New Method for Calculating Bottom Hole Pressure of Horizontal Well in Fracture-cavity Reservoir Based on Point Source Function


 Accurate determination of unsteady bottom hole pressure helps to monitor and predict well production in real-time. On the premise of fully considering the seepage characteristics of carbonate rock, a new source function suitable for the seepage of carbonate rock is established. It enlarges the application scope of source function theory and lays a theoretical foundation for solving the seepage problem of carbonate rock. This paper presents the process of solving bottom hole pressure step by step. Step 1: Based on the triple media model, the Pedrosa permeability calculation formula is applied to establish the seepage model of the triple media reservoir considering the formation stress sensitivity. Step 2: By perturbation transform and Laplace transform, the point source function considering stress sensitivity in carbonate reservoir is obtained in Laplace space. The point source function in the infinite plate reservoir is obtained by the principle of mirror image and superposition. Step 3: The method of solving the horizontal well pressure under the constant pressure boundary is established. Through literature comparison and numerical simulation, the rationality of the proposed method is verified. Simultaneously, the sensitivity analysis of pressure and pressure derivative is carried out, and the influences of fracture number, fracture angle, fracture half-length, skin factor, horizontal well segment length, and horizontal well segment spacing on pressure and pressure derivative are analyzed in detail. Considering fracture orientation and stress sensitivity, we divide the triple media fracture-vuggy reservoir fluid flow into five stages. The number of fractures and fracture direction mainly affect stage C. In contrast, the length of horizontal subsection and skin factor mainly affect stage B. Stage D is more obvious when the fracture half-length and the horizontal sublevel interval of the horizontal well are small.


Introduction
Source function is one of the most important methods to solve the problem of reservoir seepage. In 1963, Warren first introduced source function into the field of reservoir seepage (Warren, 1963). In 1973, Ramey further extended the application range of source function and established point source function, line source function, and surface source function under different reservoir boundary conditions (Ramey, 1973(Ramey, , 1974. In 1987, Kuchuk and Ozkan applied Laplace transform to establish point source function, line source function, and surface source function to solve the seepage problem of the dual-medium reservoir under different reservoir boundaries (Kuchuk, 1987; Ozkan, 1987). These above source functions are suitable to solve the reservoir seepage problems in vertical wells, horizontal wells, multi-branch wells, and fractured horizontal wells. What's more, they can provide an important theoretical basis for oil well productivity prediction, well test interpretation, and understanding the development laws of different well types. However, with the development of carbonate rocks, the traditional source function's application is greatly limited. Carbonate rock is a triple media reservoir with a certain stress-sensitive effect. Still, the existing source function can not consider the influence of stress sensitivity on the seepage of the oil reservoir in triple media.
As for bottom hole pressure solution, domestic and foreign scholars have done a lot of research and made great progress. In 1993, Onur used the typical curve method to solve the analytical solution of bottom hole pressure for a naturally fractured reservoir (Onur, 1993). In 1999, Zhang approximately treated the fractured vertical well as a single fracture and derived the analytical solution of the bottom hole pressure of the fractured vertical well (Zhang, 1999). In 2000, Bui used the Laplace transform to calculate the bottom hole pressure of a fractured vertical well for a partially perforated reservoir (Bui, 2000). In  Triple medium pressure equation: Considering the fluid flow in the fracture, matrix, and karst cave, based on the positional relationship of the well, karst cave, and fracture, the pressure and flow rate at the intersection of the fracture, matrix, and karst cave is equal to obtain the pressure distribution.
Although the influence of fractures on bottom hole pressure is considered in the literature, the orientation of fractures is generally treated as simple, such as fractures parallel to each other or perpendicular to the horizontal section of the horizontal well.
Fractures are of various orientations and sizes, and skin factor varies from fracture to fracture. Therefore, it is necessary to consider the actual situation of fracture and establish a pressure-solving model. This paper presents a new way to obtain the point source function. Firstly based on the triple media model, the Pedrosa permeability calculation formula is applied to establish the triple media reservoir seepage model considering the formation stress sensitivity. Secondly, by perturbation transform and Laplace transform, the point source function considering stress sensitivity in carbonate reservoir is obtained in Laplace space. Then, the point source function of the infinite plate fracture-cavity reservoir is obtained by the mirror image and superposition principle. Based on the new point source function, this paper presents "three steps" to solve the pressure of a horizontal well under the constant pressure boundary.  ①The oil reservoir is composed of the cave system, matrix system, and fracture system. The cave and matrix system are the main storage space, and the fracture system is the main flow channel. All the well production comes from the influx of the fracture system; ②Considering the permeability sensitivity of natural fracture system, it is assumed that the permeability of matrix system is constant; ③There is a point source in the oil reservoir. The initial pressure is equal everywhere in the oil reservoir, and the reservoir temperature is constant during the production process of the oil well.
④The reservoir fluid is slightly compressible, and the influence of gravity and capillary force is not considered. The porosity of the reservoir and fluid viscosity is constant.

Mathematical model
Based on the above assumptions, the center of a single fracture is located at (x0，y0， z0), and the governing equation of the fracture is shown as follows: (1) f(x,y,z) is the source (sink) phase, and its expression is shown as below.
Where δ is the Dirac Delta function.
If    i p p p , then the change trend of p and p is consistent. The pressure term p in the governing equation (1) can be directly replaced with pressure difference term p .
Similarly, the governing equation of the matrix system can be obtained as the The governing equation of the karst cave system is obtained as the equation (6) ( Where, kx, ky and kz are the permeability in x, y and z directions, mD; μ is the viscosity of crude oil, mPa.s; ∆p is pressure difference, MPa;  is the porosity, decimal; c is the compressibility coefficient, Pa -1 ; Subscript "m" is the matrix system; Subscript "f" is natural fracture system; Subscript "v" refers to the karst cave system.
When considering the stress sensitivity of the natural fracture system, the natural fracture permeability decreases with the decrease of formation pressure. The natural fracture permeability can be expressed as: Where kxi is the initial permeability of the natural fracture system in the x direction, 10 -3 μm 2 ； pi is the initial formation pressure, MPa; α is the permeability modulus, MPa -1 。 2 Model solving

Solution research
Literature survey found that there are four main methods to solve the equation (4).

Dimensionless transformation
The dimensionless transformation is defined as follows: Based on the dimensionless transformation above, Equation (4) can be summarized as: Matrix system dimensionless equation: Cave system dimensionless equation: The internal boundary condition: The outer boundary condition: The initial conditions:

Equation Solving
The Laplace transform of equation (19), (22) and (23) is obtained as follows: Substitute equation (28) and (29) into Equation (27), we can get: 2 22 14 Equation (30) is a strongly nonlinear partial differential equation, and a perturbation transformation is introduced: The derivative of Equation (32) with respect to rD can be obtained as follows: The derivative of Equation (33) with respect to tD can be obtained as follows: Substitute equation (33) and (34) into Equation (30), we can get: In the form of power series, the  and   Through internal and external boundary conditions, A and B can be solved: Then, the solution of instantaneous point source in the Laplace space of fracture-vuggy triple medium reservoir is: The above equation is the pressure distribution calculation formula of instantaneous point source function per unit strength, and the point source is located at the origin of coordinates.

(1) Point source function is not at the origin of coordinates
If the point source function is not at the origin of coordinates, but at (xwD、ywD、 zwD), first define the following expression: Then the calculation formula of pressure distribution generated by the unit source function is as follows: If the instantaneous source function intensity is not 1, then the pressure calculation formula is as follows: If: Then, the time-space solution of Equation (48) is: By applying the superposition principle, the solution of continuous point source function can be obtained as follows: Based on the property of convolution, Laplace transform of Equation (51), and we can get: This equation is not convenient to solve. Possion relation, trigonometric function and difference product formula can be used to transform the above equation, and the following equation can be obtained: The boundary types of fracture-cavity reservoirs are diverse, so we will not discuss them one by one here. ① Firstly, it is assumed that the length and height of a certain fracture are 2Lfi and hwi. Then, according to the boundary type of fracture-vuggy reservoir (the above pressure boundary is taken as an example to illustrate), Formula (57) integrates x in the interval of (xw-Lf， xw+Lf) , and then integrates z in the interval of (zw-hw/2， zw+hw/2) .
The expression of the pressure of a single fracture in the horizontal section of the horizontal well can be obtained.
② According to the superposition principle, the pressure generated by each fracture in the corresponding horizontal section is superimposed to obtain the Laplace space solution of the total pressure.
③ According to the solution of Laplace space, the Stehfest method is used to carry out numerical inversion, and the numerical solution of real space is obtained.  4. Figure 4 shows that the results of the two have a good consistency. Riley et al. is a special case of the model in this paper, so the model in this paper is correct.   initial vertical permeability 500mD 500mD 500mD 500mD 500mD The E300 module in Eclipse 2017 is designed for triple-media fracture-vuggy method is adopted to divide meshing and ensure that each fracture has at least three meshings to describe the heterogeneity of formation fluid. If the fracture is short, local mesh encryption should be performed at the fracture, as shown in figure 5(b). The plane mesh step size is 20m, and the vertical mesh step size is 5m. Then the total number of grids is 50*50*16=40000. Other parameters required for numerical simulation are shown in table 3. The numerical simulation and calculation results in this paper are shown in table 4. Table 4 shows that the relative error is controlled within 5% under the basic error, which is consistent with the allowable error range, indicating that the method we provide is reliable.

Flow Characteristics Analysis
A fracture-vuggy reservoir with a rectangular outer boundary has a horizontal well in the center. The number of fractures is 4. The length of the horizontal section of the horizontal well is 400m. kx = ky, kz/kx =100. The wellbore storage factor is 0.001, and the skin factor is 1. Based on the basic parameters in Table 1, the dimensionless pressure and pressure derivative were calculated, as shown in figure 6. Figure 6 shows that the fluid flow process of the triple media fracture-vuggy reservoir can be divided into five stages, considering fracture orientation and stress sensitivity.
Stage A: The wellbore storage phase. The pressure and pressure derivative curves coincide, and the asymptotic analysis shows that the slope of the curve is about 1.
This stage is mainly affected by the wellbore storage effect.
Stage B: The karst cave fluid flows to the fracture stage. Generally, the permeability of the karst cave system is greater than that of the fracture system. As the fracture system supplies fluid to the wellbore, the pressure of the fracture system drops.
Then the karst cave system first supplies fluid to the fracture to supplement the pressure of the fracture system. A "dent" appears in the curve.

Horizontal segment spacing of horizontal well
When the total length of the reservoir shot by the horizontal well is fixed, the entire drainage area communicating with the fractures outside the wellbore is certain. The influence of horizontal segment spacing on the type curves is shown in figure 7. Figure   7 shows that horizontal segment spacing mainly affects stages B and C. The larger the interval between the horizontal segments of the horizontal well, the longer it takes for fractures to supply the wellbore. The more difficult the channeling flow between the cave and fracture will be.
When the horizontal segment spacing is large, a new "platform" appears on the pressure derivative curve in stage C, with a horizontal line of about 0.25. The asymptotic analysis shows that the new "platform" height depends on the number of segments in the horizontal well. In general, when the number of segments in the horizontal well is n, the corresponding horizontal line of the new "platform" is about 1/n.
Infield well testing, it isn't easy to have stage E if the test time is not long enough.
Therefore, during well test interpretation, the actual permeability can be obtained by multiplying the permeability k explained in stage C by the number of segments of horizontal wells.

Horizontal well segment length
The influence of different horizontal segment lengths on the type curves is shown in figure 8. Figure 8 shows that segment length mainly affects stage B. The longer the segment length is, the lower the "hump" is, and the earlier stage B is. However, the shorter the duration of stage B is, and the lower the value of the pressure derivative is and more quickly recovered to 0.5. The larger the reservoir interval opened by horizontal wells, the more favorable the flow of reservoir fluid to the wellbore, and the earlier the well reservoir effect will disappear. When the total length of the horizontal segment is fixed, the influence of each segment's length and distribution mode on the type curves is shown in figure 9. Figure 9 shows that when the section length at both ends of the horizontal well increases, the pressure drop generated becomes smaller, conducive to horizontal well production.
Therefore, in the actual perforation process, it is suggested to increase the production section at the toe and foot ends of the horizontal well to increase the supply of fractures to the horizontal wellbore and improve the horizontal wellbore productivity.

Skin factor
The influence of the skin factor on the type curves is shown in figure 10. Figure 10 shows that the larger the skin factor is, the more serious the wellbore pollution is.
What's more, the longer the duration of stage A is, and the later the appearance of stage B is, and the higher the "hump" is. Stage C is covered in varying degrees, and the skin factor affects the radial flow of the fracture. When the sum of the total skin factor of each segment is constant, the influence of the size and distribution of the skin factor of each segment on the type curves is shown in figure 11. Figure 11 shows that the skin factor causes different flow rates in each section, and the non-uniformly distributed skin factor produces different pressure responses at the bottom of the well.
When the pollution at both ends of the horizontal well is relatively serious, the pressure drop loss caused is greater, which is not conducive to improving the production of the horizontal well. Therefore, when using horizontal wells to implement reservoir stimulation measures, special attention should be paid to modifying the formations at both ends of the horizontal well to eliminate pollution near the wellbore.

Fracture number
The influence of the fracture number on the type curves is shown in figure 12. Figure 12 shows that the fracture number mainly affects stage C. When there are many fractures, and the fractured-vuggy reservoir has a fracture network, stage D will be covered by stage C. To a certain extent, the matrix does not directly supply the caves.
Instead, the matrix directly supplies the fractures, mainly caused by the development of a fracture network in the reservoir. The more developed the fracture network, the seepage characteristics of the triple-medium fracture-cavity reservoir are more similar to the dual-media. The matrix/cavities in the reservoir directly supply the fractures and then flow to the wellbore. This flow mode is more conducive to pressure transmission and production increase.

Fracture direction
When the angle between the fracture and the horizontal wellbore changes, different fracture angles have different effects on the type curves, as shown in Figure 13. Figure   13 shows that the fracture direction mainly affects stage C. When the fracture is perpendicular to the wellbore, the pressure drop is smaller, and the pressure transmission is faster, which is more conducive to improving the productivity of horizontal wells. Because the drainage area is larger, and more fluid in the reservoir flows through the fractures to the wellbore at the same time. The perforating gun should be parallel to the horizontal wellbore during fracturing as much as possible, and then fractures perpendicular to the wellbore should be generated.

Fracture half-length
The effect of the fracture half-length on the type curves is shown in figure 14. Figure 14 shows that the fracture half-length mainly affects stage B and stage C. When the fracture half-length is longer, the pressure drop is smaller, which is most conducive to production. When the fracture half-length is very small (≤50m), stage D is more obvious because the fracture half-length is small, the fluid in the matrix cannot flow directly to the fracture. It has to flow to the cave first and then flows to the fracture through the cave. The drainage area in the reservoir is limited, and the pressure spreads more slowly. Therefore, in the design of hydraulic fracturing, the fracture half-length should be greater than 50m as far as possible.

Conclusion
In this paper, a new point source function for triple-medium fractured-vuggy reservoirs is established, and the steps for solving the bottom hole pressure of horizontal wells are given. Through literature comparison and numerical simulation methods, the rationality of the method in this paper is verified. The influence of fracture number, fracture angle, fracture half-length, skin factor, horizontal well segment length, and horizontal well segment spacing on pressure and pressure derivative is analyzed in detail. The study believes special attention should be paid to reforming the formations at both ends of the horizontal well to eliminate pollution near the wellbore when using horizontal wells to implement reservoir production stimulation measures. When designing pressure construction, the perforating gun should be made parallel to the horizontal wellbore as much as possible, and the fracture half-length formed should be greater than 50m. When designing horizontal well fracturing, it is necessary to increase the production section at the toe and toe ends of the horizontal well, expand the oil drainage area, and accelerate the supply of fractures to the horizontal wellbore to achieve the purpose of increasing the productivity of the horizontal well.