A Semianalytical Approach for Production of Oil from Bottom Water Drive Tight Oil Reservoirs with Complex Hydraulic Fractures

Dai, Shuhui; Chen, Chunqiang; Wang, Hao; Fu, Zhaohui; Liu, Dameng

doi:https://doi.org/10.1155/2019/5186831

Journal of Chemistry

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 5186831 | https://doi.org/10.1155/2019/5186831

A Semianalytical Approach for Production of Oil from Bottom Water Drive Tight Oil Reservoirs with Complex Hydraulic Fractures

Shuhui Dai,^1,2Chunqiang Chen,¹Hao Wang,³Zhaohui Fu,⁴and Dameng Liu²

Academic Editor: Fateme Razeai

Received28 Aug 2018

Revised04 Dec 2018

Accepted18 Dec 2018

Published06 Mar 2019

Abstract

In order to improve the recovery rate, fractured horizontal wells are widely used in tight reservoirs. In general, a complex fracture system is generated in tight reservoirs to improve oil production. Based on the fractal theory, a semianalytical model is presented to simulate a complex fracture system by using the volumetric source method and superposition principle, and the solutions are determined iteratively. The reliability of this model is validated through numerical simulation (Eclipse 2011); it shows that the result from this method is identical with that from the numerical simulation. The study on influence factors of this model was focused on matrix permeability, fractal dimension, and half-length of main fracture. The results show that (1) the production increases with the increase of half-length of main fracture and matrix permeability during the initial production stage, but the production difference becomes smaller in different half-length of main fractures and matrix permeability in middle and later stages and (2) the cumulative production increases with the increase of fractal dimension, and the increments of cumulative production in different fractal dimensions gradually increase during the initial production stage, but the increments tend to be stable in middle and later stages. This study can be used for forecasting the oil production from bottom water tight oil reservoirs with complex hydraulic fractures. It can also guide in optimization of horizontal well fracturing design to improve oil production and oil recovery in bottom water tight oil reservoirs.

1. Introduction

With the increasingly lack of energy sources and strict requirement of customers, more attention is being paid on unconventional resources by researchers and scholars [1–7]. Among these reserves, tight oil reservoirs have larger proportion. Because of the low permeability and low porosity of tight oil reservoirs, it is necessary to improve production by the hydraulic fracturing technology. In recent years, the fracture geometry is considered as constant fracture width, and the directions of fractures are perpendicular to the horizontal wellbore [8–10]. Also, analytical and semianalytical approaches were used to predict the production of vertical fractures in such systems [11–15]. A hybrid numerical/analytical model was presented to simulate the pressure transient with a finite-conductivity fracture [16]. A numerical model for predicting the production was presented by microseismic data in a shale gas reservoir [17]. Based on the planar vertical fractures, many models were presented for production [18–20].

However, the actual hydraulic fracture geometry is more complex than it was assumed earlier. The complexity of main fractures and branch fractures was revealed by the field microseismic data [21]. A semianalytical model was presented to produce a point source from complex hydraulic fracture networks in the Barnett shale [22]. Based on the nonplanar hydraulic fracture geometry and point source, a semianalytical model was presented to predict the production [23].

In the last few years, numerous attempts have been made to model complex fracture geometry using a fractal theory. The induced and natural fractures were taken as orthogonal fracture networks in shale gas reservoirs [17], and they were used to build a numerical model to study the stimulated reservoir volume (SRV). A semianalytical model was built to study the complex well performance by using a point source and fractal theory in a tight oil reservoir (Figure 1) [24]. Then, a semianalytical fractal model was built for production by using a fractal-tree network model [25]. A fractal induced fracture network model was also proposed to study SRV heterogeneity in tight oil reservoirs [26].

In this study, a semianalytical model for fracture network production is presented by the volumetric source and fractal theory. The model is verified by the results of commercial numerical simulation software (Eclipse 2011). Then, the influencing factors of matrix permeability, half-length of main fracture, and fractal dimensions are studied. This work is helpful in understanding the effect of fracture geometry on production in tight oil reservoirs.

2. Methodology

A horizontal well is located in a box-shaped tight reservoir with bottom water drive, and the well has been fractured. Fractures are considered as fractal geometry, and fractures fully penetrate the reservoir. It should be noted that the energy of bottom water decreases. The specific assumptions are as follows:(1)The horizontal well is located in the middle of a box type reservoir. Five boundaries of the reservoir are closed and the bottom boundary has a constant pressure.(2)The reservoir is anisotropic, homogeneous, and slightly compressible.(3)The reservoir contains a single-phase fluid, and the fluid flow is steady.(4)The production of all the fractures is equal to the horizontal well.

2.1. Complex Fracture Characterization

It is assumed that the fractal model obeys strict self-similarity, and the starting point coordinate of the fracture is (0, 0) (Figure 2). Then, the coordinate of the ith subtube of the k-level is (x_ij, y_ij), and j is the number of fracture segments. The bifurcate angle of the fractal fractures is δ_k and the coordinate of the end point is (m_ij, n_ij) as shown in Figure 2. This subtube generates the level of (k + 1) with (2i − 1)th and 2ith subtubes in which . The coordinates of the fractal fracture network system can be expressed as [24]where l_k is the length of k-level branch fractures, m is the maximum level of branch fractures, k is the level of branch fractures, n is the maximum number of subtubes, i is the number of subtubes, n is the total number of fracture segments, j is the number of fracture segments, and δ_k is the bifurcate angle.

Fracture distribution shows fractal characteristics; the length and radius of a fracture can be represented as [27]

The following result can be obtained from Equation (2):where d_k is the diameter of k-level branch fractures, l₀ is the primary-level fracture length, and d₀ is the primary-level fracture diameter.

Because this model shows self-similar fractal characteristics, it is assumed that the similar dimensions of the length of the bifurcated pipe and the diameter of the bifurcated pipe are D_l and D_d:where D_l is the dimensionless primary-level fracture length and D_d is the dimensionless primary-level fracture diameter.

2.2. Reservoir Flow

Figure 3 shows the schematic of a volumetric source system. In this paper, the porous media is assumed to be an anisotropic, homogeneous reservoir and slightly compressible in a box-shaped reservoir.

As illustrated in Figure 3, x_e, y_e, and z_e are the size of x, y, and z in three directions of the reservoir. There is a volumetric source in the tight oil reservoir and the strength is q. The central point coordinate of the volume source is (c_x, c_y, c_z), and the size of the volume source in three directions is , , and .

The volumetric source model for tight oil reservoirs with bottom water drive can be expressed as

The boundary conditions arewhere is the Heaviside function:where p_i is the pressure of bottom water drive, is the tight oil reservoir pressure, q is the production rate in a tight oil reservoir, k is the formation permeability, and V_source is the physical dimension of volumetric source.

Detailed analysis and solution process of this equation are given by Luo et al. [29]. The solution at the tip of the ith subtube in k-level branch fractures can be expressed aswhere

Equation (9) represents the solution of homogeneous reservoirs; however, reservoirs are usually anisotropic. Hence, the correction method of Besson [30] is introduced. The specific expression is as follows:

It is assumed that the cylindrical wellbore is equivalent to a rectangular wellbore, and the assumption does not affect steady flow [31, 32]. Then, the dimensions of the three-dimensional direction can be converted as

Substituting Equations (11)–(15) in Equation (9), we can obtain the pressure of segment i of k-level branch fractures:

It is assumed that the pressure drop coefficient of the reservoir is , then can be expressed as

Then, Equation (17) can be given as

2.3. Fracture Flow

Figure 2 shows the flow model of a fracture system. The fluid flows through the branch fractures to the main fracture and then to the wellbore. It is assumed that the wellbore has infinite conductivity and the fracture is a one-dimensional Darcy flow, then the pressure drop from the fracture to the wellbore can be expressed as

It is assumed that the pressure drop coefficient of each segment of the fracture is I_ij, then I_ij can be expressed as

The fracture system satisfies the material balance. For the whole fracture system, the inflow volume is equal to the outflow:

Combining Equations (19)–(21), the pressure drop of fracture can be calculated aswhere l_ij is the fracture segment of k-level branch fractures, q_ij is the flow rate at the bottom hole of k-level branch fractures, is the fracture width of k-level branch fractures, k_fij is the hydraulic fracture permeability of k-level branch fractures, p_wf is the horizontal well bottom hole pressure, and M is the number of segments of k-level branch fractures.

2.4. Coupling Model and Solution Method

In Figure 2, we can see a fractal bifurcated tree fracture system and the level of branch joints is 2. Each fracture is divided into 3 segments, and there are 21 fracture elements in the fracture system. Equations (20) and (22) constitute a continuity equation of reservoir-fractures-wellbore coupling. The continuity equations are transformed into matrix form and can be expressed as

Since the flow in the fracture is a one-dimensional Darcy flow, the linear equation (23) can be solved by the Gauss–Seidel method.

3. Results and Discussion

3.1. Model Validation

This model proposed in this paper is verified against a reservoir simulation model (Eclipse 2011) using data of a tight oil reservoir with bottom water drive. A horizontal well is located in the tight oil reservoir with bottom water drive. The basic data of the tight oil reservoir with bottom water drive is listed in Table 1.

The production rate solutions obtained from this paper model and numerical simulation (Eclipse 2011) are compared to check the accuracy of the proposed model. The grid dimension of this model is 120 × 60 × 42, and the grid sizes of x, y, and z in three directions are 10 m, 10 m, and 10 m, respectively. As shown in Figure 4, the complex fracture network is embodied by local grid refinement (LGR) in the reservoir simulation model (Eclipse 2011). As a note, reservoir characteristics and fluid properties in the reservoir simulation model (Eclipse 2011) are consistent with the present model. The result can be seen in Figure 5, which shows a good agreement between the proposed model result and the reservoir simulation.

3.2. Matrix Permeability

Figure 6 shows production at a 300-day period with different matrix permeability. As in Figure 6, the production increases with the matrix permeability, but the increments gradually decrease. This is because the production of fractures increases with the increase of matrix permeability during the initial production stage, which leads to the greater difference of the production in different matrix permeability. But as time goes on, the production difference becomes smaller in different matrix permeability.

3.3. Half-Length of Main Fracture

Figure 7 shows production at a 300-day period with different half-length of main fractures. As shown in Figure 7, the production increases with the increase of half-length, but the increment gradually decreases, and the contact area between the fracture system and the reservoir become larger with the increase of half-length. So the production increases with the increase of fracture half-length. Therefore, in order to obtain a better tight oil production, the half-length of fractures should be optimized.

3.4. Fractal Dimension

Figure 8 shows a cumulative production at a 300-day period with different fractal dimensions and also that the cumulative production increases with the increase of fractal dimensions. This is because the fracture system is more complex with the increase of fractal dimensions. Moreover, the complex fracture system makes the contact area between reservoir and fracture become larger. Therefore, in order to ensure good fracture results, the complexity of fractures should be increased as far as possible.

4. Conclusion

Based on the fractal theory and the volumetric source method, a semianalytical model is developed to predict the production from bottom water drive tight oil reservoirs with complex hydraulic fractures. The reliability of this model is validated through a numerical simulation (Eclipse 2011), which shows that the result from this method is identical with that of the numerical simulation. The study on influence factors of this model was focused on model validation, fractal dimension, and half-length of main fracture. The results show that (1) the production increases with the increase of half-length of main fracture and matrix permeability during the initial production stage, but the production difference becomes smaller in different half-length of main fractures and matrix permeability in middle and later stages and (2) the cumulative production increases with the increase of fractal dimensions, and the increments of cumulative production in different fractal dimensions gradually increase during the initial production stage, but the increments tend to be stable in middle and later stages. The main purpose of this paper is to study the production of a single fluid. However, a part of the fracking fluid is always left behind [33]. Hence, the next stage of our research will focus on the production of oil/water two-phase flow in bottom water drive tight oil reservoirs with complex hydraulic fractures.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are grateful to the School of Energy, China University of Geosciences (Beijing), for supporting this finding.

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Copyright © 2019 Shuhui Dai et al. 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.

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