Semianalytical Coupled Model of Transient Pressure Behavior for Horizontal Well with Complex Fracture Networks in Tight Oil Reservoirs

Abstract

Due to a large number of natural fractures in tight oil reservoir, many complex fracture networks are generated during fracturing operation. There are five kinds of flow media in the reservoir: “matrix, natural fracture, hydraulic fracture network, perforation hole, and horizontal wellbore”. How to establish the seepage model of liquid in multiscale medium is a challenging problem. Firstly, this paper establishes the dual medium seepage model based on source function theory, principle of superposition, and Laplace transformation and then uses the “star-triangle” transform method to establish the transient pressure behavior model in the complex fracture network. After that, perforating seepage model and variable mass flow in horizontal wellbore were established. Finally, continuous condition was used to couple the seepage model of dual medium seepage model, transient pressure behavior model in the complex fracture network, perforation seepage model, and the variable mass seepage model in horizontal wellbore, to establish a semianalytical coupled seepage model for horizontal well in tight reservoir. This paper provides theoretical basis for field application of horizontal well with complex fracture networks.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Domestic and foreign development practice proves that horizontal drilling technology and large-scale fracturing technique are the core technologies of successful development of tight reservoir [1]. Complex fracture networks have been developed around horizontal well after large-scale fracturing (as shown in Figure 1). Fractures in the network are irregular intersecting, and the cross types of fractures can be divided into four categories (as shown in Figure 2). There are five kinds of multiscale flow media in tight oil reservoir after fracturing, contains nanoscale matrix, micron scale natural fracture, millimeter scale hydraulic fracture (fracture width), centimeter scale perforation hole, and meter scale horizontal wellbore. The highly discrete multiscale media leads to a great change in the seepage of tight reservoir. In recent years, scholars at home and abroad have established a large number of seepage models for horizontal wells with complex fracture networks in tight oil reservoirs. These models can be divided into three types: analytical model, semianalytical model, and numerical model. Each model is described as below.

[figure omitted; refer to PDF]

[figures omitted; refer to PDF]

The first type is analytical model. These model is also called linear flow model [2–5]. The assumption of linear flow is too ideal and leads to inaccurate calculation results.

The second type is semianalytical model which was developed by using the source function method [6–9]. Ren Zongxiao et al., 2016, established transient pressure behavior models for horizontal wells with complex fracture networks, respectively. The work of Ren Zongxiao et al., 2016, just considered the “Seepage model of dual medium reservoir”. In the new article, we not only consider the “Seepage model of dual medium reservoir” but also consider the “Seepage model in complex fracture networks”, “Seepage model of Perforation”, and “Variable mass pipe seepage model in horizontal wellbore” which is more accurate than the previous paper.

The third type is numerical model, which mainly includes the discrete fracture network (DFN), discrete fracture model (DFM), and embedded discrete fracture Model (EDFM). The characteristics and representative results of each model are shown in Table 1.

Table 1

Summary table of numerical model.

Model name	Discrete diagram	Representative research	Advantages	Disadvantages
DFN		Yao et al., 2010 [10]; Dreuzy et al., 2012 [11]; Habibi et al., 2014 [12];	Accurately characterize the orientation and dimensions of the fracture. The meshing method is simple, reducing the number of solving nodes	There is a solution node on the fracture, and there is no solution node in the matrix, which leads to a rough description of fluid seepage in the matrix.
DFM		Baca et al., 1984 [13]; Karimi-Fard et al., 2004; Yan et al., 2016 [14]; Zhang et al., 2017 [15]; Zhang et al., 2017 [16]; Zhang et al., 2017 [17]	Both the fracture and matrix have solution nodes. The matrix mesh is divided along the fracture shape to describe the discrete characteristics of the fractures.	Fine partitioning of the grid leads to the explosion of computing grid, resulting in large computation and low efficiency.
EDFM		Lee et al., 2000 [18]; Li & Lee, 2008 [19]; Yan Xia et al., 2014 [20]; Jiang et al., 2016 [21]	The effect of medium and small length fissures on seepage flow is treated by equivalent grid permeability, and large scale fractures are described separately. Embedded processing reduces the number of discrete grids.	This method needs to record the relative position between fracture and grid. It can increase the data storage in the tight oil reservoir which containing large scale cracks.

The numerical model has the advantage of describing the heterogeneity of multiscale media. But it also has disadvantages such as complex modeling process and unstable calculation result, which limits its application in practical engineering problems. As with the semianalytical model, the numerical model does not consider the impact of perforation flow and the flow in horizontal well.

In conclusion, there is still lacking multiscale coupled seepage model in tight oil reservoir considering “matrix, natural fracture, hydraulic fracture, perforation, and horizontal well”. Based on the method of source function, superposition principle, and star-triangle transform, this paper aims to establish a precise and efficient seepage model of horizontal well with complex fracture networks. Using the new model, the seepage law of horizontal well with complex fracture networks is revealed, providing theoretical basis for the application of fracturing horizontal wells.

2. Physical Model of Multiscale Media Flow in Tight Oil Reservoir

The tight oil reservoir contains a large number of natural fractures and is assumed to belong to dual medium reservoir (as shown in Figure 3). The dual medium model is composed of uniformly distributed matrix system and natural fracture system [22].

[figures omitted; refer to PDF]

The liquid in tight reservoir flows through four parts: double media (as shown in Figure 4(a)), complex fracture networks (as shown in Figure 4(b)), perforation, and horizontal well (as shown in Figure 4(c)).

[figure omitted; refer to PDF]

To facilitate the establishment of the mathematical model of physical model in Figure 4, the following assumptions are made:

$(1)$ Tight oil reservoir is infinite plate with impermeable boundaries.

$(2)$ All fractures in the complex fracture networks penetrate reservoir; liquid in fractures satisfies 1D Darcy seepage.

$(3)$ Fluid flow is considered as single phase flow and production of the well is constant.

$(4)$ Fluid is compressible; the reservoir temperature remains constant during production.

$(5)$ Ignore the effect of liquid capillary force and gravity.

3. Establishment of Seepage Coupling Model for Horizontal Well with Complex Fracture Networks in Tight Oil Reservoir

The multiscale medium flow process in tight oil reservoirs consists of four parts: dual media flow, internal flow in complex fractures, perforation flow, and horizontal wellbore variable mass flow. A simple volume fracturing horizontal well (as shown in Figure 5) is taken as an example to establish the mathematical model for four parts flow, respectively.

[figure omitted; refer to PDF]

3.1. Physical Model of Reservoir

In 1991, Ozkan et al. [23] established a surface source function in dual medium reservoir in Laplace space, which solved the problem of seepage with hydraulic fracture in dual medium reservoirs. However, the Ozkan source function requires that the fracture must be vertical or parallel to the horizontal wellbore and cannot solve the seepage problem of the inclined fracture in the complex fracture network. Therefore, Ren et al. [24] established the surface source function of the inclined fracture based on Ozkan surface source function. The expression of a single inclined fracture seepage solution in Laplace space is [24]$$\begin{array}{}\text{(1)}& {\overline{p}}_{fD}={\overline{q}}_{fD}{{\displaystyle \int }}_{-\mathrm{1}}^{\mathrm{1}}{K}_{\mathrm{0}}\left[\sqrt{sf\left(s\right)}\sqrt{{\left[\left(x_D-x_{wD}\right)\cos \theta +\left(y_D-y_{wD}\right)\sin \theta -u\right]}^{\mathrm{2}}+{\left[\left(y_D-y_{wD}\right)\cos \theta -\left(x_D-x_{wD}\right)\sin \theta \right]}^{\mathrm{2}}}\right]du\end{array}$$

In (1), the dimensional variable is defined as follows:$$\begin{array}{c}\text{(2)}& p_{jD}=\frac{\mathrm{2}\pi k_{ini}h\left(p_{ini}-p_j\right)}{\mu Q_w}\hspace{1em}j=\frac{f}{w}{q}_{fD}=\frac{q_f}{Q_w}\\ x_D=\frac{x}{l_f}\\ y_D=\frac{y}{l_f}\end{array}$$

There are 16 fractures in Figure 5. Considering fractures interaction [25], the pressure drop of fracture in Figure 5 can be expressed as follows:$$\begin{array}{}\text{(3)}& {\overline{p}}_{fDi}={\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{16}}{\overline{q}}_{fDj}{\overline{p}}_{fDi,j}\hspace{1em}i=\mathrm{1,2},\mathrm{3,4},\mathrm{5}\dots ,\mathrm{16}\end{array}$$

Regardless the flow resistance inner fracture, the dimensionless pressure drop of fractures is equal to the horizontal well bore pressure drop.$$\begin{array}{}\text{(4)}& {\overline{p}}_{fDi}={\overline{p}}_{wD}\end{array}$$

The sum of dimensional production of fractures is 1.$$\begin{array}{}\text{(5)}& {\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{16}}{\overline{q}}_{fDj}=\frac{\mathrm{1}}{\mathrm{s}}\end{array}$$

Equations (3)-(5) can be transformed into matrix equations as follows:$$\begin{array}{}\text{(6)}& \left[\begin{array}{ccccc}{\overline{p}}_{fD\mathrm{1,1}}& {\overline{p}}_{fD\mathrm{1,2}}& \cdots & {\overline{p}}_{fD\mathrm{1,16}}& -\mathrm{1}\\ {\overline{p}}_{fD\mathrm{2,1}}& {\overline{p}}_{fD\mathrm{2,2}}& \cdots & {\overline{p}}_{fD\mathrm{2,16}}& -\mathrm{1}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\overline{p}}_{fD\mathrm{16,1}}& {\overline{p}}_{fD\mathrm{16,2}}& \cdots & {\overline{p}}_{fD\mathrm{16,16}}& -\mathrm{1}\\ \mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}& \mathrm{0}\end{array}\right]\left[\begin{array}{c}{\overline{q}}_{fD\mathrm{1}}\\ {\overline{q}}_{fD\mathrm{2}}\\ ⋮\\ {\overline{q}}_{fD\mathrm{16}}\\ {\overline{p}}_{wD}\end{array}\right]=\left[\begin{array}{c}\mathrm{0}\\ \mathrm{0}\\ ⋮\\ \mathrm{0}\\ \frac{\mathrm{1}}{s}\end{array}\right]\end{array}$$

Let$$\begin{array}{c}\text{(7)}& \mathbf{A}=\left[\begin{array}{cccc}{\overline{p}}_{fD\mathrm{1,1}}& {\overline{p}}_{fD\mathrm{1,2}}& \cdots & {\overline{p}}_{fD\mathrm{1,16}}\\ {\overline{p}}_{fD\mathrm{2,1}}& {\overline{p}}_{fD\mathrm{2,2}}& \cdots & {\overline{p}}_{fD\mathrm{2,16}}\\ ⋮& ⋮& \ddots & ⋮\\ {\overline{p}}_{fD\mathrm{16,1}}& {\overline{p}}_{fD\mathrm{16,2}}& \cdots & {\overline{p}}_{fD\mathrm{16,16}}\end{array}\right]\mathbf{q}=\left[\begin{array}{c}{\overline{q}}_{fD\mathrm{1}}\\ {\overline{q}}_{fD\mathrm{2}}\\ ⋮\\ {\overline{q}}_{fD\mathrm{16}}\end{array}\right]\\ \mathbf{O}={\left[\begin{array}{c}\mathrm{0}\\ \mathrm{0}\\ ⋮\\ \mathrm{0}\end{array}\right]}_{\mathrm{16}\times \mathrm{1}}\\ \text{(8)}& \mathbf{I}\mathbf{I}={\left[\begin{array}{cccc}\mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}\end{array}\right]}_{\mathrm{1}\times \mathrm{16}}\end{array}$$

Then (6) is written as a matrix vector equation:$$\begin{array}{}\text{(9)}& \left[\begin{array}{cc}\mathbf{A}& -\mathbf{I}{\mathbf{I}}^{\mathrm{T}}\\ \mathbf{I}\mathbf{I}& \mathrm{0}\end{array}\right]\left[\begin{array}{c}\mathbf{q}\\ {\overline{p}}_{wD}\end{array}\right]=\left[\begin{array}{c}\mathbf{O}\\ \frac{\mathrm{1}}{s}\end{array}\right]\end{array}$$

3.2. Seepage Model in Complex Fracture Networks

Figure 2 shows the typical intersection of fractures in complex fracture networks. How to determine fracture conductivity is the key to establishing the seepage model inner complex fracture networks. In 2004, Karimi used the “star-triangle” transformation method to calculate the fracture conductivity. The conductivity expression between fracture 1 and fracture 2 in Figure 2(a) can be written as follows (Karimi, 2004):$$\begin{array}{c}\text{(10)}& T_{\mathrm{12}}=\frac{{\gamma }_{\mathrm{1}}{\gamma }_{\mathrm{2}}}{{\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}}{\gamma }_i=\frac{A_i{k}_i}{\mu D_i}{\mathbf{n}}_i{\mathbf{m}}_i\hspace{1em}i=\mathrm{1,2}\end{array}$$

Karimi (2004) also gave the formula for calculating the conductivity of multiple intersection fractures as follows:$$\begin{array}{}\text{(11)}& T_{i,j}=\frac{{\gamma }_i{\gamma }_j}{{\sum }_{k=\mathrm{1}}^n{\gamma }_k}\end{array}$$

Based on the fracture conductivity, assuming one-dimensional Darcy flow in the fracture, the production formula for fracture 1 in Figure 2(a) is calculated as follows:$$\begin{array}{}\text{(12)}& q_{f\mathrm{1}}=T_{\mathrm{12}}\left(p_{f\mathrm{2}}-p_{f\mathrm{1}}\right)\end{array}$$

Equation (12) can be nondimensionalized by (2), and the Laplace transform is obtained as follows:$$\begin{array}{}\text{(13)}& {\overline{q}}_{fD\mathrm{1}}=K_{\mathrm{12}}\left({\overline{p}}_{fD\mathrm{2}}-{\overline{p}}_{fD\mathrm{1}}\right)\end{array}$$

Similar to (13) the seepage control equation for fracture 1 in the Y-shaped fracture intersected in Figure 2(b) can be written as$$\begin{array}{}\text{(14)}& {\overline{q}}_{fD\mathrm{1}}=-{\overline{p}}_{fD\mathrm{1}}\left(K_{\mathrm{1,2}}+K_{\mathrm{1,3}}\right)+K_{\mathrm{1,2}}{\overline{p}}_{fD\mathrm{2}}+K_{\mathrm{1,3}}{\overline{p}}_{fD\mathrm{3}}\hspace{1em}{K}_{i,j}=\frac{T_{ij}\mu }{\mathrm{2}\pi k_{ini}h}\end{array}$$

Similarly, in the Y-shaped fractures, the flow expressions of fracture 2 and fracture 3 are$$\begin{array}{c}\text{(15)}& {\overline{q}}_{fD\mathrm{2}}=K_{\mathrm{2,1}}{\overline{p}}_{fD\mathrm{1}}-{\overline{p}}_{fD\mathrm{2}}\left(K_{\mathrm{2,1}}+K_{\mathrm{2,3}}\right)+K_{\mathrm{2,3}}{\overline{p}}_{fD\mathrm{3}}{\overline{q}}_{fD\mathrm{3}}=K_{\mathrm{3,1}}{\overline{p}}_{fD\mathrm{1}}+K_{\mathrm{3,2}}{\overline{p}}_{fD\mathrm{2}}-{\overline{p}}_{fD\mathrm{3}}\left(K_{\mathrm{3,1}}+K_{\mathrm{3,2}}\right)\end{array}$$

Equations (14) and (15) are written as matrix equation:$$\begin{array}{}\text{(16)}& \left[\begin{array}{ccc}-{{\displaystyle \sum }}_{k=\mathrm{2}}^{\mathrm{3}}{K}_{\mathrm{1},k}& K_{\mathrm{1,2}}& K_{\mathrm{1,3}}\\ K_{\mathrm{2,1}}& -{{\displaystyle \sum }}_{k=\mathrm{1,3}}^{}{K}_{\mathrm{2},k}& K_{\mathrm{2,3}}\\ K_{\mathrm{3,1}}& K_{\mathrm{3,2}}& -{{\displaystyle \sum }}_{k=\mathrm{1}}^{\mathrm{2}}{K}_{\mathrm{3},k}\end{array}\right]\left[\begin{array}{c}{\overline{p}}_{fD\mathrm{1}}\\ {\overline{p}}_{fD\mathrm{2}}\\ {\overline{p}}_{fD\mathrm{3}}\end{array}\right]=\left[\begin{array}{c}{\overline{q}}_{fD\mathrm{1}}\\ {\overline{q}}_{fD\mathrm{2}}\\ {\overline{q}}_{fD\mathrm{3}}\end{array}\right]\end{array}$$

Equation (16) is the seepage model for a simpler complex fracture networks with three fractures. Similarly, we can write the seepage matrix equation for the complex fracture networks in Figures 2(c), 2(b), and 5. Since the seepage matrix equation of fracture in Figure 5 is too large, only its matrix vector form is given here:$$\begin{array}{}\text{(17)}& \mathbf{K}\mathbf{p}=\mathbf{q}\text{(18)}& {\mathbf{q}}^{\mathrm{T}}=\left[\begin{array}{ccccc}{\overline{q}}_{fD\mathrm{1}}& {\overline{q}}_{fD\mathrm{2}}& {\overline{q}}_{fD\mathrm{3}}& \cdots & {\overline{q}}_{fD\mathrm{16}}\end{array}\right]\text{(19)}& {\mathbf{p}}^{\mathrm{T}}=\left[\begin{array}{ccccc}{\overline{p}}_{fD\mathrm{1}}& {\overline{p}}_{fD\mathrm{2}}& {\overline{p}}_{fD\mathrm{3}}& \cdots & {\overline{p}}_{fD\mathrm{16}}\end{array}\right]\end{array}$$

In (19), $\mathbf{K}$ is a 16×16 order conductivity matrix.

3.3. Seepage Model of Perforation

Fluid entering the complex fracture networks flows through the perforation hole into the horizontal wellbore (as shown in Figure 4(c)). As shown in Figure 5, horizontal wells have perforation holes in fracture 2, fracture 7, and fracture 14. The pressure drop caused by fluid passing through these perforation holes is shown as below [26]:$$\begin{array}{c}\text{(20)}& \Delta p_{per\mathrm{2}}=r_{\mathrm{2}}{q}_{per\mathrm{2}}\Delta p_{per\mathrm{7}}=r_{\mathrm{7}}{q}_{per\mathrm{7}}\\ \Delta p_{per\mathrm{14}}=r_{\mathrm{14}}{q}_{per\mathrm{14}}\\ \text{(21)}& r_i=\frac{l_{peri}\mu }{A_{peri}{k}_{peri}{N}_i}\hspace{1em}i=\mathrm{2,7},\mathrm{14}\end{array}$$

Ignore pressure loss in horizontal wellbore, then$$\begin{array}{c}\text{(22)}& p_w=p_{per\mathrm{2}}^{\prime }-\Delta p_{per\mathrm{2}}{p}_w=p_{per\mathrm{7}}^{\prime }-\Delta p_{per\mathrm{7}}\\ p_w=p_{per\mathrm{14}}^{\prime }-\Delta p_{per\mathrm{14}}\end{array}$$

Equation (22) can be nondimensionalized by (2) and written as Laplace space expression:$$\begin{array}{c}\text{(23)}& {\overline{p}}_{wD}={\overline{p}}_{perD\mathrm{2}}^{\prime }+R_{\mathrm{2}}{\overline{q}}_{perD\mathrm{2}}{\overline{p}}_{wD}={\overline{p}}_{perD\mathrm{7}}^{\prime }+R_{\mathrm{7}}{\overline{q}}_{perD\mathrm{7}}\\ {\overline{p}}_{wD}={\overline{p}}_{perD\mathrm{14}}^{\prime }+R_{\mathrm{14}}{\overline{q}}_{perD\mathrm{14}}\\ \text{(24)}& R_i=\frac{\mathrm{2}\pi k_{if}h{L}_{peri}}{A_{peri}{k}_{peri}{N}_i}\hspace{1em}i=\mathrm{2,7},\mathrm{14}\end{array}$$

According to the conditions of constant production, the sum of the perforation production is$$\begin{array}{}\text{(25)}& {\overline{q}}_{perD\mathrm{2}}+{\overline{q}}_{perD\mathrm{7}}+{\overline{q}}_{perD\mathrm{14}}=\frac{\mathrm{1}}{s}\end{array}$$

In (25), ${\mathrm{q}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{2}}$ is the liquid flow at the perforation corresponding to the fracture 2, m³/s; ${\mathrm{q}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{7}}$ is the liquid flow at the perforation corresponding to the fracture 7, m³/s; ${\mathrm{q}}_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{14}}$ is the liquid flow at the perforation corresponding to the fracture 14, m³/s.

3.4. Research on Variable Mass Pipe Seepage Model in Horizontal Wellbore

The inflow of liquid at the perforation hole causes the flow of liquid in the horizontal wellbore to be variable mass flow. Fluid rate in horizontal wells is increasing from the toe of the horizontal well to the root (as shown in Figure 4(c)). The pressure drop in horizontal well is composed of frictional pressure drop of the wall, liquid’s accelerating pressure drop, liquid mixing pressure drop, and liquid’s gravity pressure drop. However, scholars at home and abroad have pointed out that the wall friction pressure drop is the main component of the pressure drop in horizontal well. Therefore, we mainly study the wall friction pressure drop; the frictional pressure drop satisfies the following flow equation [27]:$$\begin{array}{c}\text{(26)}& p_{per\left(i+\mathrm{1}\right)}-p_{peri}=F_i{{\displaystyle \sum }}_{\mathrm{1}}^i{q}_{peri}{F}_i=\frac{\mathrm{8}{f}_w\rho \Delta l_{wi}}{{\pi }^{\mathrm{2}}{D}_w^{\mathrm{3}}}\\ \hspace{1em}\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{N}\end{array}$$

Equation (26) is nondimensionalized by (2) and written as Laplace space expression:$$\begin{array}{c}\text{(27)}& {\overline{p}}_{perDi}={\overline{p}}_{perD\left(i+\mathrm{1}\right)}+M_i{{\displaystyle \sum }}_{\mathrm{1}}^i{\overline{q}}_{perDi}{M}_i=\frac{\mathrm{16}{f}_w{k}_{if}h\rho \mu }{{\pi }^{\mathrm{2}}\Delta l_{wi}{D}_w^{\mathrm{4}}}\\ \hspace{1em}\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{N}\end{array}$$

For the specific case in Figure 5, the relationship between ${\overline{p}}_{perD\mathrm{2}}$, ${\overline{p}}_{perD\mathrm{7}}$, ${\overline{p}}_{perD\mathrm{14}}$, and ${\overline{p}}_{wD}$ is shown as follows:$$\begin{array}{}\text{(28)}& {\overline{p}}_{perD\mathrm{7}}={\overline{p}}_{perD\mathrm{14}}+M_{\mathrm{14}}{\overline{q}}_{perD\mathrm{14}}\text{(29)}& {\overline{p}}_{perD\mathrm{2}}={\overline{p}}_{perD\mathrm{7}}+M_{\mathrm{7}}\left({\overline{q}}_{perD\mathrm{7}}+{\overline{q}}_{perD\mathrm{14}}\right)\text{(30)}& {\overline{p}}_{wD}={\overline{p}}_{perD\mathrm{2}}+M_{\mathrm{2}}\left({\overline{q}}_{perD\mathrm{2}}+{\overline{q}}_{perD\mathrm{7}}+{\overline{q}}_{perD\mathrm{14}}\right)\end{array}$$

What needs special attention here is that ${\overline{p}}_{perD\mathrm{2}}$ and ${\overline{p}}_{perD\mathrm{2}}^{\prime }$ are not the same pressure, and the difference between them is perforation pressure drop $R_{\mathrm{2}}{\overline{q}}_{perD\mathrm{2}}$. The difference between ${\overline{p}}_{perD\mathrm{7}}$ and ${\overline{p}}_{perD\mathrm{7}}^{\prime }$ is perforation pressure drop $R_{\mathrm{7}}{\overline{q}}_{perD\mathrm{7}}$. The difference between ${\overline{p}}_{perD\mathrm{14}}$ and ${\overline{p}}_{perD\mathrm{14}}^{\prime }$ is perforation pressure drop $R_{\mathrm{14}}{\overline{q}}_{perD\mathrm{14}}$.

3.5. Coupled Seepage Model of Horizontal Wells with Complex Fracture Networks in Tight Reservoirs

According to continuity condition, the transient pressure behavior model of hydraulic horizontal wells can be obtained as (31) by coupling (9), (17), (23), (25), and (28)~(30).(31) A - I O O O O O O O - I K O 2 O 7 O 14 O O O O Θ Θ 1 1 1 0 0 0 0 Θ Θ 2 R 2 0 0 - 1 0 0 0 Θ Θ 7 0 R 7 0 0 - 1 0 0 Θ Θ 14 0 0 R 14 0 0 - 1 0 Θ Θ M 2 M 2 M 2 1 0 0 - 1 Θ Θ 0 M 7 M 7 - 1 1 0 0 Θ Θ 0 0 M 14 0 - 1 1 0 q p q ¯ p e r D 2 q ¯ p e r D 7 q ¯ p e r D 14 p ¯ p e r D 2 p ¯ p e r D 7 p ¯ p e r D 14 p ¯ w D = O O 1 s 0 0 0 0 0 0

In (31),$$\begin{array}{}\text{(32)}& {\mathbf{O}}_{\mathrm{2}}^{\mathrm{T}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]\text{(33)}& {\mathbf{O}}_{\mathrm{7}}^{\mathrm{T}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]\text{(34)}& {\mathbf{O}}_{\mathrm{14}}^{\mathrm{T}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}\end{array}\right]\text{(35)}& {\mathbf{\Theta }}_{\mathrm{2}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]\text{(36)}& {\mathbf{\Theta }}_{\mathrm{7}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]\text{(37)}& {\mathbf{\Theta }}_{\mathrm{14}}=\left[\begin{array}{cccccccccccccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}\end{array}\right]\end{array}$$

Equation (31) takes into account the dual medium flow, seepage in complex fracture networks, perforation flow, and variable mass flow in horizontal wellbore. A total of 39 unknowns in (31) include ${\overline{q}}_{fDi}$ (i=1,2,…,16), ${\overline{p}}_{Di}$ (i=1,2,…,16), ${\overline{q}}_{perD\mathrm{2}}$, ${\overline{q}}_{perD\mathrm{7}}$, ${\overline{q}}_{perD\mathrm{14}}$, ${\overline{p}}_{perD\mathrm{2}}$, ${\overline{p}}_{perD\mathrm{7}}$, ${\overline{p}}_{perD\mathrm{14}}$, and ${\overline{p}}_{wD}$. The number of equations in the matrix is also 39, so (31) is solvable. The Gauss-Jordan elimination method can be used to find the solution of the unknown in the Laplace space. Using the numerical inversion of Stehfest [28], the Laplace space solution can be transformed into time space solution.

4. Analysis of Model Results

4.1. Validation of Model Results

Chen et al. [8] established a seepage model of horizontal well with orthogonal fractures, which takes into account the seepage within the fracture network but ignores the dual medium flow, perforation flow, and variable mass flow in the horizontal wellbore. The model built in this paper is simplified to Chen ZM model for verification. The basic data used is shown in Table 2.

Table 2

Basic data of tight oil reservoir.

Parameter	Value	Unit
The thickness of the reservoir	12	m
Viscosity of oil	1	mPa.s
Length of main fracture	400	m
Length of secondary	40	m
Cross flow coefficient	2×10−4	Real
Storage ratio	0.16	Real
Production	20	m3/d
Porosity	0.1	Real
Volume factor	1	m3/m3
Number of main fractures	3	Real
Number of secondary fractures	12	Real
The compression coefficient	4×10−4	MPa−1
Formation pressure	26	MPa

From Figure 6, it can be seen that the results obtained in this paper are basically the same as those of the Chen ZM model, but the two models have slightly different results in the early stages of production. This is due to the fact that the direction of fluid flow in the fracture network is artificially specified in the Chen ZM model, and the direction of liquid flow in this paper is automatically selected based on the pressure in the fracture network. The different treatment methods of the flow direction of the liquid in fracture network are the main reasons for the minor differences in the development data in the early stage of production.

[figure omitted; refer to PDF]

4.2. Flow Regimes

Take a simple fracturing horizontal well as an example. The fractures in the network are orthogonal and the geometry of the fracture is shown in Figure 7. Ignore pressure drop in perforation hole and horizontal well. The other basic parameters of the tight reservoir are shown in Table 3.

Table 3

Basic data of tight oil reservoir.

Parameter	Value	Unit
Reservoir thickness	10	m
kx	0.1	mD
ky	0.1	mD
Fracture permeability	106	mD
Cross flow coefficient	10−6	Real
Reservoir pressure	12	MPa
Production	15.9	m3/d
Reservoir porosity	0.1	Real
Volume factor	1	m3/m3
Oil viscosity	1	mPa.s
Compression factor	4.35×10−4	MPa−1
Storage ratio	0.03	Real

[figure omitted; refer to PDF]

Based on the reservoir data in Table 3, the transient pressure behavior of the fractured horizontal well in Figure 7 is solved. The double logarithm well test curves are shown in Figure 8.

[figure omitted; refer to PDF]

As can be seen from Figure 8, the derivative of dimensionless pressure curve shows different laws as development progresses. Combining the previous understanding of the well test curve, the seepage of horizontal well with complex fracture networks can be divided into eight flow regimes: Stage I is linear flow in complex fractures, Stage II is bilinear flow, Stage III is reservoir linear flow, Stage IV is transition flow, Stage V is fracture interference flow, Stage VI is the pseudo-radial flow of the natural fracture system, Stage VII is cross flow, and Stage VIII is pseudo-radial flow in the entire system. The characteristic of each flow regime is summarized in these papers [24, 29]. For the simplicity of the paper, we do not repeat the characteristic of each flow regime.

4.3. Parameter Sensitivity Analysis

4.3.1. Sensitivity Analysis of the Space between Complex Fracture Networks

There are 24 fractures in two orthogonal fracture network (as shown in Figure 9). The other basic data of tight oil reservoir is shown in Table 3. When the distance between the two fracture networks is 200m, 500m, and 1000m, respectively, the well test curves are shown in Figure 10.

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

It can be seen from Figure 10 that the space between the two fracture networks mainly affects the fracture interference flow and has no influence on other flow regimes. With the increase of spacing, the pressure drop curve and the derivative of pressure drop curve are gradually reduced, and the “hump” shaped bump also decreased, indicating that the fracture interference is weaker.

4.3.2. Sensitivity Analysis of Perforation Pressure Drop Coefficient

When the perforation pressure drop coefficient R is equal to 0, 0.1, 1, and 10, the change in the bottom-hole pressure drop of horizontal well is shown in Figure 12. The physical model used for the calculation is shown in Figure 11, and the basic data is shown in Table 3.

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

From Figure 12 we can see that the perforation coefficient has a great influence on the pressure drop in the early stage of horizontal well production. With the increase of perforation coefficient, the larger is the pressure drop in the early stage of production. The change of perforation coefficient has almost no effect on the derivative curve of pressure drop, indicating that the value of the perforation coefficient does not affect the seepage law of fractured horizontal wells.

4.3.3. Sensitivity Analysis of Pipe Flow Pressure Drop Coefficient

Three cases were studied in which the pipe flow pressure drop coefficient M was set to 10⁻¹, 10⁻³, and 10⁻⁶ individually. The well testing curves are shown as Figure 13. The physical model used for the calculation is shown in Figure 11, and the basic data is shown in Table 3.

[figure omitted; refer to PDF]

As can be seen from Figure 13, the pipe flow pressure drop coefficient mainly affects the linear flow in the fracture, the bilinear flow, and the linear flow in the formation. With the increase of the pipe flow pressure drop coefficient, the longer the linear flow in the fracture, the shorter the linear flow in formation and bilinear flow, and it has no influence on the later flow regimes.

4.3.4. Sensitivity Analysis of Permeability of Fracture Network

Assuming that all fractures have the same permeability, five cases were studied in which the fracture permeability was set to 10⁸mD, 10⁷mD, 10⁶mD, 10⁵mD, and 10⁴mD individually. The well testing curves are shown as Figure 14. The physical model used for the calculation is shown in Figure 11, and the basic data is shown in Table 3.

[figure omitted; refer to PDF]

From Figure 14 it can be seen that fracture permeability mainly affects the linear flow in the fracture, the bilinear flow, and the linear flow in the formation. With the increasing of permeability, the duration of the linear flow in fracture and bilinear flow is getting shorter and shorter, and the linear flow in formation is getting longer and longer. The linear flow in fracture and bilinear flow disappeared, when fracture permeability increased to 10⁸mD. It indicates that the fracture permeability is extremely high, and the liquid in fracture reaches the horizontal wellbore almost instantaneously. At this time, the fracture can be regarded as an infinite conductivity fracture

4.4. Transient Pressure Behavior of Horizontal Well with Unorthogonal Fracture Networks

Fractures intersect irregularly in complex fracture networks as shown in Figure 15. Other basic data of tight oil reservoir is shown in Table 3.

[figure omitted; refer to PDF]

Due to the irregular intersection of fractures in the fracture network, the regularity of the linear flow in the fracture, the bilinear flow, and the linear flow of the formation in the well test curve is not obvious. Figure 17 shows the production of each fracture at six moments (${\mathrm{t}}_{\mathrm{D}\mathrm{1}}$~${\mathrm{t}}_{\mathrm{D}\mathrm{6}}$ in Figure 16). The variation of the fracture production in Figure 17 is basically the same as that of Figure 9, which is not repeated here.

[figure omitted; refer to PDF]

[figures omitted; refer to PDF]

5. Conclusion

Based on the methods of source function, superposition principle, Laplace transform, and “star-triangle” transformation, this paper establishes dual media seepage model, seepage model in complex fracture networks, seepage model of perforation, and variable mass seepage model in horizontal wellbore. Using continuity condition, the four models are coupled together to establish a multiscale semianalytical seepage model for horizontal fractured wells in tight oil reservoirs. The following conclusions and understandings can be obtained through this study:

$(1)$ The transient pressure behavior model of horizontal well in this paper considers the influence of dual media flow, seepage model in complex fracture networks, perforation flow, and variable mass seepage model in horizontal wellbore, which is the most well established model at present.

$(2)$ The influence of the spacing of the fracture networks on fractures interference flow was studied. The results show that the smaller the space between the fracture networks is, the larger the “hump” shaped is, and the more the interference between the fractures is.

$(3)$ The fracture permeability mainly affects the linear flow in the fracture, the bilinear flow, and the linear flow of the formation. When the fracture permeability increases to a certain value, the fracture can be considered as an infinite conductivity fracture, the linear flow in the fracture, and the bilinear flow disappear.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation (no. 51804258, no. 51674200, no. 51674198, no. 51874241, and no. 41502311)

Glossary

Nomenclature

A:Area, m²

c:Centroid

D:Distance between the center of fracture and the interface

f:Friction coefficient of pipe wall

h:The thickness of the reservoir, m

k:Permeability, m²

${\mathrm{K}}_{\mathrm{0}}$ :Bessel function

l:Length, m

M:Coefficient of horizontal well pressure drop under unit flow

n:The internal normal unit vector of fracture

N:The number of perforation

p:Pressure, pa

q:Flow rate of fracture, m³/s

Q:Flow rate, m²/s

R:Coefficient of perforation pressure drop under unit flow

s:Laplace variable

TFN:The total number of fracture

T:Transmissibility between fractures, m³/(Pa.s)

x, y, z:Coordinate orientation

$\mu$ :Oil viscosity, pa▪s

$\omega$ :Storage ratio

$\theta$ :Inclined angle

$\lambda$ :Cross flow coefficient.

D:Dimensionless

f:Fracture

ini:Initial

i, j:Number

per:Perforation

w:Wellbore.

“$\stackrel{\stackrel{-}{}}{\hspace{0.17em}}$”:Laplace variable.

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