In TGRs, the pore size is in nano-scale. Non-Darcy flow has a significant impact on the apparent permeability [11Wang, J.; Liu, H.; Wang, L.; Zhang, H.; Luo, H.; Gao, Y. Apparent permeability for gas transport in nanopores of organic shale reservoirs including multiple effects. Int. J. Coal Geol., 2015, 152, 50-62.
[http://dx.doi.org/10.1016/j.coal.2015.10.004] ]. The apparent permeability of TGRs significantly deviates from the intrinsic permeability because gas flow in such formation occurs by various mechanisms different from the viscous flow regime represented by Darcy’s law [12Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena, 2^nd ed; John Wiley & Sons: New York, USA, 2002. -14Roy, S.; Raju, R.; Chuang, H.F.; Cruden, B.A.; Meyyappan, M. Modeling gas flow through microchannels and nanopores. J. Appl. Phys., 2003, 93(8), 4870-4879.
[http://dx.doi.org/10.1063/1.1559936] ]. Gas slippage effect or Klinkenberg effect is incorporated into this model by modifying the apparent permeability of gas flow [15Wu, Y.; Pruess, K.; Persoff, P. Gas flow in porous media with Klinkenberg effects. Transp. Porous Media, 1998, 32(1), 117-137.
[http://dx.doi.org/10.1023/A:1006535211684] ]:

(2)

in which,

(3)

where K_∞ is the absolute permeability; c is a proportional coefficient; λ is the molecular mean free path; is the mean gas pressure; r is the hydraulic radius.

The impacts of effective stress on gas permeability in tight formation were studied by Soeder, Bustin et al. and Wang and Reed [16Bustin, R.M.; Bustin, A.M.; Cui, A.; Ross, D.; Pathi, V.M. Impact of Shale Properties on Pore Structure and Storage Characteristics. Proceedings in the SPE Shale Gas Production Conference, November 16-18, Fort Worth, Texas, USA, 2008.-18Wang, F.P.; Reed, R.M. Pore Networks and Fluid Flow in Gas Shales. Proceeding in the SPE Annual Technical Conference and Exhibition, October 4-7, New Orleans, Louisiana, 2009.]. The stress-dependent relationship between permeability and effective mean stress for the first type of rock (Type-I rock) called by Raghavan and Chin [19Raghavan, R.; Chin, L.Y. Productivity changes in reservoirs with stress-dependent permeability. SPE Reservoir Eval. Eng., 2004, 7(4), 308-315.
[http://dx.doi.org/10.2118/88870-PA] ] is in well agreement with the experimental data,

(4)

Thus, the absolute permeability at the initial scenario should be:

(5)

where K₀ is the permeability at effective mean stress σ_m=0; η is a parameter; σ_m is the mean effective stress, which is mainly related to the vertical overburden load and reservoir pressure by [19Raghavan, R.; Chin, L.Y. Productivity changes in reservoirs with stress-dependent permeability. SPE Reservoir Eval. Eng., 2004, 7(4), 308-315.
[http://dx.doi.org/10.2118/88870-PA] ]:

(6)

and the initial mean effective stress σ_m,i by:

(7)

where is the total mean stress pressure, which is a function of σ_ob,v and is nearly a constant value for the system studied. Substituting Eqs. (6) and (7) into Eqs. (4) and (5) for σ_m and σ_m,i, respectively, the absolute permeability considering stress dependence effect becomes [20Wang, J.; Luo, H.; Liu, H.; Ji, Y.; Cao, F.; Li, Z.; Sepehrnoori, K. Variations of Gas Flow Regimes and Petro-physical Properties During Gas Production Considering Volume Consumed by Adsorbed Gas and Stress Dependence Effect in Shale Gas Reservoirs. Proceeding in the SPE Annual Technical Conference and Exhibition, Houston, Texas, USA, 2015.].

(8)

According to the theory of equivalent fracture conductivity, the relationship between the hydraulic fracture and its equivalent refined grid used in simulation is:

(9)

where K_hf is the permeability of the hydraulic fracture; w_hf is the fracture aperture; is the equivalent permeability of the hydraulic fracture because it is usually handled by a refined grid; Δx_hf is the length of the refined grid along the well.

According to experimental results [21Kamenov, A.; Zhu, D.; Hill, A.D.; Zhang, J. Laboratory Measurement of Hydraulic Fracture Conductivities in the Barnett Shale. Proceeding in the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA, 2013.], the relationship between the effective permeability and gas pressure can be written as:

(10)

Similarly handling Eq. (10) with Eq. (8) yields:

(11)

where α_s is stress sensitivity coefficient for hydraulic fracture; K_hf(p_g) is the permeability of hydraulic fracture considering stress effect. Applying Eqs. (9) and (11) yields:

(12)

In order to consider the high-velocity non-Darcy effect, the Forchheimer equation is usually used [22Forchheimer, P. Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing, 1901, 45, 1782-1788.-24Li, D.; Engler, T.W. Literature Review on Correlations of the Non-darcy Coefficient. Proceeding in the SPE Permian Basin Oil and Gas Recovery Conference Midland, Texas, USA, 2001.]:

(13)

By introducing the equivalent permeability and applying the Darcy's law, the following equation is obtained:

(14)

Applying Eqs. (12), (13), and (14) yields:

(15)

where β is the high-velocity non-darcy coefficient. The correction of Evans and Civan [25Evans, R.D.; Civan, F. Characterization of non-darcy multi-phase flow in petroleum bearing formation, Final Report for US Department of Energy Assistant Secretary for Fossil Energy, 1994.] is used to determine the high-velocity non-Darcy coefficient,

(16)

This correlation for β matched with more than 180 data points including those for propped fractures and was found to match the data very well with the correlation coefficient of 0.974 [26Rubin, B. Accurate Simulation of Non-Darcy Flow in Stimulated Fractured Shale Reservoirs. Proceeding in the SPE Western Regional Meeting, Anaheim, California, USA, 2010.
[http://dx.doi.org/10.2118/132093-MS] ].

Table 1
OED cases for comparing horizontal well with vertical well.

Fig. (2) The fitting results of incremental ratios for horizontal wells with different lengths.

Based on the features of horizontal well and sensitivity analysis, the permeability, porosity, and reservoir thickness are the dominant factors of the IR of horizontal well to vertical well. Moreover, the length of the horizontal well is the key parameter of drilling cost and IRs. Thus, we first choose permeability, porosity, and reservoir thickness to set up simulation cases using Orthogonal Experimental Design (OED), as shown in Table 1. Then, the horizontal wells with different well lengths (L_H) for developing TGRs are performed to obtain the IRs.

The model of the IRs of horizontal well to vertical well is obtained based on the simulation results using data regression. The fitting results of IRs for horizontal wells with different lengths are shown in Fig. (2). We can see that as the permeability and reservoir thickness increase, the IRs decreases; as the porosity increases, the IRs increases. In order to extend the application of Eq. (17), we further obtain the relations between the coefficients a, b, c and L_D based on Fig. (3), which are shown as follows:

(17)

(18)

(19)

(20)

where R_H/V is the IR of horizontal well to vertical well; K is the intrinsic permeability; H is reservoir thickness; φ is the intrinsic porosity; L_D is the dimensionless length of horizontal well, which equals to the ratio of well length to reservoir length; a, b, and c are the coefficients. Then, the above models are used to predict the IRs of different TGRs, and the comparison of predicted value with simulated value is shown in Fig. (4). Thus, the above predicted model is validated.

Fig. (3) Coefficients a and b vs. Dimensionless length LD.

Fig. (4) Comparisons between predicted value and simulated value.

Based on the features of fractured-horizontal well and sensitivity analysis, the permeability, porosity, reservoir thickness, and hydraulic fracture parameters are the dominant factors of the IRs of fractured-horizontal well to horizontal well. But the hydraulic fracture parameters also significantly affect the cost. Hence, we firstly choose permeability, porosity and reservoir thickness to set up simulation cases using Orthogonal Experimental Design (OED), as shown in Table 2. Then, we compare the IRs of fractured-horizontal well to horizontal well with different products of fracture number (n_f) and half-length (h_f) for developing TGR.

Fig. (5) The fitting results of IRs for horizontal wells with different hydraulic fractures nfhf..

The model of the IRs of fractured-horizontal well to horizontal well is obtained based on the simulation results. The fitting results of IRs with different products of n_f and h_f are shown in Fig. (5). We can see that as the permeability increases, the IRs decreases; as the porosity and reservoir thickness increase, the IRs increases. In order to extend the application of Eq. (21), we obtained the relation between the coefficients a, b, c and n_fh_f based on Fig. (6), which are shown as follows:

Table 2
OED cases for comparing fractured-horizontal well with horizontal well.

Fig. (6) Comparisons between calculated value and simulated value.

(21)

(22)

(23)

(24)

where R_Hf/Hw is the IR of fractured-horizontal well to horizontal well; n_f is the fracture number; h_f is the half-length of the hydraulic fracture; a, b, and c are the coefficients. Then, the above models are used to predict the IRs of different TGRs with fractured-horizontal well, and the comparison of predicted value with simulated value is shown in Fig. (7). We can see that the agreement is pretty well.

Fig. (7) Coefficients a, b, and c vs. nfhf..

It is an essential problem to determine the well pattern or horizontal segment length for developing TGR. High profit is the ultimate objective of TGR development. From the economic perspective, horizontal well is not always better than vertical well, and a long horizontal well is not always better than a short horizontal well as well; Moreover, hydraulic fracturing is not always necessary due to its high cost.

Firstly, we study how to optimize the length of horizontal well. The cost of a horizontal well includes initial and production investment. In this paper, the initial investment which is the main difference between horizontal well and vertical well is taken into account. The initial investment is the sum of vertical segment cost, inclined segment cost, horizontal segment cost, and the well completion cost of horizontal segment [27Hu, J.; Zhou, Z.; Li, X. A method of determining the horizontal-well lateral length with optimal economic value. Natural Gas Industry, 2014, 34(12), 142-146.]:

(25)

Thus, the cost ratio (CR) of horizontal well to vertical well is as follows:

(26)

where C_tH is the initial investment; C_V, C_I, C_H are the vertical segment cost, inclined segment cost, horizontal segment cost per meter, respectively; r_h is incremental coefficient of well completion cost of horizontal segment per square meter.

A higher cost must require a higher gas production. We introduce the coefficient ξ as the ratio of minimum incremental ratio of production (PIR) to the incremental ratio of cost (CIR). In fact, ξ is related to the gas price, and a higher gas price permits a smaller ξ. ξ can be described as:

(27)

Hence, the following relation must be met for achieving an ideal profit with horizontal well,

(28)

We first use the above models to study the feasibility of horizontal well in different reservoirs with different scenarios. The related parameters are shown in Table 3. Then, we can obtain the plate of different reservoir thicknesses with different ξ, as shown in Fig. (8). We can see that the plate can be used to determine the reasonable well length for different reservoirs. Thin reservoir is more suitable to use a horizontal well, and longer well does not mean better profit.

Table 3
Parameters for studying the feasibility of horizontal well in different reservoirs.

Fig. (8) Dimensionless well length vs. Production ratios of different thicknesses and ξ.

As it is discussed above, hydraulic fractures also need be designed. The cost of hydraulic fracturing is higher than 5,000,000 $ in USA, which is twice of well drilling. Similarly, we can compare the initial investment of fractured-horizontal well with horizontal well. The initial investment is the sum of vertical segment cost, inclined segment cost, horizontal segment cost, and hydraulic fracturing:

(29)

Thus, the cost ratio of fractured-horizontal well to horizontal well is as follows,

(30)

where C_F is the cost of hydraulic fracturing per meter. Similarly, the following relation must be met for achieving an ideal profit with fractured-horizontal well,

(31)

We first use the above models to study the feasibility of fractured-horizontal well in different reservoirs with different scenarios. The related parameters are shown in Table 4. Then, we can obtain the plate of different permeability with different ξ, as shown in Fig. (9). We can find that the plate can be used to determine the hydraulic fracture parameters; and the lower the permeability is, the more necessary to carry out hydraulic fracturing.

Table 4
Parameters for studying the feasibility of horizontal well in different reservoirs.

Fig. (9) Product of fracture number and half-length vs. Production ratios of different permeability and ξ.

Two TGRs in China are used to perform the design of different well patterns. The parameter is shown in Table 5.

Fig. (10) Production ratios vs. Dimenssionless well length in TGR A.

Fig. (11) Production ratios vs. Dimenssionless well length in TGR B.

Table 5
Parameters of two TGRs in China.

In TGR A, the permeability is 0.01md, which is relatively high for gas flow into well, so hydraulic fracturing may be not necessary, and vertical well and horizontal well are compared. Fig. (10) shows the production ratios of horizontal well to vertical well with different well lengths. We can see that when L_D is less than 0.2, the production ratio is less than 1, vertical well is apparently better. For ξ=0.3, L_D must be larger than 0.4 to satisfy the requirement that R_H/V> R_H/V^min; Similarly, L_D must be larger than 0.6 for ξ=0.4; whereas horizontal well is always unreasonable for ξ=0.5.

In TGR B, the permeability is 0.002md. We first evaluate the feasibility of horizontal well. Fig. (11) shows the production ratios of horizontal well to vertical well with different well lengths. We can see that R_H/V is much greater than R_H/V^min even for ξ=1.0, so horizontal well is significantly necessary. Then, we evaluate the feasibility of fractured-horizontal well. Fig. (12) shows the production ratios of fractured-horizontal well to horizontal well with different products of fracture number and half-length. We can see that when ξ=0.3, the product should be less than 900 to satisfy the requirement that R_Hf/Hw> R_Hf/Hw^min; Similarly, it must be less than 600 and 400 and for ξ=0.4 and ξ=0.5, respectively.

Fig. (12) Production ratios vs. Product of fracture number and half-length in TGR B.