J. Cent. South Univ. (2016) 23: 3322-3331

DOI: 10.1007/s11771-016-3398-0

Investigation of Barree-Conway non-Darcy flow effects on coalbed methane production

YANG Lei(杨蕾)^{1, 2}, RUI Hong-xing(芮洪兴)², ZHAO Qing-li(赵庆利)³

1. College of Science, China University of Petroleum, Qingdao 266580, China;

2. School of Mathematics, Shandong University, Ji’nan 250100, China;

3. School of Science, Shandong Jianzhu University, Ji’nan 250101, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: Coalbed gas non-Darcy flow has been observed in high permeable fracture systems, and some mathematical and numerical models have been proposed to study the effects of non-Darcy flow using Forchheimer non-Darcy model. However, experimental results show that the assumption of a constant Forchheimer factor may cause some limitations in using Forchheimer model to describe non-Darcy flow in porous media. In order to investigate the effects of non-Darcy flow on coalbed methane production, this work presents a more general coalbed gas non-Darcy flow model according to Barree-Conway equation, which could describe the entire range of relationships between flow velocity and pressure gradient from low to high flow velocity. An expanded mixed finite element method is introduced to solve the coalbed gas non-Darcy flow model, in which the gas pressure and velocity can be approximated simultaneously. Error estimate results indicate that pressure and velocity could achieve first-order convergence rate. Non-Darcy simulation results indicate that the non-Darcy effect is significant in the zone near the wellbore, and with the distance from the wellbore increasing, the non-Darcy effect becomes weak gradually. From simulation results, we have also found that the non-Darcy effect is more significant at a lower bottom-hole pressure, and the gas production from non-Darcy flow is lower than the production from Darcy flow under the same permeable condition.

Key words: non-Darcy flow; Barree-Conway model; coalbed methane production; error estimate; numerical simulation

1 Introduction

Coalbed methane (CBM) reservoirs are dual porosity systems, which consist of coal matrix and fracture network [1]. Most of coalbed gas is stored in the coal matrix as adsorbed gas, and only a small amount is stored in fracture systems as free gas. Usually, fracture systems are considered as the gas flow passage and the coal matrix is treated the source for fracture systems [2-7]. The gas transport in the dual porosity coal seam is governed by different flow mechanisms: gas diffusion process in the coal matrix is governed by Fick’s law, while gas seepage flow in fracture systems is usually described by Darcy’s law. Based on the above understandings about coalbed gas flow mechanism, a series of conventional mathematical and numerical models [8-11] have been developed, obtaining some useful computational and simulation results.

However, in recent years, the non-Darcy behaviors of coalbed gas flow have been observed in high permeable fracture systems such as the zones near the wellbore [12-13], which affects the coalbed gas extraction significantly. In 1901, according to the non-Darcy experiment data, FORCHHEIMER expanded Darcy’s linear form into a quadratic flow equation named as Forchheimer’s equation, which has been widely used in analyzing non-Darcy flow problem. For the past decades, the study about non-Darcy flow problem in porous media has been a research hotspot. Some application studies about non-Darcy flow have been carried out to study the effects of non-Darcy flow using Forchheimer model, and have obtained some useful computational and practical results. CLARKSON et al [14] conducted the production-data analysis for a single- phase coalbed gas well with Forchheimer non-Darcy flow, and found that the gas production rates were different at the early time and the later time. ZENG and ZHAO [15] studied the effects of Forchheimer non-Darcy flow on a coal seam gas well production at different bottom-hole pressures, and found that the non-Darcy flow results in a smaller production rate, a larger decline rate in the boundary-dominated period. WANG et al [12] developed a fully coupled finite element model to quantify the Forchheimer non-Darcy flow effects, and found that the non-Darcy effect is significant for high pressure drops and exists only within a small region near wellbore. In addition, some other studies about numerical method and numerical analysis for the Forchheimer non-Darcy model have also been carried out. GIRAULT and WHEELER [16] proposed a mixed finite element method for the Forchheimer problem using Crouzeix-Raviart element. RUI and PAN [17] considered a block-centered finite difference method for the Forchheimer model and second-order error estimates for both pressure and velocity were established.

Although the above studies about non-Darcy flow are mostly based on the Forchheimer model, we still need to point out that due to the assumption of a constant permeability, the Forchheimer model has some limitations in describing the non-Darcy flow in porous media. In 2004, BARREE and CONWAY [18] proposed a new non-Darcy flow model, which does not rely on the assumption of a constant Forchheimer factor β and could describe the entire range of relationships between flow velocity and pressure gradient from low to high flow velocity through porous media, including those in transitional zones. Therefore, the Barree-Conway non-Darcy model has attracted more and more attention in recent years. In 2007, through a series of intensive experiments, LOPEZ [19] verified the advantages of Barree-Conway model in describing relationships between flow rate and potential gradient, especially in transitional zones. WU et al [20] presented a general numerical model for incorporating the Barree-Conway model to simulate multiphase, multidimensional non-Darcy flow in porous media. Nevertheless, until now, we have not found any paper adopting Barree-Conway model to study the non-Darcy flow effects in coalbed gas production. Also, there is no paper conducting numerical analysis for the Barree-Conway non-Darcy flow problem.

The objective of this work is to investigate the effects of non-Darcy flow on coalbed gas well production performance, in which the non-Darcy flow is described by using the Barree-Conway model. For this purpose, an expanded mixed element approximation method is introduced to solve the coalbed gas Barree- Conway non-Darcy flow model, in which the gas pressure and velocity can be solved simultaneously, and error estimates for both pressure and velocity are established. A series of simulation scenarios were conducted to illustrate the non-Darcy flow characteristic with different non-Darcy parameters, e.g. minimum permeability ratio k_mr, characteristic length τ, diffusion coefficient D_m, etc. Furthermore, the effects of non-Darcy flow on the performance of coalbed gas well are investigated by incorporating the Barree-Conway model in non-Darcy flow simulation.

2 Coalbed gas non-Darcy flow model

In this section, considering desorption-diffusion- seepage behaviors of coalbed gas in CBM reservoirs, we derive a coalbed gas non-Darcy flow model based on the Barree-Conway equation.

2.1 Basic assumptions

1) CBM reservoirs are treated as dual porosity systems consisting of coal matrix and fracture network.

2) CBM reservoirs are isothermal and the gas viscosity is constant under isothermal conditions.

3) Gas non-Darcy flow in fracture systems is described by the Barree-Conway model.

4) Gas diffusion process in the coal matrix is pseudo-static and described by the Fick’s first diffusion law.

5) Gas absorption/adsorption is described by Langmuir adsorption isotherm.

2.2 Gas state descriptions

The equation of real gas state is

                        (1)

where p is the gas pressure; V is the gas volume; n is the amount of substance; T is the gas temperature; m is the gas mass; M is the gas molar mass; Z is the gas deviation factor; and R is the ideal gas constant. From Eq. (1), the gas density ρ can be written as

                               (2)

From Langmuir isotherm formula and the state equation of real gas, the adsorbed gas concentration C_m in coal matrix and free gas concentration C_f in fracture systems can be written respectively as

                        (3)

where p_m is the gas pressure in coal matrix; C_L is the maximum adsorption concentration of the coal matrix; p_L is the Langmuir pressure; p_f is the gas pressure in fracture systems, and f_f is the porosity of fracture systems.

2.3 Barree-Conway non-Darcy flow equation

The Barree-Conway model was proposed by BARREE and CONWAY in 2004, in which the Darcy’s law is still assumed to apply, but a apparent permeability K_a is introduced instead of conventional constant Darcy permeability K_d. The apparent permeability K_a is defined as

                   (4)

where k_mr is the minimum permeability ratio in high rate (0≤k_mr≤1); τ is the characteristic length; u is the gas flow velocity; μ is the gas viscosity and ρ is the gas density.

Substituting the apparent permeability K_a into Darcy’ law, the Barree-Conway model can be expressed as

                   (5)

When k_mr=0, the Barree-Conway model Eq. (5) is just the Forchheimer model. And when k_mr=1, the Barree-Conway model Eq. (5) is the Darcy model. We consider the case that the Darcy permeability tensor K_d is a diagonal matrix and the permeability is isotropic, k=k_x= k_y, for the two-dimensional problem. Then, from Eq. (5) we have

     (6)

Solving the quadratic Eq. (6) and dropping the negative term, the gas flow velocity u can be written as

    (7)

2.4 Continuity equation of gas flow in fracture systems

Then, according to the mass conservation law, the continuity equation of gas flow in fracture systems is

                         (8)

where q_m is the gas volume of interporosity flow from the coal matrix to fracture systems. Combining with the state equation of real gas Eq. (1) and unfolding the two terms on the left-hand side of Eq. (8), the continuity equation of gas flow in fracture systems can rewritten as

               (9)

where c_f is the compressibility factor of f_f and c_z is the compressibility factor of Z, defined as

                              (10)

2.5 Gas diffusion equation in coal matrix

According to the Fick’s first diffusion law, the pseudo-steady state equation of gas diffusion in the coal matrix can be written as

                   (11)

where C_am is the average concentration of a coal matrix block; C_E(p_f) is the adsorbent concentration of the matrix block surface and defined as τ_d is the diffusion time in a coal matrix block. Usually, the diffusion time τ_d can be expressed as

                                 (12)

where D_m is the gas diffusion coefficient in the coal matrix block, and σ is the geometrical factor.

Combining Eqs. (7), (9) and (11), we have the coalbed gas non-Darcy flow model as follows:

     (13)

where Ω is the objective region for the two-dimensional problem with boundary Ω, and [0,T] is a time interval with T∈(0,∞].

3 Expanded mixed element numerical method

In this section, we will present an expanded mixed element numerical method for the coalbed gas Barree-Conway non-Darcy flow model.

3.1 Some notations

For the convenience of subsequent analysis, we first give some notations. Define the inner product on Ω and its norm:

    (14)

Define function spaces:

      (15)

                     (16)

Let W_h×Λ_h ×V_hW×Λ×V be the mixed finite element function spaces such as RT_k with index k and discretization parameter h.

3.2 Numerical scheme for non-Darcy flow in fracture system

We introduce an auxiliary variable, defined by Thus, coalbed gas non-Darcy flow in fracture system can be described as

    (17)

Let △t>0, N=T/△t, an integer, and t_n=n△t, n=0, 1, …, N. Also let pⁿ≡p(x, t_n) for any p defined on . We define the full discrete expanded mixed element numerical scheme with backward Euler time-discretization, which is to find W_h× Λ_h×V_h such that

         (18)

3.3 Matrix form of numerical scheme and iterative algorithm

Let τ_h be a quasi-regular triangulation for the two- dimensional rectangular region Ω with mesh size h. For a triangular element T(A₁A₂A₃), let x_i=(x_i, y_i) be the coordinate of vertex A_i and γ_i be the edge opposite to vertex A_i. Let n_i be the outward unit vector on the edge γ_i and h_i be the length of the perpendicular dropped from the vertex A_i onto the edge γ_i, i = 1, 2, 3. We denote by F_h the set of all edges of the triangulation τ_h.

The scalar function space W_h is the space of piecewise constant functions:

    (19)

The dimension of the space W_h is equal to m₁, the total number of elements T_k in the triangulation τ_h. ψ_k(x) is the basis function of W_h associated with the element T_k.

The vector function spaces V_h and Λ_h are chosen to be the RT₀(Ω, τ_h), defined as

  (20)

and the dimension of vector function space is equal to m₂, the total number of edges γ_i∈F_h. f_i is the basis function of vector function space associated with the edge γ_i∈F_h.

Now, we use the basis functions ψ_k(x) (k=1, …, m₁) and f_j(x) (j=1, …, m₂) to represent and

                         (21)

Substituting Eq. (21) into Eq. (18), and setting ω_h= ψ_k(x) (k=1, …, m₁), v_h =f_j(x) (j=1, …, m₂) and π_h=f_j(x) (j=1, …, m₂), we introduce matrices and vectors as follows:

                 (22)

where

(23)

Therefore, Eq. (18) can be written in matrix form as follows:

                   (24)

Since gas diffusion Eq. (11) is an ordinary differential equation, when is known, we can get C_am(t_n) as follows:

        (25)

Then, the gas volume of interporosity flow from coal matrix to fracture systems could be calculated as follows:

        (26)

In practical calculation, alternative and iterative method is used to solve Eqs. (24)-(26), and specific calculation procedure is listed as follows:

1) For n=1, 2, …, N, when is known, q_m() can be calculated using Eq. (26).

2) Then, substitutinginto Eq. (24),can be obtained by solving Eq. (24).

3) Lastly, when is known, C_am(t_n) can be calculated using Eq. (25).

4 Simulation results and evaluation of non- Darcy effects

In this section, we carry out numerical simulation by using the lowest order Raviart-Thomas mixed finite elements RT₀ for the coalbed gas non-Darcy flow problem.

4.1 Error and convergence analysis

In order to verify the convergence, the test region is selected as unit square, i.e., Ω=[0, 1]×[0, 1], and 6 levels for uniform mesh size h=1/4,1/8,1/16,1/32,1/64,1/128 are computed to estimate the convergence rate. Since we can not get the analytical solution p_f, u and λ for problem Eq. (13), we use and as the criterion of convergence for the numerical solutions p_h, u_h and λ_h. The relative error and convergence rate estimates for p_h, u_h and λ_h are listed in Table 1, Table 2 and Table 3, respectively. We can find that the numerical solutions p_h, u_h and λ_h could achieve first-order convergence rate (see Fig. 1).

4.2 Non-Darcy dynamic characteristics of coalbed gas flow

For a simple application, the scale of objective region Ω is 200 m×200 m, and the mesh size h=10 m.

Table 1 Relative errors and convergence rates for p_h

Table 2 Relative errors and convergence rates for u_h

Table 3 Relative errors and convergence rates for λ_h

Fig. 1 Convergence rates for , and

There is a production well at the center of region Ω with constant bottom-hole pressure p_w. Simulation parameters are listed in Table 4.

Substituting the parameter value into calculation procedure, we can get the numerical solution, so as to analyze the dynamic characteristic of coalbed gas non-Darcy flow in CBM reservoirs.

Table 4 Parameters for simulation

4.2.1 Distribution characteristic of non-Darcy effects

From the description of non-Darcy flow phenomenon, we know that non-Darcy effect is dependent on the gas flow velocity. Figure 2 presents a typical velocity distribution of gas flow at the stable stage (t=10⁴ h). In Fig. 2, we have found that the coalbed gas flows toward production well, and flow velocity is much higher in the zone near the wellbore than the zone faraway from wellbore. Therefore, we could give a possible conclusion that the non-Darcy effect may be more significant in the zone near the wellbore than the zone faraway from wellbore.

Fig. 2 Gas flow velocity distribution in CBM reservoirs

In order to further illustrate the distribution characteristic of non-Darcy effect, Fig. 3 presents the distribution of apparent permeability K_a in the Barree-Conway non-Darcy model. From the definition of apparent permeability K_a, we know that K_a is dependent on the properties of porous media and pore fluid including permeability, characteristic length, viscosity, density and flow velocity, and is a decreasing function of flow velocity. Therefore, K_a could reflect the effects of inertial flow resistance on the Darcy permeability K_d, and could be as an indicator for the non-Darcy effect. In Fig. 3, we have found that K_a increases gradually from 0.186 to 0.2 with the distance from the wellbore increasing, which again indicates that the non-Darcy effect is more significant in the zone near the wellbore than the zone faraway from wellbore. This also verifies that the Barree-Conway model has advantages in describing the non-Darcy flow characteristic in transitional zones, being capable of overcoming the limitation of constant permeability in Forchheimer model.

Fig. 3 Distribution characteristic of apparent permeability K_a

4.2.2 Non-Darcy pressure dynamic characteristics

In order to illustrate the decrement of reservoir driving energy during production process, Figs. 4 and 5 present the evolution of gas average pressure in fracture systems with time t, and show the effects of coal matrix adsorption capacity C_L and bottom-hole pressure p_w on average pressure, respectively. In Fig. 4, the model has the initial reservoir pressure of 10 MPa, and C_L is taken as 32, 36 and 40 kg/m³, respectively, so as to analyze the effects of C_L on average pressure in fracture systems. In Fig. 4, we have found that pressure drop curves could be divided into 3 stages: Firstly, at the initial stage, because the fracture systems could not get sufficient pressure compensation instantly from coal matrix or outer boundary, the pressure drop speed increases gradually such that the average pressure drops sharply from 10 MPa to about 9.9 MPa. Also, at this stage, the effect of C_L is not evident and pressure drop curves with different C_L are almost coincident. Secondly, at the transition stage, with the pressure compensation from the outer boundary and coal matrix increasing, the average pressure in fracture systems drop speed slows down and pressure drop curves become apart. Lastly, at the stable stage, pressure drop and pressure compensation gradually achieve a balance state, and average pressure in fracture systems tends to a constant value. From the definition of coal matrix adsorption capacity C_L, we know that the coal matrix with larger C_L could supply more gas compensation for fracture systems, such that fracture systems could maintain higher pressure. Therefore, at the stable stage, we have found that the larger the C_L is, the higher the average pressure will be.

In Fig. 5, the bottom-hole pressure p_w is set as 4.5, 5.5 and 6.5 MPa, respectively, so as to illustrate the effects of p_w on average pressure in fracture systems. We have found that the effect of p_w is not evident and pressure drop curves with different p_w are almost coincident at early time. Then, at the later stage, in Fig. 5 we have found that the pressure drop curves become apart, and the difference of average pressure with different p_w becomes wider. Also, the larger the p_w is, the higher the average pressure will be.

Fig. 4 Gas average pressure of CBM reservoir with different C_L values

Fig. 5 Gas average pressure of CBM reservoir with different p_w values

4.2.3 Dynamic process of coalbed gas desorption

In Figs. 6 and 7, we show the evolution of gas average desorption rate with time t. In Fig. 6, the coal matrix adsorption capacity C_L is also taken as 32, 36 and 40 kg/m³, respectively, so as to analyze the effects of C_L on gas desorption process. According to equilibrium desorption model, we know that gas desorption rate is dependent on the coal matrix adsorption capacity C_L, fracture pressure p_f diffusion coefficient D_m. During gas desorption process, there are 2 stages. At the first stage, duo to quick drawdown of fracture pressure, gas concentration C_E(p_f) on matrix surface declines sharply, and then the gas desorption rate increases gradually to the maximum. Then, at the second stage, as the fracture pressure drop speed slows down and achieves a stable state, the desorption rate gradually declines and tends to a constant value at last. From the definition of coal matrix adsorption capacity, the value of C_L directly determines the initial adsorbed gas concentration C₀ in coal matrix. Therefore, in Fig. 6, we have also found that when C_L is larger, gas desorption rate will be higher. Also, the effects of C_L on desorption rate become more significant with time going on. In Fig. 7, the bottom-hole pressure p_w is set as 4.5, 5.5 and 6.5 MPa, respectively, so as to illustrate the effects of p_w on gas desorption rate. In Fig. 7, we have found that the variation of gas desorption rate is similar to the case in Fig. 6. We have also found that the higher the p_w is, the lower the desorption rate will be.

Fig. 6 Gas desorption rate of coal matrix with different C_L values

Fig. 7 Gas desorption rate of coal matrix with different p_w values

4.3 Effects of non-Darcy on coalbed gas well production performance

In this section, through incorporating the Barree- Conway equation in non-Darcy flow simulation, we investigate the effects of non-Darcy flow on the performance of coalbed gas well, and analyze the sensitivity of non-Darcy parameters, e.g. minimum permeability ratio k_mr, characteristic length τ, diffusion coefficient D_m, etc. Main simulation parameters are listed in Table 4.

4.3.1 Non-Darcy effect with different diffusion coefficients D_m

In Fig. 8, the bottom-hole pressure p_w is set as 6 MPa, and the diffusion coefficient D_m is taken as 10^-5, 10^-7 and 10^{-9 m3}/s, respectively, so as to illustrate the effects of D_m on gas production rate. In Fig. 8, we have found that production rate increases sharply and achieves the peak rate (2236 m³/h) after starting production. Then, as the gas pressure decreases, the production rate also declines and tends to be a stable state at last. Also, after a sharp decreasing, due to the gas compensation from coal matrix, the production rate shows a modest recovery in the curve (D_m=10^-5). In addition, due to the delay phenomenon of gas desorption, we have found that the effects of diffusion coefficient D_m on production rate are not evident at early time, and then the effects become more and more significant at later stage. Also, the larger the D_m, the higher the gas production rate. Figure 9 presents the gas cumulative production varying with time t, and shows the effects of diffusion coefficient. In Fig. 9, we have found that the effect of D_m on production is very evident, and becomes more and more significant with time going on. This illustrates that the CBM reservoirs with higher diffusion coefficient may supply more gas production.

Fig. 8 Gas production rate with different D_m values

Fig. 9 Gas cumulative production with different D_m values

4.3.2 Non-Darcy effect with different characteristic lengths τ

In Figs. 10 and 11, characteristic length τ is taken as 1×10⁵, 5×10⁵ and 1×10⁶ m^-1, respectively. In Figs. 10 and 11, we have found that the curves with different τ are almost coincident at early time, and then with the time going on, the effects of τ on production rate and cumulative production become evident at the transitional stage. Nonetheless, at the stable stage, due to the smaller and smaller difference of velocity between Darcy flow and non-Darcy flow, non-Darcy effect becomes weak gradually. Also, we have found that the larger the τ is, the higher the production rate and cumulative production will be. In addition, according to the definition of Reynolds number R_e=ρv/μτ, we know that when τ is increasing, the inertial flow resistance decreases and apparent permeability K_a increases, resulting in non-Darcy effect becoming weak. Therefore, from the above analysis, we can conclude that non-Darcy effect will cause the gas production declining. That is to say, under high flow velocity condition, if we do not consider the non-Darcy effect, we may overestimate the coalbed gas production.

4.3.3 Non-Darcy effects at different bottom-hole pressures

In Figs. 12 and 13, we set the bottom-hole pressure p_w as 4.5, 5.5 and 6.5 MPa, respectively, so as to illustrate the effects of non-Darcy on gas production at different bottom-hole pressure. In Fig. 12, we have found that the lower the bottom-hole pressure is, the higher the production rate will be. Also, with the bottom-hole pressure decreasing from 6.5 to 4.5 MPa, the peak production rate increases from 1956 to 3075 m³/h. In addition, comparing the production rates of Darcy (k_mr= 1) flow and non-Darcy (k_mr=0) flow, we have found that the effects of non-Darcy are not evident at early time, and with the time going on, the effects become significant at the transitional stage. Further, at the stable stage, the non-Darcy effect becomes weak gradually. Comparing the effects of non-Darcy at different bottom-hole pressures (p_w=4.5, 5.5 and 6.5 MPa), it has been found that when p_w is lower, the effects of non-Darcy on production rates will be more significant. The maximum differences between Darcy and non-Darcy production rates are 77 m³/h (at p_w=4.5 MPa), 55 m³/h (at p_w=5.5 MPa) and 36 m³/h (at p_w=6.5 MPa), respectively.

Fig. 10 Gas production rate with different τ values

Fig. 11 Gas cumulative production with different τ values

Fig. 12 Effects of pw and k_mr on gas production rate

Fig. 13 Effects of p_w and k_mr on gas cumulative production

Similarly, in Fig. 13, it has been found that the lower the bottom-hole pressure is, the higher the cumulative production will be. Also, with the time going on, effects of non-Darcy on cumulative production will become more and more significant. In addition, we have also found that when bottom-hole pressure is lower, the non-Darcy effects will be more evident. At different bottom-hole pressures (p_w=4.5, 5.5 and 6.5 MPa), the difference of ultimate production (t=10⁴ h) between Darcy and non-Darcy are 9.75×10⁵ m³ (at p_w=4.5MPa), 5.55×10⁵ m³ (at p_w=5.5 MPa) and 3.62×10⁵ m³ (at p_w= 6.5 MPa), respectively. This indicates that it is very necessary to consider the non-Darcy effects on coalbed gas production, especially at a lower bottom-hole pressure. Otherwise, the gas production may be overestimated under Darcy condition.

5 Conclusions

1) Through incorporating the Barree-Conway equation in coalbed gas flow problem, we present a more general coalbed gas non-Darcy flow model, in which both the desorption-diffusion-seepage behaviors of coalbed gas and effects of non-Darcy flow have been considered. Further, an expanded mixed element method is introduced to solve the Barree-Conway non-Darcy flow model.

2) Error estimate results indicate that the numerical solutions p_h and u_h could achieve first-order convergence rate. Non-Darcy simulation results have shown that the non-Darcy flow affects the dynamic characteristics of gas pressure, velocity and desorption rate, and the non-Darcy effects are more significant in the zone near wellbore. Simulation results also indicate that the non-Darcy effect affects the coalbed methane production rate and cumulative production, and the non-Darcy effect is more significant at a lower bottom-hole pressure. To be specific, the gas production from non-Darcy flow is lower than the production from Darcy flow under the same permeable condition, and the difference of production rates from both Darcy flow and non-Darcy flow becomes narrower due to their smaller and smaller difference of velocity, at stable stage.

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(Edited by YANG Hua)

Foundation item: Projects(91330106, 11171190) supported by the National Natural Science Foundation of China; Projects(15CX05065A, 15CX05003A) supported by the Fundamental Research Funds for the Central Universities, China

Received date: 2015-07-08; Accepted date: 2015-12-11

Corresponding author: RUI Hong-xing, Professor, PhD; Tel: +86-531-88369816; E-mail: hxrui@sdu.edu.cn