## Abstract

The purpose of this study is to introduce a new three-linear flow model for capturing the dynamic behavior of water flooding with different fracture occurrences in carbonate reservoirs. Low-angle and high-angle fractures with different occurrences are usually developed in carbonate reservoirs. It is difficult to simulate the water injection development process and the law of water flooding is unclear, due to the large variation of the fracture dip. Based on the characteristics of water flooding displacement streamlines in fractured cores with different occurrences, the matrix is discretized into a number of one-dimensional linear subregions, and the channeling effect between each subregion is considered in this paper. The fractures are divided into the same number of fracture cells along with the matrix subregion, and the conduction effect between the fracture cells is considered. The fractured core injection-production system is divided into three areas of linear flow: The injected fluid flows horizontally and linearly from the matrix area at the inlet end of the core to the fracture and then linearly diverts from the fracture area. Finally, the matrix area at the outlet end of the core also presents a horizontal linear flow pattern. Thus, a trilinear flow model for water flooding oil in fractured cores with different occurrences is established. The modified BL equation is used to construct the matrix water-flooding analytical solution, and the fracture system establishes a finite-volume numerical solution, forming a high-efficiency semianalytical solution method for water-flooding BL-CVF. Compared with traditional numerical simulation methods, the accuracy is over 86%, the model is easy to construct, and the calculation efficiency is high. In addition, it can flexibly portray cracks at any dip angle, calculate various indicators of water flooding, and simulate the pressure field and saturation field, with great application effect. The research results show that the greater the fracture dip angle, the higher the oil displacement efficiency. When the fracture dip angle is above 45°, the fracture occurrence has almost no effect on the oil displacement efficiency. The water breakthrough time of through fractures is earlier than that of nonthrough fractures, and the oil displacement efficiency and injection pressure are more significantly affected by the fracture permeability. With the increase of fracture permeability, the oil displacement efficiency and the injection pressure of perforated fractured cores dropped drastically. The findings of this study can help for better understanding of the water drive law and optimizing its parameters in cores with different fracture occurrences. The three-linear flow model has strong adaptability and can accurately solve low-permeability reservoirs and high-angle fractures, but there are some errors for high-permeability reservoirs with long fractures.

## 1. Introduction

Carbonate reservoirs have various types of fractures with different occurrences—mainly concentrated in high-angle fractures and low-angle fractures, which have a great impact on oilfield water injection development. This fractured carbonate reservoir is very different from conventional reservoirs in terms of fracture development characteristics, fluid distribution rules, reservoir heterogeneity, and displacement development characteristics, leading to water flooding in fractured carbonate reservoirs. The injected water flooding, which is easy to run along the fractures, is directional, causing the water cut of oil wells in the fracture development area to rise at a high rate after water breakthrough.

The research is different from conventional sandstone reservoirs, and the research difficulty is much higher than the latter. For conventional sandstone reservoirs, generally based on the Buckley-Leverett equation [1] to study the theory of water flooding oil, through the BL equation, a one-dimensional linear water flooding analytical solution is established. Mohsen [2] corrected the variability of the BL equation. They used the Welge method instead of the conservation of mass method to solve the position of the water flooding front. Drid and Tiab [3] established a water-gas alternating (WAG) flooding model based on the BL equation, which broadened the scope of application of the BL equation. The analytical method has good adaptability to conventional sandstone reservoirs and has been widely used. However, for carbonate reservoirs, widely developed fractures affect the characteristics of water flooding and need to be considered separately. Therefore, a semianalytical method is introduced. Zhou et al. [4] discretized the fractures into several nodes and then numbered them; gave the fluid flow equations of fractures, wellbore, and reservoir; obtained the semianalytical solution under the discrete fracture network; and explored the evaluation of productivity production. Some scholars regard cracks as infinite diversion [5] and use point source function or line source function to solve them to obtain a semianalytical solution. By establishing a mathematical model of seepage flow and solving it in Laplace space [6, 7], the solution speed is improved. The semianalytical method can flexibly describe various fracture network forms, with high accuracy and fast speed, but there are still certain limitations in solving the flow in fractures [8], and it cannot restore the real seepage characteristics, so it is necessary to deal with fractures in carbonate reservoirs. Some scholars choose to use numerical simulation methods with better flexibility. Numerical simulation methods divide the fracture and matrix into grids. The main methods for simulating the flow of discrete fracture networks are the finite difference method, finite volume method, and finite element method. Lee et al. [9, 10] used the finite difference method to equate small-scale fractures as a continuum, and large-scale fractures were embedded in the flow simulation of discrete fracture networks. The finite element method was first applied to discrete fracture single-phase flow simulation [11–13] and later extended to discrete fracture multiphase flow simulation [14–19], which can be better applied to the oilfield water injection development process. Karimi-Fard and others [20] applied the finite volume method to realize the oil-water two-phase seepage of discrete fractures and dealt with the flow relationship at the intersection of fractures. Numerical simulation methods have good flexibility and can also deal with various complicated reservoir seepage problems, but there will also be problems such as a large number of grids and low computational efficiency.

Based on the water flooding streamline characteristics of fractured cores, the core is divided into several linear subregions, and a core-scale water flooding semianalytical model of trilinear flow is constructed in this paper. The BL equation is used to establish the matrix flow analytical solution, and the finite volume method is used to establish the numerical solution of fracture flow which has formed a BL-CVF hybrid high-efficiency solution method for water flooding oil, with high calculation accuracy; it can realize the calculation of water flooding indicators under any fracture occurrence and key seepage parameters and can realize the water flooding oil saturation field, and the simulation of the pressure field provides a new method for water flooding in carbonate reservoirs.

First, the physical model and assumptions are presented. Next, the three-linear flow mathematical model of injection zone, fracture zone, and production zone is established. Then, we develop a semianalytical solution based on BL-CVF. After that, by comparison with a numerical simulation, the accuracy of the “trilinear flow” model is verified. Last, we explore the water drive law and parameter optimization of cores with different fracture occurrences.

The main innovation of this paper is the establishment of a semianalytical model suitable for exploring the dynamic law of water flooding in cores with different fracture occurrences. Compared with the experimental method, it is impossible to describe arbitrary fracture occurrences. The analytical solution is difficult to couple the matrix and fractures. For the numerical solution, the number of grids is large and difficult to handle. The semianalytical model in this paper can more accurately describe any occurrence of fractures and realize the exploration of core water flooding rules.

## 2. Physical Model and Assumptions

### 2.1. Streamline Characteristics of Water Flooding

For columnar carbonate cores, in order to explore the influence of fractures on fluid migration and water flooding characteristics, different fracture occurrences (Figure 1) can be set to compare and analyze the differences in saturation field and streamline field and explore the law of water-oil displacement in carbonate rock. In particular, when analyzing the streamline characteristics of water flooding, certain characteristics were found: the injected fluid flows linearly from the matrix at the inlet end of the core to the fracture and then flows linearly from the fracture zone, and finally, the outlet end of the core also presents a horizontal linear flow pattern. This feature is called the “trilinear flow” (Figure 2). It can also be seen from the low-angle nonpenetrating fracture streamline diagram that the streamlines in the core are mainly bent at the cracks, and the rest are almost parallel and linear. Based on the above facts, this paper established a three-linear flow model to realize the simulation of water flooding oil saturation field and pressure field.

### 2.2. Basic Assumption

According to the trilinear flow characteristics of cores with different fracture occurrences, the following two basic assumptions can be established: (1) The core matrix is divided into several one-dimensional linear subregions from top to bottom (Figure 3), and the channeling flow between each subregion is considered. (2) Fractures are divided into the same number of fracture cells along the matrix subregion, considering the conductivity between the fracture cells.

Based on the above basic assumptions and given the basic parameters of the core, the physical model of the trilinear flow can be established. The basic parameters of the core mainly include the following aspects: (1) Carbonate columnar core with length $L$ and diameter $d$, the core matrix is isotropic, the permeability is $Km$, the inside of the fracture is also isotropic, the permeability is $Kf$, and the side of the core is a closed boundary. (2) The core and fluid are incompressible and flow is isothermal, which conforms to Darcy’s law. (3) The capillary force and gravity are ignored. (4) At the time $t=0$, the core is saturated with oil and the irreducible water saturation is $Swc$, and the entire core pressure is equal everywhere, which is the initial pressure $Pi$. (5) Considering the core water flooding process, the core is placed horizontally, the injection end of the core is constant flow $Q$ injected water, the outlet end is constant pressure $Pw$, and the oil and water viscosity are $\mu o$ and $\mu w$, respectively.

## 3. Mathematical Model of Water Flooding Trilinear Flow

This chapter establishes the three-linear flow mathematical model of the injection zone, fracture zone, and production zone. Based on the above basic assumptions, the oil-water two-phase governing equations of each linear subregion can be listed, and the mathematical model can be obtained by combining the initial conditions, inner boundary conditions, and outer boundary conditions.

### 3.1. Two-Phase Flow of Oil and Water in the Injection Zone

Water is injected by constant flow $Q$ to the injection zone, and the flow is linear in the matrix (Figure 4).

### 3.2. Fracture Oil-Water Flow Equation

The fracture zone connects the matrix on the left and right sides of the core. In addition to the linear flow in the fracture (Figure 5), it also includes the fluid flowing in from the matrix at the left end and the fluid flowing out from the matrix at the right end.

### 3.3. Oil-Water Flow Equation in Production Area

In the production area, oil and water are injected at the same time, and the total amount of oil and water injected varies with time. Water flooding in the matrix satisfies linear flow (Figure 6).

Initial conditions and inner and outer boundary conditions of the oil-water flow in the three areas are provided in the Appendix.

## 4. Semianalytical Solution Based on BL-CVF

For each linear subregion, we can first assume that the flow rate between the subregions is a known value, so as to obtain the solution of each subregion, and then according to the continuous flow conditions, the subregions are coupled to solve the problem. As a result, the two-dimensional problem of water flooding with fractured cores is transformed into a one-dimensional linear subregion coupling solution problem.

### 4.1. Analytical Solution of Oil-Water Flow in Injection and Production Area Based on BL Equation

*Injection area*: in the linear subregion, since the injection zone is a single-phase water injection, according to the BL equation, the position of a certain saturation surface at time $t$ is

*Production area*: in the linear subregion, the production area is the same injection of oil and water flowing from the fractures. The position of a certain saturation surface is

### 4.2. Numerical Solution of Oil-Water Flow in Fracture Based on CVF

The fracture flow increases the one-dimensional linear oil-water two-phase flow flowing in from the injection zone and flowing out from the production zone, which is solved by the finite volume method.

Therefore, when the oil and water flows $qo,L$ and $qw,L$ in the injection zone and the oil and water flows $qo,R$ and $qw,R$ in the production zone are determined, the fracture pressure $Pf$ and the fracture water saturation $SwF$ can be solved.

### 4.3. Three-Zone Coupling Solution Process

The process of the three-zone coupling solution is shown in Figure 8. By inputting the initial parameters, the linear flow equations of the injection zone, the production zone, and the fracture zone, are solved so as to realize the calculation of various indicators of water drive under any fracture occurrence and key seepage parameters through iterative calculation, and the simulation of water drive oil saturation field and pressure field is realized.

## 5. Model Verification and Analysis

In order to verify the accuracy of the “trilinear flow” model, this chapter will carry out comparative analysis from numerical simulation. In the verification of the numerical simulation, the errors of the new model and the numerical simulation in the pressure field, saturation field, and water flooding index were analyzed, and some suggestions for improvement were put forward.

### 5.1. Initial Parameters

According to the coupling solution process given in Section 4.3, the first is the input of initial parameters. Table 1 shows the basic parameters of the trilinear flow model. Figure 9 shows the relative permeability curves of the matrix and fractures, and according to different relative permeability curves, the matrix and fractures establish the relative permeability zones.

### 5.2. Numerical Simulation Verification

The water drive trilinear flow model can simulate the pressure and saturation fields, compare the new model with the numerical simulation results (Figures 10 and 11), and analyze the errors of the two fields. It can be seen from the analysis that the laws of the two fields are the same, but due to the linear seepage considered by the new model, compared with the bending of the streamline at the actual fracture, the phenomenon of fracture channeling becomes weaker, the waterline advancement speed becomes slower, and the propagation speed of the pressure drop becomes slower.

Through the comparison of production characteristic curves, the calculation results of the calculation examples show that the oil displacement efficiency prediction is relatively accurate. Although there is a certain error in the value, the relative error is not large, only 9%. The oil production and water production predictions are in good agreement with the numerical model, which reaches 86% (Figures 12–14).

## 6. Water Flooding Characteristics and Production Law

Numerical simulations verify the correctness and certain accuracy of the trilinear flow model. Therefore, the model can explore the water drive law and parameter optimization of cores with different fracture occurrences. Next, a design to change the occurrence of core fractures and fracture permeability is made in order to explore the laws of water flooding.

### 6.1. Different Fracture Occurrence

In order to explore the influence of different fracture occurrences on water flooding characteristics and production rules, only the fracture occurrence is changed here, and the remaining basic parameters are the same as Table 1. Table 2 shows the schematic diagrams of different fracture occurrences and the oil saturation field at the same time. Different fracture occurrences lead to significant differences in the oil saturation field. As the inclination angle of the fracture increases, the effect of the fracture decreases, the degree of flow along the fracture decreases, and the water breakthrough time decreases accordingly.

Aiming at the 8 fracture occurrences designed, this paper uses the trilinear flow model to explore the influence of different fracture dip angles on oil displacement efficiency, water cut, and core injection pressure, as shown in Figures 15 and 16. By observing the oil displacement efficiency curve, it is found that through fractures (0°, 10°, 15°) have the lowest oil displacement efficiency. When the fracture dip is above 45°, the fracture occurrence has almost no effect on the oil displacement efficiency. The change of water cut is analyzed, the water breakthrough time is early, but the increase of water cut is gradual. On the contrary, the water breakthrough time of nonperforated fractures is late, and the water cut changes from 0 to more than 90%; namely, no oil is produced after the core outlet end hits water.

### 6.2. Different Fracture Permeability

The ratio of fracture permeability to matrix permeability affects the law of water flooding. In this paper, only the fracture permeability is changed and the matrix permeability remains unchanged to explore how different types of fracture permeability affect the mechanism of water flooding. The set fracture permeability, four fracture occurrences, and the oil saturation field at the same time are shown in Table 3. The table shows that for each type of fracture occurrence, as the fracture permeability increases, the fracture channeling phenomenon becomes more significant, and the water breakthrough time is earlier. Under the same fracture permeability, as the fracture dip increases, the waterline advances more uniformly, and the oil displacement efficiency is higher.

Based on the trilinear flow model, 6 types of fracture permeability are designed, combined with 4 fracture occurrences, to further analyze the influence of different types of fracture permeability on the law of water flooding. Table 4 reflects the water-free oil production period with different types of fracture permeability; it can be concluded that the water breakthrough time of nonpenetrating fractures is late, and the size of the fracture permeability has little effect on the water breakthrough time. The water breakthrough time of the penetrating fracture is early, and as the fracture permeability increases, the water breakthrough time is greatly reduced. Similarly, the difference in water-free oil displacement efficiency of the four fracture occurrences can also be analyzed (Figure 17): the water-free oil displacement efficiency of through fractures is low and the range of changes is large. The oil displacement efficiency is significantly affected by the fracture permeability, while nonperforated fractures are in contrast. Figure 18 reflects the variation of core injection pressure with fracture permeability, and the injection pressure of nonthrough fractures is significantly higher than that of through fractures.

## 7. Summary and Conclusions

This paper proposes a new three-linear flow model for water flooding with different fracture occurrences in carbonate reservoirs and validates and applies the model, from which the following conclusions can be drawn:

- (1)
The injected fluid flows linearly from the matrix at the inlet end of the core to the fracture and then flows linearly from the fracture zone. Finally, the outlet end of the core also presents a horizontal linear flow pattern. The flow lines in the core are mainly bent at the cracks, and the rest are almost parallel and linear

- (2)
Comparing the semianalytical model and numerical simulation, it is found that the water drive dynamics of the two are consistent, and the coincidence degree of oil and water production is 86%, which shows the accuracy of the trilinear flow model. The model is easy to construct and can describe fractures with any dip. It can also realize the calculation of various indicators of water flooding and the simulation of pressure field and saturation field, with good on-site adaptability

- (3)
The larger the fracture dip, the higher the oil displacement efficiency; when the fracture dip is above 45°, the fracture occurrence has almost no effect on the oil displacement efficiency. The water breakthrough time of cores penetrating fractures is early, but the water cut increases gradually; when the fracture dip is 20°, the water cut is in the middle

- (4)
Fracture permeability has little effect on the oil displacement efficiency and injection pressure of nonperforated fractured cores but has a large impact on the oil displacement efficiency and injection pressure of perforated fractured cores. With the increase of fracture permeability, the oil displacement efficiency and the injection pressure of perforated fractured cores dropped drastically

- (5)
The semianalytical water flooding model established in this paper has good adaptability to cores with low-permeability and high-angle fractures, because their streamlines are almost horizontally linear. For high-permeability cores or low-angle fractures, the streamlines will bend, which increases the model error

### Appendix

The inner and outer boundaries of the crack are closed boundaries.

## Nomenclature

- $kro$:
Relative permeability of the oil phase

- $krw$:
Relative permeability of the water phase

- $x$:
Linear flow direction of the fluid, m

- $qw$:
Water injection volume, m

^{3}/s - $Bo$:
Crude oil volume coefficient

- $Bw$:
Water volume coefficient

- $pm$:
Matrix pressure, MPa

- $qo,m$:
The amount of oil flowing between subregions per unit volume

- $qw,m$:
The amount of water flowing between subregions per unit volume

- $\varphi $:
Porosity

- $Soi$:
The initial oil saturation

- $xf$:
Position of the water flooding front, m

- $Swf$:
Water saturation of the front

- $fwLL$:
The water cut at $LL$ on a certain time

- $qi$:
Flow rate in the injection zone and the production zone

- $LL$:
The position of the fracture, m

- $qo,L$:
The amount of oil supplied to the injection zone per unit volume

- $qo,R$:
The amount of oil supplied to the production zone per unit volume

- $qw,L$:
The amount of water supplied to the injection zone per unit volume

- $qw,R$:
The amount of water supplied to the production area per unit volume

- $fw\u2032sw$:
The $\u2202fw/\u2202Sw$ when water saturation is $Sw$

- $x0$:
Position of the original oil-water interface, m

- $RLt$:
The flow resistance in porous medium at the injection end

- $pf$:
Pressure of the fracture, MPa

- $\epsilon $:
Direction in which the fracture extends, m

- $wh$:
The area of the subregion, m

^{2} - $SwF$:
Water saturation at the crack

- $tf$:
Water breakthrough time in the crack

- $Xi$:
Fracture pressure and saturation.

## Data Availability

The data used to support the findings of this study are included within the article.

## Conflicts of Interest

The authors declare that they have no conflicts of interest to report regarding the present study.

## Acknowledgments

The authors thank the National Science and Technology Major Project of China (No. 2017ZX05030-002) for the financial support.