## Abstract

The lateral broadband fracturing (LBF) technique for vertical well in tight oil reservoir has been proven to provide much larger stimulated reservoir volume (SRV) by widening its width. Although this new fracturing technique has been successfully applied to the development of tight oil reservoirs in Ordos Basin, China, there is still a lack of models and methods to characterize the imbibition of matrix-fracture system, which is heterogeneous permeability distribution. In this paper, a multilinear fractal model considering Nuclear Magnetic Resonance-based research on imbibition (MFMI) is established to characterize the flow characteristics of lateral broadband fracturing vertical wells (LVWs) in tight oil reservoirs by combining the dual-porosity fractal model considering imbibition and the quad-linear flow model. Due to the application of LBF, the nonuniform distribution of fracture system is characterized by fractal porosity and fractal permeability. In addition, the imbibition in SRV of tight oil reservoir is quantitatively characterized by Nuclear Magnetic Resonance (NMR) data. And the production performance of LVWs is quantified by the MFMI. By using the Laplace transformation, Bessel function, iteration, and Stehfest numerical inversion algorithms, the approximate analytic solutions of our established model, including primary hydraulic fractures, SRV, and unstimulated reservoir volume (USRV), are derived. The solutions of pressure and production are used to compare and analyze in order to discuss the influence of parameters related to lateral broadband fracturing (such as fractal parameters, reservoir parameters, and width of SRV) on flow behavior of LVWs in a tight oil reservoir. The modeling results show that the fractal parameters of fracture system have great effect on the fluid flow in LVWs, and LBF contributes to imbibition production.

## 1. Introduction

In the past decade, in the success of development of tight oil reservoir by large-scale volume fracturing techniques in the US, China’s oilfields have begun to attach great importance to the development of tight oil reservoirs and formed suitable fracturing techniques for their actual situation [1]. The lateral broadband fracturing (LBF) technique is a new one of the large-scale volume-fracturing techniques for vertical wells in tight reservoirs, which, by applying the degradable fiber and temporary plugging ball force to the fracturing fluid from a low stress area to a high stress area, reconstruct unreconstructed area more completely and increase the width of the SRV, so as to realize the maximum of wellbore coverage and contact with oil reservoir to increase production and enhance the oil recovery. Moreover, the technology of lateral broadband fracturing has been successfully applied in Changqing oilfield [2, 3].

In the early stage of the development of some tight oil reservoirs in Changqing oilfield, vertical wells were adopted. However, the old vertical well network cannot effectively drive the all reservoirs, because of large row spacing, big threshold pressure gradient, and small width of fracturing area, so that there is a large amount of remaining oil distribution between the well rows, forming a banded zone of residual oil. Hence, there is no room to apply a large-scale volume-fractured horizontal well to develop these zones of residual oil, and the LBF for old vertical wells is chosen for economic, applicable, and efficient. Therefore, clarifying the impact of LBF on fluid flow behavior and oil recovery is an important basis for formulating reservoir development plan.

In recent years, a large number of methods have been introduced to solve the mathematical model of the fluid flow around large-scale volume-fractured horizontal/vertical well.

Lee and Brockenbrough [4] first established a trilinear flow model, which lays a foundation for the follow-up study of volume-fracturing wells, in order to study the fluid flow behavior of conventional fracturing well. Ozkan et al. [5] and Brown et al. [6] established a trilinear flow model considering the influence of SRV by modeling dual-porosity model in order to study the multiple fractures along horizontal wells in unconventional reservoirs. Based on the trilinear flow model, Stalgorova and Mattar [7] established a five-region flow model to characterize the USRV more accurately. However, these models mentioned above do not consider the heterogeneous distribution characteristics of fractures in SRV region.

To solve this problem, Cossio et al. [8] introduced the fractal permeability and fractal porosity in a trilinear flow model based on the fractal dual-porosity systems study of Chang and Yortsos [9]. Wang et al. [10–12] and Wang et al. [13] study the impacts of fracture heterogeneity on well productivity by establishing the fractal-fractured horizontal wells. Su et al. and Sheng et al. [14, 15] introduced the fractal parameters in a hydraulic fractured vertical well model in TORs. Sheng et al. [16] proposed a new method to calculate tortuosity index of fractal induced-fracture and presented the fractal dimension of induced-fracture aperture to describe the distribution of fractal induced-fracture aperture. However, Sheng’s model only considers that the reservoir fracturing area and oil drainage area are circular and does not consider the square distribution characteristics of volume fracturing caused by reservoir stress direction.

In addition, since the tight oil reservoirs after volume fracturing can be similar to fractured reservoirs, in which imbibition plays an important role in improving the oil recovery, the imbibition factor cannot be ignored in the flow model of volume-fracturing vertical well in tight oil reservoir. Su et al. [17] presented a multilinear model considering imbibition for horizontal well, in which the imbibition is represented in terms of capillary pressure between matrix and fracture. Wang et al. [18, 19] presented a multilinear fractal model for multiple-fracturing horizontal wells in tight oil reservoirs by combining fractal parameters and imbibition parameters and studied the influencing factors on the spacing of volume-fracturing horizontal well. In terms of experiments, Yang et al. [20] analyzed the influencing factors of imbibition through the spectrum of NMR experimental. Although these models consider the imbibition phenomenon in the development of tight oil reservoir, the simulation results are roughly consistent with the experimental results in trend and cannot be quantitatively analyzed.

Therefore, in this paper, a multilinear fractal model considering NMR-based research on imbibition (MFMI) is established by combining the dual-porosity fractal model and the quad-linear flow model to present the flow behaviors and production contribution rate of each region of LVWs.

## 2. Mathematical Model

### 2.1. Mathematical Characterization of Fractal Characteristics in the SRV Region of LVWs

In the conventional dual-porosity model, the fractures in the SRV region are treated as a fracture network formed by uniform sand filling, in which fracturing fracture conductivity is the same (Figure 1(b)). However, in the actual fracturing process, due to attenuating of fracturing energy propagation, the limitation of the stress direction, and the uneven distribution of the sand filling, the conductivity of the volume-fracturing fracture is reduced with the increase of the distance from the well (Figure 1(a)). Moreover, since the LBF largely overcomes the obstacle from the stress direction, the width of the SRV region is increased, and the difference of the fracture heterogeneity is increased. Therefore, the conventional dual-porosity model does not apply to the SRV region with the LBF.

As shown in Figure 1, when $\theta m=0.05$ and $df=1.95$, in SRV region, the closer to the well in SRV region, the higher the fracture permeability, and the farther away from the well, the lower the fracture permeability, which is consistent with the actual fracture distribution. When $\theta m=0$ and $df=2$, the fracture permeability is homogeneous. The fractal dual-porosity model is the same as the conventional dual-porosity model, which make the fractal dual-porosity model more generalized.

### 2.2. Mathematical Characterization of NMR-Based Research on Imbibition

Because the tight reservoir after LBF can be regarded as a fractured reservoir, in which imbibition is an important production mechanism, the imbibition factor cannot be ignored in the dual-porosity model.

The results show that under the action of capillary force, the residual fracturing fluid (wetting phase) in the fracture will displace the oil (nonwetting phase) in the matrix by imbibition, the amount of crossflow between matrix and fracture in SRV region is the sum of the amount of imbibition caused by capillary force and displacement caused by differential pressure [20].

^{3}), $Qcrossflowp$ is the cumulative production of crossflow by pressure difference, $Qcrossflowc$ is the cumulative production of crossflow by imbibition, $t$ is the production time, $VII$ is the volume of SVR region, $qcrossflowp$ is production of crossflow by pressure difference (kg/m

^{3}/d), $\alpha $ is shape factor of matrix (m

^{-2}), $km$ is permeability of matrix (mD), $pm$ is matrix pressure, MPa, $pf$ is fracture pressure (MPa), and $\mu $ is viscosity of oil (MPa·s).

The transverse relaxation time ($T2$) in the NMR spectrum is a characteristic parameter of the fluid transfer energy. The $T2$ value of the fluid at the wall of small pores and large pores is small, while the $T2$ value of the fluid in the middle of the large pores is large, so it can be used as the $T2$ cutoff value that separates the NMR spectra, the left part represents the fluid signal at the walls of the small and large pore channels, and the right part represents the fluid signal in the middle of the large pore channel.

Through NMR experiment, we can test the NMR signals in the states of saturated water and irreducible water and after water flooding and make corresponding NMR spectra and calculate the imbibition production and displacement production by area, and the final recovery ratio from imbibition and displacement can be given, as shown in Figure 2(a) [20].

### 2.3. Mathematical Model of MFMI

For tight oil reservoirs, a semianalytical model for LVWs is established, which combines the dual-porosity fractal model considering imbibition and the quad-linear flow model.

#### 2.3.1. Physical Assumptions

In our model, a rectangular ideal reservoir with wellbore as the center is established, in which the fracture length is $2xf$, the fracture width is $2wf$, the fracturing area width is $2yf$, the seepage model length is $2xf+2dG$, and the seepage model width is $2wf+2dG$, as shown in Figure 3. And the reservoir characteristics are assumed as follows:

- (1)
The crude oil, water, and fracturing fluid in reservoir are simplified as the single-phase fluid flow of crude oil

- (2)
Liquid in the matrix flows only into the fractures

- (3)
There is a threshold pressure gradient in the unfractured region, and the boundary of the porous flow in the unfractured region is determined by the threshold pressure gradient, and the distance from the outer boundary of the unfractured area to the inner boundary is equal to the ratio of the differential pressure on both sides to the starting pressure

- (4)
It is assumed that the flow in the model is linear in $x\u2009$and $y$ directions, the parallel primary fracture direction is the $x$ direction, and the vertical primary fracture direction is the $y$ direction

- (5)
It is assumed that the Mass fractal dimensions of fracture and matrix in our dual-porosity model are the same ($dm=df$), and the matrix is homogeneous ($\theta m=0$)

According to the reservoir characteristics and porous flow direction, it can be divided into four regions including primarily fracture (Region I), SRV region (Region II), and USRV regions (Regions III and IV).

#### 2.3.2. Model Descriptions and Boundary Conditions

- (1)
Porous flow in primary fracture (Region I):

The fluid in Region I flows into the wellbore along the $X$ direction.

- (2)
Multiscale porous flow in SRV region (Region II):

- (3)
Porous flow in unstimulated reservoir (Region III, Region IV):

^{3}/d), $p1$ is the pressure of Region I (MPa), $p2$ is the pressure of Region II (MPa), $p3$ is the pressure of Region III (MPa), $p4$ is the pressure of Region IV (MPa), $pi$ is the initial reservoir pressure (MPa), $G$ is the threshold pressure gradient (MPa/m), $Cl$ is the compressibility of fluid (MPa

^{−1}), $k1$ is the fracture permeability of Region I (mD), $\phi 1$ is the porosity permeability of Region I, $h\u2009$is the thickness of reservoir, $wf$. is the width of primarily fracture, $ct$ is the total compressibility of USRV region (MPa

^{−1}), $ct1$ is the total compressibility of Region I (MPa

^{−1}), $ctm$ is total compressibility in matrix (MPa

^{−1}), and $ctf$ is total compressibility in fracture networks (MPa

^{−1}).

## 3. Methodology

In this part, the MFMI for LVWs we established is solved and verified with the actual production data from tight oil reservoir in Ordos Basin, China.

### 3.1. Semianalytic Solutions

### 3.2. Model Verifications

## 4. Results and Discussion

### 4.1. Flow Regime Analysis

According to the data in Table 1, the curves of pressure, derivative pressure, production rate, and derivative production rate can be calculated by MFMI. In the light of the special slope of derivative pressure curve, the porous flow process of conventional fracturing wells can be divided into four regimes by four typical periods, while the porous flow LVWs can be divided into seven regimes by five typical periods, as shown in Figure 5. LBF adds three stages (the bilinear flow of fracture system in SRV, the similar boundary response in SRV, and the stage of crossflow) on the basis of the four stages with conventional fracturing (the wellbore storage period, the linear flow in primary fractures, and the boundary response of the USRV).

### 4.2. Sensitivity Analysis of LBF Widths

Figure 6 shows that the width of SRV caused by LBF has an important influence on the process of horizontal well flow. When the width of SRV increases, the differential pressure increases, and the production rate increases significantly.

From the analysis of productivity contribution of four regions divided by our model, the fracturing width is positively correlated with the production from Regions II and IV and negatively correlated with the production from Region III. This is because with the increase of fracturing width, the area of Region II and Region IV increases, and Region III is farther away from the well resulting in the reduction of driving differential pressure. In addition, Figure 7 shows that with the increase of width, the oil recovery from imbibition also increases greatly. Imbibition mainly occurs in Region II, and when the area of Region II increases, more crude oil will be produced by imbibition.

### 4.3. Sensitivity Analysis of Fractal Parameters

Figures 8 and 9 show that the fractal parameters have an important influence on the process of horizontal well flow. The different values of fractal parameters result in the different curves of pressure, pressure derivative, production, and production derivative.

Figure 8 shows that when the $df$ increases, the fracture complexity increases in SRV, the pressure curve increases, the differential pressure decreases, and the production rate decreases significantly. And with the increase of LBF width, $df$ has a greater impact on flow behavior of LVWs in tight oil reservoir. In addition, it is observed from Figure 9 that when the $\theta f$ increases, the path of fluid flowing to wellbore by fracture is farther, the pressure curve decreases, the differential pressure increases, and the production rate decreases significantly. And with the increase of LBF width, $\theta f$ has also a greater impact on flow behavior. These modeling results are consistent with the physical properties of the fractal parameters. In addition, Figure 10 shows that with the increase of $df$ and decrease of $\theta f$_{,}, the oil recovery from imbibition in SVR also increases greatly.

### 4.4. Sensitivity Analysis of Wettability

Eight cores from different blocks in tight oil reservoirs were selected for NMR displacement experiment, and their NMR spectra were obtained for analysis, as shown in Figure 11. On this basis, the ratio of recovery of imbibition to recovery of displacement and water-wetting index are calculated according to Equations (3) and (4), as shown in Table 2.

It is observed from Figure 12(b) that with the increase of reservoir hydrophilicity, the oil recovery from imbibition in SVR also increases greatly. The larger the water-wetting index of the reservoir, the greater the capillary force between the oil-water interface in the pores of the reservoir, and the stronger the imbibition effect, so the imbibition oil production increases with the increase of the water-wetting index.

## 5. Conclusions

- (1)
In our paper, fractal geometry parameters of fracture system and imbibition are combined in SRV region and proposed a multilinear fractal model considering NMR-based research on imbibition for LVWs in a tight oil reservoir

- (2)
Pressure transient and production rate derivative curves of LVWs were analyzed. Compared with conventional fracturing vertical wells, LBF can make the flow behavior of fracturing vertical wells from four stages to seven stages, which is conducive to the recovery of tight oil. From the analysis of productivity contribution of four regions divided by our model, with the fracturing width of SRV region increasing, the production contribution of Region II (multiscale porous flow in SRV region) and Region IV (porous flow in unstimulated reservoir) increased significantly

- (3)
The impact of LBF on production of fracturing vertical wells in tight oil reservoirs is analyzed and discussed. The increase of SRV area caused by LBF promotes the effect of fractal parameters and imbibition on oil recovery. The result showed that fractal parameters of fracture system have great effects on the stages of bilinear flow of fracture system in SRV and crossflow; the larger the Mass fractal dimension of fracture ($df$) in SRV region or the smaller the conductivity index of fluid flow in matrix ($\theta f$) is, the larger the production will be. And the larger the $df$ or the smaller the $\theta f$ or the more hydrophilic the reservoir in SRV region is, the larger oil recovery from imbibition is

### Appendix

^{3}/d).

The porous flow equations of LVWs can be converted to dimensionless, and the Laplace transform can be given.

- (1)
The Laplace transform of porous flow equations of Region III can be given by

- (2)
The Laplace transform of porous flow equations of Region IV can be given by

- (3)
The Laplace transform of porous flow equations of Region II can be given by

- (4)
The Laplace transform of porous flow equations of Region I can be given by

## Data Availability

The oil field geological description, except the information presented in the manuscript, used to support the findings of this study is restricted by the Safety Law of Petrel China. However, other data such as the depth or reservoir fluid description used to support the findings are currently under embargo while the research findings are commercialized. The requests for data, 12 months after publication of this article, will be considered by the corresponding author.

## Conflicts of Interest

All authors declare that there are no conflicts of interest regarding the publication of this article.

## Acknowledgments

This work was supported in part by the Natural Science Project of Fujian Province, China (Grant nos. 2020J01860, 2021J011028, 2022J05236 and 2022J011128), in part by the Scientific Research Foundation for the Ph.D., Minjiang University (no. MJY19032), in part by the National Natural Science Foundation of China (Grant no. 52172327), in part by the Science and Technology Planning Project of Fuzhou (Grant no. 2021-S-236), in part by the National Science and Technology Major Special Support Program (Grant no. 2017ZX05064), in part by the Natural Science Foundation of Chongqing (cstc20jcyj-msxmx0216), and in part by the Bayu Scholars Program.