e{}^{3+\theta -D}-{\left(\frac{w}{2}\right)}^{3+\theta -D}}\left(x^{3+\theta -D}-x_e{}^{3+\theta -D}\right)$$

(60)

The flow expression of region II is:

$$q_2^{\prime }=\frac{\left(p_i-\frac{p_w+p_0^{\prime }}{2}\right)}{R_2^{\prime }}$$

(61)

In the above formula, $R_2^{\prime }=\frac{\mu B\frac{w}{2}\left[{\left(\frac{x_e}{w/2}\right)}^{3+\theta -D}-1\right]}{\overline{k_{20}}{y}_{f}h\left(3+\theta -D\right)}$.

(3)

Unstimulated volume area at the reservoir boundary (Region III)

The seepage flow mathematical model in region III is:

$$\{\begin{array}{l}\frac{d^2{P}_3}{d{r}^2}+\frac{1}{r}\frac{d{P}_3}{dr}=0\\ P_3|{}_{r=w/2}=p_0^{\prime }\\ P_3|{}_{r=r_e}=p_i\end{array}$$

(62)

Solving the mathematical model of seepage flow, the pressure at any point in the area was obtained as:

$$P_3=p_0^{\prime }+\frac{p_i-p_0^{\prime }}{\ln \frac{2r_e}{w}}\ln \frac{2r}{w}$$

(63)

The productivity formula in the area is:

$$q_3^{\prime }=\frac{\left[p_i-p_0^{\prime }-G\left(r_e-\frac{w}{2}\right)\right]}{R_3^{\prime }}$$

(64)

where $R_3^{\prime }=\frac{2\mu B\ln \frac{2r_e}{w}}{\pi k_{3}h}$.

(4)

Production capacity of horizontal wells

Based on the production formulas in the three regions, the pressure expression at the tip of the primary fracture was obtained as follows:

$$p_0^{\prime }=\frac{\left(R_1^{\prime }{R}_2^{\prime }+R_1^{\prime }{R}_3^{\prime }\right)p_i+\left(R_2^{\prime }{R}_3^{\prime }-\frac{R_1^{\prime }{R}_3^{\prime }}{2}\right)p_w-R_1^{\prime }{R}_2^{\prime }G\left(r_e-\frac{w}{2}\right)}{R_1^{\prime }{R}_2^{\prime }+\frac{R_1^{\prime }{R}_3^{\prime }}{2}+R_2^{\prime }{R}_3^{\prime }}$$

(65)

The production formula of a single-fracture horizontal well is:

$$q_0{}^{\prime }=4q_1^{\prime }=4\frac{p_i-p_w-\frac{R_2^{\prime }}{R_2^{\prime }+R_3^{\prime }}G\left(r_e-\frac{w}{2}\right)}{\frac{R_1^{\prime }{R}_2^{\prime }}{R_2^{\prime }+R_3^{\prime }}+\frac{1}{2}\frac{R_1^{\prime }{R}_3^{\prime }}{R_2^{\prime }+R_3^{\prime }}+\frac{R_2^{\prime }{R}_3^{\prime }}{R_2^{\prime }+R_3^{\prime }}}$$

(66)

For staged multi-cluster fractured horizontal wells, the production expression of a single horizontal well is:

$$Q^{\prime }=n\left(z-1\right)q_0^{\prime }\left(\frac{x_c}{2}\right)+\left(n-1\right)q_0^{\prime }\left(\frac{x_b}{2}\right)+q_0^{\prime }\left(\frac{x_a}{2}\right)$$

(67)

2.3. Flow Chart of Established Productivity Model

The process of establishing and solving the productivity model is demonstrated in Figure 7.

3. Results and Discussion

3.1. Model Verification

(1)

Oilfield examples

To verify the reliability of the established steady-state productivity model, three horizontal wells of different blocks in S oilfield were selected for verification. The main parameters of three horizontal wells (A1, B1, C1) were shown in

Table 1. Primary parameters of well A1 in block A.

Symbol	Physical Meaning	Value	Physical Unit
$p_i$	Reservoir pressure	21.00	$\mathrm{MPa}$
$p_w$	Bottom hole pressure	12.20	$\mathrm{MPa}$
$l$	Length of horizontal section	900	$\mathrm{m}$
$B$	Crude oil volume factor	1.313
$h$	Reservoir thickness	12.5	$\mathrm{m}$
$y_e$	Half-width of reservoir	300	$\mathrm{m}$
$\mu$	Crude oil viscosity	7.88	$\mathrm{mPa}·\mathrm{s}$
$\rho$	Crude oil density	0.866	$\mathrm{g}/{\mathrm{cm}}^3$
$S_o$	Oil saturation	57	%
$k_3$	Matrix permeability	0.2	$\mathrm{mD}$
$k_{10}$	Initial permeability of primary fracture	2400	$\mathrm{mD}$
$k_{20}$	Initial permeability of fracture network	35	$\mathrm{mD}$
$y_f$	Half-length of primary fracture	100	$\mathrm{m}$
$w$	Width of primary fracture	0.5	$\mathrm{cm}$
$n$	Number of fracturing sections	13
$z$	Cluster number in each section	5
$x_b$	Distance between two fracturing sections	18	$\mathrm{m}$
$x_c$	Distance between two cluster fractures	8	$\mathrm{m}$
$D$	Fractal dimension of fracture network	1.7
$\theta$	Abnormal diffusion coefficient	0.1
$\alpha$	Medium deformation coefficient	0.08	${\mathrm{MPa}}^{-1}$
$G$	Threshold pressure gradient	0.06	$\mathrm{MPa}/\mathrm{m}$

,

Table 2. Primary parameters of well B1 in block B.

Symbol	Physical Meaning	Value	Physical Unit
$p_i$	Reservoir pressure	20	$\mathrm{MPa}$
$p_w$	Bottom hole pressure	5.8	$\mathrm{MPa}$
$l$	Length of horizontal section	770	$\mathrm{m}$
$B$	Crude oil volume factor	1.050
$h$	Reservoir thickness	17.3	$\mathrm{m}$
$y_e$	Half-width of reservoir	400	$\mathrm{m}$
$\mu$	Crude oil viscosity	58.8	$\mathrm{mPa}·\mathrm{s}$
$\rho$	Crude oil density	0.8991	$\mathrm{g}/{\mathrm{cm}}^3$
$S_o$	Oil saturation	69.1	%
$k_3$	Matrix permeability	0.36	$\mathrm{mD}$
$k_{10}$	Initial permeability of primary fracture	12	$\mathrm{D}$
$k_{20}$	Initial permeability of fracture network	25	$\mathrm{mD}$
$y_f$	Half-length of primary fracture	200	$\mathrm{m}$
$w$	Width of primary fracture	1	$\mathrm{cm}$
$n$	Number of fracturing sections	10
$z$	Cluster number in each section	3
$x_b$	Distance between two fracturing sections	30	$\mathrm{m}$
$x_c$	Distance between two cluster fractures	15	$\mathrm{m}$
$D$	Fractal dimension of fracture network	1.8
$\theta$	Abnormal diffusion coefficient	0.12
$\alpha$	Medium deformation coefficient	0.06	${\mathrm{MPa}}^{-1}$
$G$	Threshold pressure gradient	0.08	$\mathrm{MPa}/\mathrm{m}$

and

Table 3. Primary parameters of well C1 in block C.

Symbol	Physical Meaning	Value	Physical Unit
$p_i$	Reservoir pressure	22.27	$\mathrm{MPa}$
$p_w$	Bottom hole pressure	13.95	$\mathrm{MPa}$
$l$	Length of horizontal section	1350	$\mathrm{m}$
$B$	Crude oil volume factor	1.162
$h$	Reservoir thickness	8.3	$\mathrm{m}$
$y_e$	Half-width of reservoir	300	$\mathrm{m}$
$\mu$	Crude oil viscosity	11.7	$\mathrm{mPa}·\mathrm{s}$
$\rho$	Crude oil density	0.876	$\mathrm{g}/{\mathrm{cm}}^3$
$S_o$	Oil saturation	69	%
$k_3$	Matrix permeability	0.002	$\mathrm{mD}$
$k_{10}$	Initial permeability of primary fracture	2200	$\mathrm{mD}$
$k_{20}$	Initial permeability of fracture network	25	$\mathrm{mD}$
$y_f$	Half-length of primary fracture	120	$\mathrm{m}$
$w$	Width of primary fracture	0.4	$\mathrm{cm}$
$n$	Number of fracturing sections	20
$z$	Cluster number in each section	10
$x_b$	Distance between two fracturing sections	10	$\mathrm{m}$
$x_c$	Distance between two cluster fractures	6	$\mathrm{m}$
$D$	Fractal dimension of fracture network	1.65
$\theta$	Abnormal diffusion coefficient	0.1
$\alpha$	Medium deformation coefficient	0.18	${\mathrm{MPa}}^{-1}$
$G$	Threshold pressure gradient	0.10	$\mathrm{MPa}/\mathrm{m}$

, respectively. We deduced two steady-state productivity formulas for the primary fracture of finite conductivity under two different conditions in the above study, one of which only considers the influence of the threshold pressure gradient and another one further considers the deformation characteristics of the porous medium. Substituting the data in following tables into the two production formulas, respectively, we obtained the verification results as shown in

Table 4. Model verification results.

Research Block	Oil Well	Actual Production $\mathit{q} ($$\mathbf{t}/\mathbf{d})$	Model Production $\mathit{Q} ($$\mathbf{t}/\mathbf{d})$	Model Production ${\mathit{Q}}^{\prime } ($$\mathbf{t}/\mathbf{d})$	Error 1	Error 2
A	A1	13.6000	14.0416	13.9802	3.25%	2.80%
B	B1	14.5000	15.0994	14.8913	4.13%	2.70%
C	C1	15.0000	15.5344	15.4938	3.56%	3.29%

.

In Table 4, the actual production $q$ is the average production capacity of the selected horizontal wells in the steady production stage (about two hundred days to one year after production), the model production $Q$ represents the steady-state production capacity when only the impact of the threshold pressure gradient is considered in the model, and the model production $Q^{\prime }$ indicates the stable production when further considering the deformation characteristics of porous media. Errors 1 and 2 correspond to the error results of model production $Q$ and $Q^{\prime }$, respectively, compared with the actual production of horizontal wells. It can be seen from Table 4 that the error between the model results and the actual production value does not exceed 5%, which indicates that the steady-state productivity model proposed in this study for staged fractured horizontal wells in tight oil reservoirs can accurately predict the average productivity of horizontal wells in stable production stage within the allowable range of error, thus proving the reliability and applicability of our deliverability model. From the comparison of the model production $Q$ and $Q^{\prime }$ with the actual production, it can be known that the calculation error of the steady-state productivity model considering the deformation characteristics of porous media is reduced to within 4%, which is closer to the actual production situation of the horizontal well in the studied oil block. It shows that the deformation characteristic of porous media is an important factor in establishing a productivity model for staged fractured horizontal wells in tight oil reservoirs, which further proves the reliability of the built productivity model.

(2)

Model comparison

In order to further verify the applicability of the productivity model proposed in this study, we made a comparison between our model and other two models established in the reference [24] and paper [37] based on the basic parameters of the horizontal well A1. In order to distinguish the three models, we used Models 1, 2, and 3 to represent the productivity model in this study, the model established in reference [24], and the evaluation model proposed in paper [37].

For obtaining the production calculated by Model 2, there is an equation [44] indicating the relationship between the dimensionless bottom hole pressure and the dimensionless production of a single fracture should be helpful. The comparison results between the three models are shown in

Table 5. Model comparison results.

Actual Production $\mathit{q} ($$\mathbf{t}/\mathbf{d})$	Model Production ${\mathit{Q}}_1 ($$\mathbf{t}/\mathbf{d})$	Model Production ${\mathit{Q}}_2 ($$\mathbf{t}/\mathbf{d})$	Model Production ${\mathit{Q}}_3 ($$\mathbf{t}/\mathbf{d})$	Error 1	Error 2	Error 3
13.6000	13.9802	15.6912	14.5221	2.80%	15.38	6.78%

.

$${\overline{q}}_D=\frac{1}{s^2{\overline{p}}_{wfD}}$$

(68)

In the above table, $q$ is the actual production of horizontal well A1. $Q_1$, $Q_2$, and $Q_3$ represent the production calculated by Models 1 to 3, respectively. Models 2 and 3 are unsteady-state productivity models in which the horizontal well production is related to production time. Therefore, $Q_2$ and $Q_3$ were obtained by averaging the production results in the stable production stage. For Models 2 and 3, the stable production state probably starts from days 200 to 400. Errors 1 to 3 correspond to the error results of the three models compared with the actual production of horizontal well A1. From the comparison results shown in Table 5, we can know that the calculation result of Model 1 is the closest to the actual production, which further proves the reliability of our productivity model. The fracture network is an idealized dual porous medium structure in Model 3, and some factors (including the deformation characteristics of porous medium) are not considered, which leads to a highest error compared to Models 2 and 3.

3.2. Analysis of Sensitive Factors

Adopting the main parameters of the horizontal well A1 in Table 1 as the simulation parameters, the influences of the fracturing and reservoir parameters, including the density of fractures within 100 m and the threshold pressure gradient on the productivity of horizontal wells, were studied by the established productivity model (considering the deformation characteristics of porous media), which is expected to provide a theoretical basis for increasing the production of tight oil and realizing effective exploitation of tight oil reservoirs.

(1)

Fracture conductivity

The ability of the fracture to allow fluid to pass through is described as fracture conductivity, which is expressed by the product of fracture width and fracture permeability. The value of conductivity of the primary fracture was changed with other parameters in Table 1 maintaining the same values, and the productivity of a single-fracture horizontal well under the conditions of infinite and finite conductivity fracture was calculated, respectively. Next, the relationship between the production of a single-fracture horizontal well and the conductivity of the primary fracture was demonstrated in Figure 8.

It can be seen from Figure 8 that the production of a single-fracture horizontal well remains a fixed value with the change of the value of fracture conductivity for the infinite conductivity fracture. Nevertheless, the productivity of a single-fracture horizontal well increases with the rising of the conductivity of the primary fracture for the finite conductivity fracture, and the growth rate decreases with the increase of fracture conductivity. Figure 8 shows that when the value of fracture conductivity is small, there is a big difference in the production of a single-fracture horizontal well under the situations of finite and infinite conductivity fracture. Only when the conductivity of the fracture rises to a certain value is the production of the finite conductivity fracture close to the productivity of the infinite conductivity fractures. Therefore, for the production prediction of a single-fracture horizontal well, the fracture can be regarded as an infinite conductivity fracture if the fracture conductivity reaches a certain value. However, multiple effective fractures are generated after large-scale volume fracturing of horizontal wells. If the fractures are viewed as infinite conductivity for productivity prediction of horizontal wells, the calculation error created by each fracture will cause great error after superposition, which will significantly deviate the predicted production from the actual production. Therefore, for staged fractured horizontal wells, artificial fractures cannot be regarded as fractures of infinite conductivity for prediction and analysis of the production capacity.

(2)

Fracture network permeability

Changing the value of the fracture network permeability, we obtained the relationship between the production of a single horizontal well and the permeability of the fracture network, as shown in Figure 9.

It can be known from the above figure that the daily production of a single horizontal well rises with the increase of the fracture network permeability, and the growth slows down with the continuous raising of the fracture network permeability. The permeability of fracture network represents the ability of fluid flow. The higher the fracture network permeability, the greater the fluid seepage velocity, thus leading to the rising of horizontal well production. The increase of the horizontal well production obviously slows down when the value of fracture network permeability rises to 70 $\mathrm{mD}$, and then the productivity nearly remains unchanged. Therefore, the fracture network permeability has a significant impact on the production of horizontal wells within a limited range. When the fracture network permeability reaches a certain value, the production of horizontal wells will hardly increase with it.

(3)

Threshold pressure gradient

The relationship between the daily production of a single horizontal well and the threshold pressure gradient was demonstrated in the following figure by changing the value of the threshold pressure gradient.

As shown in Figure 10, the threshold pressure gradient has a significant effect on the horizontal well production, and the larger threshold pressure gradient results in the smaller daily production. It is because the fluid can fast flow only after overcoming the threshold pressure, and a larger threshold pressure gradient leads to greater difficulty for fluid to flow. Therefore, the threshold pressure gradient is an important sensitive factor affecting the production of horizontal wells, which cannot be ignored in the productivity prediction of staged fractured horizontal wells in tight reservoirs.

(4)

Media deformation coefficient

Changing the value of the media deformation coefficient with other parameters in Table 1 unaltered, the relationship between the production of horizontal wells with finite conductivity fractures and the medium deformation coefficient was depicted in Figure 11.

As shown in the above figure, the daily production of the horizontal wells is negatively correlated with the medium deformation coefficient, and the production decreases with the increase of the medium deformation coefficient. The increase of the medium deformation coefficient indicates that the deformation degree of the fracture system is enhanced, and the fracture is closed because of extrusion, resulting in the weakening of fluid flow capacity and the decline of the horizontal well production. It shows that the deformation characteristic of porous media is an important factor that is non-ignorable in the productivity prediction and analysis of staged fractured horizontal wells in tight reservoirs.

(5)

Fractal dimension

Altering the value of the fractal dimension, the relationship between the daily production of horizontal wells and the fractal dimension was demonstrated as follows.

The fractal dimension is a parameter used to quantitatively characterize the complexity of fracture network [42]. The larger the value of the fractal dimension, the more natural fractures in tight reservoir. Under this situation, there is a greater possibility that the primary fracture will connect with natural fractures to generate more secondary fractures, thus resulting in the more complex structure of fracture network and more channels of fluid seepage. Therefore, the production of horizontal wells increases with the rise of the fractal dimension, which is consistent with the change trend of production shown in Figure 12.

(6)

Fracture density within 100 m

The number of fracturing sections was set as 7, with other parameters in Table 1 unchanged. Substituting the simulation parameters into the steady-state productivity formula of finite conductivity fractures, the relation between the fracture density within 100$\mathrm{m}$ and the horizontal well production was obtained in Figure 13 by means of adjusting the number of fractures inside each fracturing section and the distance between two adjacent fractures.

As shown in Figure 13, the fracture density within 100 m has an obvious influence on the production of horizontal wells. The more fractures within one hundred meters, the greater the production of horizontal wells. Within the fixed section length, the denser the distribution of fractures, the greater the number of fractures, and the more fluid flow channels, thus resulting in the greater the daily production of horizontal wells. It is known from the above figure that the growth rate of the daily production of horizontal wells decreases with the continuous increase of the 100-m fracture density. At the same time, the denser distribution of fractures in fracturing construction represents the higher cost of fracturing. Therefore, there is a reasonable value range of fracture density, which can not only improve the development effect of tight oil reservoirs but also save the fracturing cost.

4. Conclusions

In this study, we proposed a two-dimensional heterogeneous fracture network structure model, in which the secondary fractures are non-uniformly distributed along both directions of the horizontal wellbore and the primary fracture. Considering the influence of the threshold pressure gradient and the deformation characteristics of the porous medium, a three-zone steady-state productivity model with the combination of radial and linear flow for staged fractured horizontal wells in tight oil reservoirs was established. To eliminate the strong nonlinear characteristics of the mathematical equation caused by the deformation characteristics of porous media, the fracture system permeability was simplified to the average permeability under the average pressure. The steady-state production formulas of staged fractured horizontal wells in tight oil reservoirs under infinite and finite conductivity fractures were deduced through the step-forward calculation method, which can effectively predict the actual production of horizontal wells. The productivity sensitive factors were analyzed based on the established deliverability model, and the results show that the finite conductivity fractures can be approximately treated as fractures of infinite conductivity when predicting the production of a single-fracture horizontal well within a certain range of fracture conductivity. However, for multiple-fracture horizontal wells, the fracture of finite conductivity cannot be regarded as the infinite conductivity fracture, otherwise, it will cause a great error in the production prediction of horizontal wells. The horizontal well production increases with the growth of the fractal dimension and fracture network permeability. There is a reasonable range of the value of the hundred-meter fracture density, which can not only improve the production of horizontal wells but also save the fracturing cost. The threshold pressure gradient and medium deformation characteristics are two important sensitive factors in the prediction and analysis of production for staged fractured horizontal wells in tight oil reservoirs.

Although the three-zone steady-state productivity model based on a two-dimensional heterogeneous fracture network structure can provide a theoretical basis for the formulation of rational technical policies for development of tight reservoirs, but the pressure loss in horizontal wellbore was not considered, thus leading to the limitations in optimizing the length of horizontal wells. Our future study will consider the pressure loss in horizontal wellbore and make further improvements on the productivity model.