Seidle applied the single-phase (gas) pseudo-pressure productivity equation to a dewatered coalbed according to its production characteristics (Seidle, 1993). From the definition of pseudo-pressure, the lower limit of the integral is arbitrary and can be set as zero. Then, m ( p ) = p 2 μ ¯ Z ¯ (1) where m(p) is the pseudo pressure, and Z ¯ and μ ¯ are the average deviation factor and viscosity, respectively, which can be calculated by the empirical expressions used for Chinese gas wells by Chen and Dong (2001), and shown in detail in Appendix A. In light of the work of Feng et al. (2012), the radial productivity equation can be expressed as follows: q gsc = 2 π k h T sc ( m ( p ¯ ) − m ( p wf ) ) p sc T ( ln r d r w + S t ) (2) where q _gsc is the production of CBM at surface conditions; p _sc and T _sc are the standard pressure and temperature, which are 0.1 MPa and 293.15 K, respectively; k is the permeability of the CBM reservoir; h is the thickness of the reservoir; S _t is the total skin factor; T is the reservoir temperature; p ¯ is the average reservoir pressure; p _wf is the bottom-hole pressure; r _d is the drainage radius, which is 0.472 times the physical drainage radius r _e (Al-Hussainy et al., 1966); and r _w is the radius of the wellbore.

In this study, the effects of SSE, MSE, and multi-wing fractures in the entire reservoir area are integrated into a fracturing skin factor. In addition, the SSE–MSE has a significant bearing on the speed sensitivity skin factor. For a more accurate IPR study, the total skin factor is investigated in detail and presented in Section 2.2.

For CBM wells in the Qinshui Basin, the total skin factor is composed of the speed sensitivity skin factor, the well completion skin factor, and the fracturing skin factor. The well completion skin factor can be obtained directly from in-situ construction, while the other two can be calculated based on the in-situ reservoir parameters. Hence, the total skin factor can be expressed as follows: S t = S c + S f + D q gsc (3) where S _c is the well completion skin factor, S _f is the fracturing skin factor, and D is the non-Darcy flow constant. In Eq. 3, Dq _gsc refers to the speed sensitivity skin factor. The expressions for the speed sensitivity skin factor and the fracturing skin factor will be discussed in what follows.

As shown in Figure 1A, the major and secondary hydraulic fractures intersect at the wellbore in a vertical CBM well at a certain angle. Unlike the traditional two-wing fractures, the effect of all hydraulic fractures on production should be considered in multi-wing fracture conditions (Zhang et al., 2018). Based on microseismic data (Runsheng et al., 2016), complicated (or major) hydraulic fractures primarily propagate along the main crack direction, while secondary hydraulic-fractures tend to propagate along the vertical direction. Therefore, the crossing of major and secondary hydraulic fractures represents the primary fracture morphology of complicated hydraulic fractures, which can take the form of multi-wing fractures. To investigate the fracturing skin factor and the speed sensitivity skin factor better, the production area of a single well was divided into an inner region (r ≤ r _d) and an exterior region (r _d < r ≤ r _e), shown in Figure 1B. For the calculation of gas productivity with a closed boundary, the physical drainage radius r _e should be replaced by r _d in Eq. 1. However, given that gas flows in the area with radius r _e instead of r _d in actual wells, ignoring the permeability variation and flow characteristics of the exterior region in Eq. 1 could lead to a critical calculation error. Therefore, the flow characteristics of both the inner and exterior regions should be emphasized. The permeabilities of the inner and exterior regions (i.e., k ₁ and k ₂) are both dominated by the SSE–MSE and will be discussed in Section 2.3. Due to the fact that the speed sensitivity influences the area near the wellbore, the non-Darcy flow constant should be calculated based on the permeability of the inner region (Jones, 1987). D = 2.56 × 10 - 9 k 1 γ β μ ¯ h r w (4) β = 4.52 × 10 6 k 1 1.55 (5) where γ is the relative density of the gas, and β is the inertial coefficient. In Eq. 4, the constant values are revised to unify the units.

As shown in Figure 1B, to form a simplified central symmetry fracture model, we take the wellbore as the center and the major and secondary hydraulic fractures as the long and short axes. The lengths of half the long and short axes are l ₁ and l ₂, and the included angles are θ and π−θ. Furthermore, based on the long and short axes, the multi-wing fractures control area is elliptical. According to related research (Zhiming et al., 2014; Chen et al., 2017), the radial flow between the intersecting fractures (yellow area in Figure 1A) could be transferred to unidirectional flow by conformal transformation, shown in Figure 2. For the transformed unidirectional flow, the width, thickness, and length of the flowing medium are π, h, and ξ _e. The parameter ξ _e refers to the flow length after conformal transformation.

The radial flow model is more consistent with actual production demands and more valuable to production investigations than the unidirectional flow model. Therefore, the radial flow model with its pseudo-pressure has been developed and widely accepted (Al-Hussainy et al., 1966; Sun et al., 2018), while the radial flow model with pseudo-pressure is less studied. The unidirectional flow model with pseudo-pressure is given as follows: { d 2 m ( p ) d x 2 = 0 m ( p ) | x = 0 = m ( p ¯ ) m ( p ) | x = ξ e = m ( p wf ) (6) By solving Eq. 6 and deriving a solution, the unidirectional productivity equation with pseudo-pressure is: q gsc = π h T sc ( m ( p ¯ ) − m ( p wf ) ) 2 p sc T ξ e (7) The derivation of Eq. 7 is found in Appendix B. With θ and π−θ included angles, the parameter ξ _e could be presented based on the conformal transformation. ξ e 1 = ln 4 r d π / θ ( l 1 + r w ) π / θ + ( l 2 + r w ) π / θ (8) ξ e 2 = ln 4 r d π / ( π − θ ) ( l 1 + r w ) π / ( π − θ ) + ( l 2 + r w ) π / ( π − θ ) (9) Equations 8, 9 are for four-wing fractures and were derived from the equations for multi-wing fractures with arbitrary angles and fracture lengths (Chen et al., 2017). By substituting fixed angles and fracture lengths into the multi-wing fracture equations, four-wing fracture equations could be obtained. Combining Eqs 7–9, the productivity equation for the inner region can be written as: m ( p ¯ ) − m ( p wf ) = q gsc p sc T ξ e 1 ξ e 2 π k 1 h T sc ( ξ e 1 + ξ e 2 ) (10) For the exterior region, the productivity equation is: m ( p e ) − m ( p ¯ ) = q gsc p sc T ⁡ ln r e r d π k 2 h T sc (11) Based on Eqs 10, 11, the productivity equation for the whole drainage region, with multi-wing hydraulic fractures, can be expressed as: m ( p e ) − m ( p wf ) = q gsc k 1 p sc T ξ e 1 ξ e 2 + q gsc k 1 p sc T ⁡ ln r e r d ( ξ e 1 + ξ e 2 ) π k 1 k 2 h T sc ( ξ e 1 + ξ e 2 ) (12) Without the effect of skin factors, the productivity equation of an ideal CBM well is: m ( p e ) − m ( p wf ′ ) = q gsc p sc T ⁡ ln r e r w π k 0 h T sc   (13) where k ₀ is the intrinsic permeability, and p'_wf is the ideal bottom-hole pressure.

Eqs 12, 13 can be combined to obtain the difference between the real and ideal bottom-hole pseudo-pressure. m ( p wf ) − m ( p wf ′ ) = q gsc p sc T π k 0 h T sc S f (14) where S _f is the fracturing skin factor, expressed as: S f = k 0 ξ e 1 ξ e 2 k 1 ( ξ e 1 + ξ e 2 ) + k 0 k 2 ln r e r d − ln r e r w (15) From Eqs 4, 15, the speed sensitivity skin factor and fracturing skin factor can be calculated.

Equations 3, 14, which represent permeability in the inner and exterior regions, as controlled by the SSE–MSE, are discussed here. As shown in Figure 3, the CBM well production triggers the decrease of reservoir pressure from the initial pressure p _i to the critical desorption pressure p _r, and the permeability k declines correspondingly due to the SSE (the black line in Figure 3). With the continued decrease of reservoir pressure (smaller than p _r), permeability is influenced by the combined SSE–MSE. This permeability is reversed and starts to increase (the blue line in Figure 3). Without the MSE, the coal matrix will swell because of the SSE, and permeability will continue to decrease (the red line in Figure 3). The average pressure p is also marked in Figure 3. For reservoir pressures smaller than p, permeability has a negative relationship with pressure.

The permeability model developed by Zhao et al. (2013) was selected for demonstrating the permeability variations. This permeability model is a modification of the S&D model (Shi and Durucan, 2004), requires fewer experimental test parameters, and is given in Eq. 16. The detailed derivation of the model is given in Appendix B. k d = k 0 e − 3 C f 0 { − ν 1 − ν ( p − p e ) + [ ln ( 1 + b p ) − ln ( 1 + b p r ) ] ρ c R T V L 3 ( 1 − ν ) V 0 } ( p ≤ p r ) (16) where k _d is the dynamic permeability; the C_f0 is the compressibility factor of the microfractures; υ is Poisson’s ratio; ρ _c is the density of the coal matrix; R is the universal gas constant; V _L is the Langmuir volume; b is the reciprocal Langmuir pressure; and V ₀ is the molar volume.

As shown in Figure 4, the pseudo-pressure is distributed in a funnel shape. The permeability grows with increased proximity to the wellbore. To obtain the permeability of the inner and exterior regions, special treatment should be applied. k 1 = 1 m ( p ¯ ) − m ( p wf ) ∫ m ( p wf ) m ( p ¯ ) k d d m ( p ) (17) k 2 = 1 m ( p e ) − m ( p ¯ ) ∫ m ( p ¯ ) m ( p e ) k d d m ( p ) (18) where p _e is the pressure of the closed boundary. At the closed boundary position, the pseudo-pressure distribution curve is perpendicular to the outer boundary (Al-Hussainy et al., 1966), indicating that the variation in pseudo-pressure in the exterior region is less than in the inner region, resulting in similar values of p _e and p. Therefore, to avoid seeking boundary pressure, Eq. 17 could be simplified to, k 2 = k d ( p ¯ ) (19)

The relative data of CBM Well A from the No. 3 coal seam in the South Shizhuang Block of the Qinshui Basin was used to validate the model (Table 1). Based on the microseismic crack monitoring technique, the lengths of the major and secondary hydraulic fractures and the included angle were all obtained (Table 1). The boundary is closed and the relative gas density is 0.556. The target coal seam has low water content with minimal supply from distant water. Water production, therefore, declined to zero with the increase in gas production (Figure 5). After 3 years of exploitation, Well A is considered a dry or dewatered gas well, due to the scant water production. This may be related to the underground hydrodynamic conditions and well pattern distribution (Ye et al., 2011). The dewatered gas well is a condition of CBM, which is producing gas without water, instead of being a kind of CBM well. A dewatered well, or one in a reservoir with no mobile water in the first place, can both be called a dewatered CBM well. As shown in Figure 6, after 1,000 days of production, there is temporary water production due to the desorption of new coal and the removal of irreducible water. However, compared with gas production, water production is negligible. The relationship data of p _wf and gas production for the well test are presented in Figure 6. The IPR curve with SSE–MSE for the entire region (Curve I in Figure 6), calculated by Eq. 2, is consistent with the well test data, which proves the validity of the production model for a vertical CBM well with multi-wing hydraulic fractures.

When considering only the SSE in the entire region (Curve IV in Figure 6), the absolute open flow rate is much smaller than the ones with and without SSE–MSE in the entire region (Curves I and III), which are 83.2% and 194.1% larger, respectively. These results reveal that the SSE strongly inhibits CBM productivity due to the positive relationship of permeability with pressure caused by the SSE on its own. However, when considering SSE–MSE together, CBM productivity is stimulated instead of inhibited. Thus, the MSE has a much stronger stimulating impact on productivity than the inhibiting impact of the SSE. As a result, the SSE–MSE benefits CBM productivity. Thus, the absolute open flow rate with the SSE–MSE is 60.5% larger than the ones without (Curves I and III in Figure 6). These results correspond to increasing permeability with decreasing pressure (blue line in Figure 3). When considering the SSE–MSE in the inner region only (Curve II in Figure 6), the absolute open flow rate is reduced by 21%, compared to the SSE–MSE in the whole region. These results indicate that the SSE–MSE in the exterior region has an apparent influence on gas production and should not be neglected. The permeability in the exterior region is larger than the intrinsic permeability due to the SSE–MSE (Figure 3), hence improving gas production.

For bottom-hole pressures close to the average reservoir pressure, Curves II and III are close to each other, while an obvious difference appears between Curves I and II (marked by the dotted rectangle in Figure 6). In this situation, the SSE–MSE of Well A in the exterior region has a considerable impact on productivity, while the one in the inner region does not. Due to the close values of bottom-hole pressure and the average reservoir pressure, the permeability of the inner region, k ₁, is close to k_d(p), based on Eq. 17, resulting in the overlap of Curves II and III. When considering the permeability of the exterior region k ₂, total permeability is improved as 2k _d(p), bringing larger productivity. This result further proves the importance of the exterior region.

Owing to the influence of the SSE–MSE and consideration of the drainage areas in the inner and exterior regions, the stimulation effect of multi-wing fractures varies with pressure (Figure 7). Curves a, b, c, and d represent the relationships of S _f and p _wf for different situations. Smaller S _f corresponds to a stronger stimulation effect on productivity. Without the SSE–MSE, the fracturing skin factor has no relationship with pressure (Curve b), while the fracturing skin factor with the SSE (Curve a) increases with the decline of p _wf, owing to the positive relationship of permeability controlled by the SSE and the reservoir pressure (the pressure larger than p _r in Figure 3). Greater permeability helps the fluid flow from cleats to the artificial fracture and the stimulation effect of the multi-wing fractures is enhanced.

The fracturing skin factor is smaller with the SSE–MSE than without (Curves c and d in Figure 7). These results indicate that the MSE has a greater effect on the fracturing skin factor than the SSE, and tends to lower it, benefitting CBM well production. Unlike the situation with the SSE only, the fracturing skin factor with the SSE–MSE has a positive relationship with p _wf. This is explained by the negative relationship between permeability and reservoir pressure (the pressure is lower than p _r in Figure 3). In addition to the inner region, and comparing Curves c and d, the SSE–MSE in the exterior region also has an important impact on the fracturing skin factor. The fracturing skin factor with the SSE–MSE in the entire region (Curve d), is the smallest among these four situations. This is due to additional consideration of the MSE in the exterior region, compared with Curve c. Therefore, for the gas well production system, reducing p _wf not only enlarges the producing pressure difference but also improves the stimulation effect of multi-wing fractures. When considering the SSE only, or neglecting the SSE–MSE in the exterior region, the stimulation effect of hydraulic fracturing is either overestimated or underestimated, and the optimization of the gas well production system is misdirected.

Fixing the p _wf as 1.5 MPa, the variation in fracturing skin factor in light of the included angle and length of fracture can be observed (Figure 8). The fracturing skin factor has a negative relationship with l ₂, by a similar variation rate as the arbitrary included angle. With the increase of the included angle, the fracturing skin factor decreases, and the decreasing rate gradually slows. Considering the area of the ellipse, a longer short axis or larger included angle creates a larger multi-wing fracture control area, which promotes the flow of CBM from the cleats to the fractures. Multi-wing fractures of longer length and a larger included angle produce a stronger stimulation effect. The curves with 60° and 90° angles almost overlap in Figure 8, due to the similarity of the elliptical areas with 60° and 90° angles and fixed l ₁ and l ₂ values. Similar to Figure 7, the fracturing skin factor with SSE–MSE in Figure 8 is smaller than the one without. Observing the curves with SSE–MSE in Figure 8, the variation in the fracturing skin factor with l ₂ would be less, and the diversity among these curves would be lower. These results indicate that the SSE–MSE weakens the negative impact of the fracture length and the included angle on the fracturing skin factor. Under the influence of the SSE–MSE, enhanced permeability provides more benefit than fracture length and included angle in making the gas flow more easily to the fractures. In other words, the SSE–MSE contributes more to the relative flow capacity than fracture length and included angle. Overall, a weaker SSE, a stronger MSE, longer fracture length, and a larger included angle, all increase the productivity of a CBM well.

In addition to the fracturing skin factor, the speed sensitivity skin factor also influences the productivity of a CBM well. The relationship of the non-Darcy flow constant with p _wf is similar to that of the fracturing skin factor (Figure 9). Without the SSE–MSE, the non-Darcy flow constant is not related to p _wf, due to the stable permeability (Figure 9A). Due to the positive relationship of permeability with reservoir pressure, the non-Darcy flow constant with the SSE only decreases with the increase of p _wf. However, when considering the SSE–MSE, the negative relationship of permeability and reservoir pressure results in the positive relationship of the non-Darcy flow constant and p _wf. Furthermore, compared to the curve without the SSE–MSE, the SSE increases the non-Darcy flow constant, while additional consideration of the MSE decreases and attenuates the extent of the non-Darcy flow.

Based on Eq. 3, the speed sensitivity skin factor is the product of the non-Darcy flow constant and gas production. As shown in Figure 9B, the absolute value of the speed sensitivity skin factor is much smaller than that of the fracturing skin factor. These results indicate that the positive effect of artificial fractures is much stronger than the negative effect of the non-Darcy flow with respect to CBM productivity. Nevertheless, the variation in speed sensitivity deserves further investigation for the most accurate evaluation of gas productivity. In Figure 9B, the speed sensitivity skin factor without the SSE–MSE is the largest among the three situations (with SSE–MSE, without SSE–MSE, and with SSE only). Combining Figures 6 and 9B, for the same p _wf, the non-Darcy flow constant is the largest with the SSE only, while the gas production is the smallest. In the situation with the SSE–MSE, the opposite is true. Accordingly, the curves with the SSE only and those with the SSE–MSE are similar. When considering the SSE or MSE, the non-Darcy flow effect would be weaker due to the reduced flow velocity or permeability.