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

Exploration and Development Department, China ZhenHua Oil Co., Ltd., Xicheng District, Beijing 100031, China School of Energy Resources, China University of Geosciences, Beijing, Haidian District, Beijing 100083, China +e 3rd Welllog Branch of Geology and Exploration Research Institute, CNPC Chuanqing Drilling Engineering Company Limited, Chengdu, Sichuan 610000, China Energy Research Institute, China Electric Power Planning & Engineering Institute, Xicheng District, Beijing 100120, China


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][2][3][4][5][6][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][9][10].Also, analytical and semianalytical approaches were used to predict the production of vertical fractures in such systems [11][12][13][14][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][19][20].
However, the actual hydraulic fracture geometry is more complex than it was assumed earlier.e 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.
e 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].
en, 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.
e model is verified by the results of commercial numerical simulation software (Eclipse 2011).
en, the influencing factors of matrix permeability, halflength of main fracture, and fractal dimensions are studied.
is work is helpful in understanding the effect of fracture geometry on production in tight oil reservoirs.

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.e specific assumptions are as follows: (1) e 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) e reservoir is anisotropic, homogeneous, and slightly compressible.(3) e reservoir contains a single-phase fluid, and the fluid flow is steady.(4) e production of all the fractures is equal to the horizontal well.

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).en, the coordinate of the ith subtube of the k-level is (x ij , y ij ), and j is the number of fracture segments.e 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. is subtube generates the level of (k + 1) with (2i − 1)th and 2ith subtubes in which i � 2 k .e coordinates of the fractal fracture network system can be expressed as [24] (1) 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  Journal of Chemistry 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] ( e following result can be obtained from Equation ( 2): where d k is the diameter of k-level branch fractures, l 0 is the primary-level fracture length, and d 0 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.

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.ere is a volumetric source in the tight oil reservoir and the strength is q.
e 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 2w x , 2w y , and 2w z .
e volumetric source model for tight oil reservoirs with bottom water drive can be expressed as where H(x − x 0 ) is the Heaviside function: where p i is the pressure of bottom water drive, p(x, y, z) 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].e solution at the tip of the ith subtube in k-level branch fractures can be expressed as Equation ( 9) represents the solution of homogeneous reservoirs; however, reservoirs are usually anisotropic.Hence, the correction method of Besson [30] is introduced.
e 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].
en, the dimensions of the threedimensional 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 W(x ij , y ij , z ij ), then W(x ij , y ij , z ij ) can be expressed as en, Equation ( 17) can be given as 2.3.Fracture Flow. Figure 2 shows the flow model of a fracture system.e 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 4 Journal of Chemistry 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 e 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 as where 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, w fij 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.

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.e 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.

Model Validation.
is 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.e basic data of the tight oil reservoir with bottom water drive is listed in Table 1.
e production rate solutions obtained from this paper model and numerical simulation (Eclipse 2011) are compared to check the accuracy of the proposed model.e 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.e result can be seen in Figure 5, which shows a good agreement between the proposed model result and the reservoir simulation.

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.is 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.

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.erefore, in order to obtain a better tight oil production, the half-length of fractures should be optimized.

Fractal Dimension.
Figure 8 shows a cumulative production at a 300-day period with different fractal dimensions Journal of Chemistry and also that the cumulative production increases with the increase of fractal dimensions. is 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.erefore, in order to ensure good fracture results, the complexity of fractures should be increased as far as possible.

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.e 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.e study on influence factors of this model was focused on model validation, fractal dimension, and half-length of main fracture.e 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    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.e 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.

Figure 4 :
Figure 4: A schematic diagram of fracture geometry in a numerical simulation of Eclipse 2011.

Figure 5 :Figure 6 :Figure 7 :
Figure 5: Comparison of the cumulative production from this paper model and numerical simulation (Eclipse 2011).

Figure 8 :
Figure 8: Cumulative production at a 300-day period with different fractal dimensions.

Table 1 :
Basic data of a tight oil reservoir with bottom water drive.