Figure 1 is a schematic diagram of a deviated well with varying inclination angles in a carbonate gas reservoir that has multiple heterogeneous layers. This multilayer reservoir has two layers, and for individual layers, the inner region near the inclined wellbore is a triple porous medium with fractures, vugs, and matrix, and the outer region only has a single medium with matrix. Compared with the second (lower) layer, the first layer has larger permeability and porosity, which is the main layer in this multilayer carbonate gas reservoir.

For the presented model, it is assumed that the reservoir is composed of n cylindrical and heterogeneous layers, which have closed boundaries in horizontal and vertical directions. The wells produce a constant gas flowing rate and the individual layer is penetrated completely and the flowing rate along the wellbore distributes uniformly. Since the model presented in this paper can be seen as the extension of the model developed by Meng, et al. (2018), assumptions about fluids, rocks, and flowing law for this model can be found in their paper. For any layer, the inclined wellbore is assumed to be located in the inner region. It should be noted that the range of the inner region is small whereas the deviated angle is large, thus the inclined wellbore can also penetrate the inner and outer regions simultaneously, and the above assumptions may not be reasonable. However, in this paper, these circumstances are not explored and could be studied further in the future.

In carbonate reservoirs, due to the existence of natural fractures and vugs, many laboratory experiments and pilot tests show that the stress sensitivity is serious for naturally fractured-vuggy formation (Wang, et al., 2016a; Wang, et al., 2016b; Zhao, et al., 2013). Therefore, it is crucial to consider this effect for fracture permeability of the inner region in the developed mathematical model. Meng et al. (2017) assumed that the measured permeability is indeed fracture permeability. Using curve match with data from laboratory experiments, they obtained the relationship between the dimensionless permeability and net confining pressure, as shown in Eq. 1. k k i = ( σ s − p σ s − p i ) − α (1) where α is the exponent to reflect the formation stress sensitivity. According to Meng et al. (2018) it is 0.738 for natural fractured-vuggy formation.

To validate the proposed model further and demonstrate its applicability in practice, a case study was chosen, the Anyue carbonate gas reservoir. In this case, the reservoir has two layers, and both 3D seismic interpretation and drilling data show that the deviated well penetrates the “sweet spot” for each layer, where the region surrounding the wellbore is a fractured-vuggy formation that has lots of natural fractures, and the region far from the well is a tight formation with matrix. In this multilayer reservoir, for each layer, through the method of well logging, the formation thickness, porosity, and water saturation for fractures, vugs and matrix are obtained and through the method of well-testing analysis, the permeability, inter-porosity flowing coefficients, initial pressure and temperature of the reservoir can be acquired. The well trajectory and inclination angle can be obtained from drilling data. The detailed formation and fluid information for this case are given in Table 2. It should be noted that formation anisotropy represents the horizontal-vertical permeability ratio.

Since there is a lack of separate layer test information, the well bottom-hole flowing pressure information is acquired to match calculated data with the presented model. Figure 4 shows the comparison of well bottom-hole pressure in Cartesian and semi-log coordination systems. The goal of the introduction of the semi-log scale is the presentation of the initial fitting effect. It can be seen that whether for the Cartesian or semi-log coordination system, the curves fit perfectly between the testing well bottom-hole pressure and this model, which indicates this model could calculate the well bottom-hole flowing pressure precisely, and verifies the practicality of it in examining multilayer heterogeneous carbonate gas reservoirs.

Figures 5, 6 show the influences of inner region radius for first and second formations on the production performance of deviated well and individual layers. It should be noted that in Figures 5B, 6B, the gas production rate for the first layer and second layer are represented with solid and dashed lines, respectively To clearly show the early production distribution, the semi-log scale is used in these figures.

Through the comparison of Figures 5A, 6A, it is easy to see the variation of radius for the inner region in the first layer has a greater effect on well bottom-hole pressure. The reason for this phenomenon is that the first layer has a larger thickness, porosity, and formation permeability (see Table 2), which is the main layer in this multilayer heterogeneous carbonate gas reservoir. The increase of inner region radius for this layer can greatly improve the level of well bottom-hole pressure. Since the gas production mainly derives from the inner region initially, hence, before the pressure response reaches the interface between the inner region and outer region, the production rate of gas for each layer is identical. As the inner region has larger porosity and permeability, on the condition of constant well production rate, with the increase of range for the inner region, the production rate of gas for any layer increases with the rise of inner region radius for this layer. In addition, the increase of inner region radius will delay the time of reaching this interface, then in Figures 5B, 6B, the time of emerging discrepancy for different cases increases with the inner region radius.

The influences of horizontal permeability for natural fracture in the inner region on production behaviors are given in Figures 7, 8. As the first layer is the main formation in this multilayer heterogeneous carbonate gas reservoir, the variation law in Figures 7A, 8A is analogous to the curves in Figures 5A, 6A, and the rise of fracture horizontal permeability for the first layer can increase the well bottom-hole pressure significantly, whereas the fracture horizontal permeability of the second layer has little effect on the well bottom-hole pressure. However, since the inner region contributes mostly to the well gas production at the initial stage, and the flowing capacity increases with the fracture horizontal permeability, in contrast with Figures 5B, 6B, in Figures 7B, 8B gas production rate for any layers has a huge initial gap for different scenarios. When the gas production rate from the outer region occupies the larger ratio, the impact of the inner region decreases, and the gaps for different cases reduce continuously. It should be noted that because we assumed that the deviated well produces with a constant rate, while the gas production rate for some formations increases, for other formations it would certainly decline. For example in Figure 7B, with the rise of the horizontal permeability of natural fractures for the first formation, the production rate for this formation increases, while for the second layer, the production rate decreases.

Since the outer region has a larger area than the inner region, it is important to evaluate the influences of horizontal permeability of matrix on production behaviors, which are given in Figures 9, 10. It can be seen that as the gas supply capacity of the outer region increases with the matrix horizontal permeability, although the well bottom-hole flowing pressure and production rate of gas for different layers are the same for different cases initially, when the time is larger than 30 days, whether for the first or second layers, obviously, these two production index increase with the matrix horizontal permeability of this layer. Compared with Figures 5B, 6B and Figures 7B, 8B, in Figures 9B, 10B, as the radius of the inner region and permeability of natural fractures are equal, hence, there are no discrepancies of well bottom-hole flowing pressure and production rate initially, and the moment of appearing difference are nearly the same for different cases. Furthermore, through the comparison of Figures 7A, 9A, it can be seen that the smaller variation of matrix permeability can cause the huge gaps of bottom-hole pressure, which indicates that the properties of the outer region determine the production behaviors of the deviated well.

During the exploitation of the multilayer reservoir with the deviated well, the design of the well trajectory is an important task. It is then crucial to investigate the influence of the inclination angle for each layer on the production performance of individual zone and well bottom-hole flowing pressure (Figures 11, 12). Since the contact area between the slanted wellbore and formation increases with the inclination angle, as shown in Figures 11, 12, the well bottom-hole flowing pressure and gas production rate of the first or second layer increase with the penetrated inclination angle in this layer. As it was assumed that the slanted wellbore is located only in the inner region, the variation of inclination angle influences the production rate distribution at the early stage (Figures 11B, 12B), which is analogous to Figures 7B, 8B. Figure 10 shows the increase of inclination angle for the first layer with larger gas reserve and permeability, the well bottom-hole pressure can be improved largely. Therefore, to keep the well bottom-hole pressure at a high level, the penetrated inclination angle must be greater than a certain value. In addition, as shown in Figure 12, the inclination angle for the second layer has few influences on the well bottom-hole flowing pressure, whereas it can greatly increase the production rate of gas for this layer, which demonstrates the gas recovery of tight formation can be enhanced by enlarging the inclination angle in this layer.

To determine the critical values of inclination angle for the first and second layers, we plot the curves of the inclination angle for the first layer versus bottom-hole flowing pressure and the inclination angle for the second layer versus cumulative gas production, which is shown in Figure 13.

Note that the bottom-hole flowing pressure and the cumulative gas production are collected on the 1,000th day. While the inclination angle is smaller than 60°, there is a linear relationship between the inclination angle and bottom-hole flowing pressure or cumulative gas production. When the inclination angle is larger than 60°, the bottom-hole flowing pressure and cumulative gas production increase drastically, whether it is the first layer or the second layer, the critical value of inclination angle for these two layers is 60°. To keep the bottom-hole pressure at a relatively high level and enhance the gas recovery of the tight layer, the penetrated inclination angle should be greater than 60°, and be enlarged as much as possible.