An improved transport model of shale gas considering three-phase adsorption mechanism in nanopores

Journal of Petroleum Science and Engineering 182 (2019) 106291

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An improved transport model of shale gas considering three-phase adsorption mechanism in nanopores

T

Chaohua Guoa,*, Hongji Liua, Liying Xub, Qihang Zhouc a

Key Laboratory of Tectonics and Petroleum Resources, China University of Geosciences, Ministry of Education, Wuhan 430074, Hubei, China Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, Hubei, China c School of Automation, China University of Geosciences, Wuhan 430074, Hubei, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Shale gas Three-phase adsorption Porosity modification Gas transport model Apparent permeability

For the shale gas transport model in nanopores, researchers recently pay more and more attention to the existence of water phase. However, the effect of water film on the gas transport is still not clearly understood. In this paper, we considered the existence of water film on inorganic shale pore surface and introduced the adsorption mechanism of methane-water film-shale clay three-phase (gas-liquid-solid) to the shale gas transport model. Then, we modified the expression of porosity and derived a new transport equation of shale gas considering the adsorption mechanism of three-phases in nanopores. Using the finite element method, we solved the equation and analyzed the effect of water film. Finally, the range of water film effect was analyzed by defining the offset ratio. The results show that: (1) The existence of water film has negative effects on physical quantities, such as effective porosity and apparent permeability. When the water molecular coverage ratio reaches to 1, the decrease of porosity can be about 18.1%; (2) The existence of water film reduces the gas production rate and accumulative gas production. When the water molecular coverage ratio reaches to 1, the gas accumulative production decreases about 49.6%; (3) The pore radius which has significant water film effect is very small. Furthermore, the smaller of the pore size, the more significant the water film effect is. Also, it can be found that the effect of water film on porosity can be ignored with when pore radius exceeds 29 nm. When the pore radius is larger than 11 nm, the effect of water film on kapp/kd can be ignored.

1. Introduction Shale gas is one kind of unconventional natural gas which exists in organic and inorganic shale and shale intercalation. And it is an efficient and clean energy resource compared with traditional petroleum and coal (Zhang et al., 2019). Shale gas mainly exists in three forms: free gas existed in matrix pores and natural fractures; adsorption gas on the surface of kerogen, clay, or matrix pores; and dissolved gas in kerogen, asphaltene, and etc. Since the fraction of dissolved gas is much lower than that of free gas and adsorption gas, it can be approximately ignored. Shale reservoir is not only a source rock but also a reservoir, which is a typical “primary storage” model (Zhang et al., 2004). And it has the physical characteristics of low porosity and ultra-low permeability. Its porosity is generally less than 5.2% and permeability is generally lower than 0.001mD (Loucks et al., 2009). The formation mechanism of shale gas reservoir is very complex (Xu et al., 2009; Li et al., 2018), which has both adsorbability and piston-type reservoir formation mechanism (Luo et al., 2017).

*

Compared with conventional reservoirs, shale gas has complex flow mechanisms which are different from conventional reservoirs, including adsorption-desorption, diffusion, and slippage effect, and etc. (Li et al., 2018; Kast and Hohenthanner, 2004; Gordon et al., 1946). The adsorption and desorption is the interaction between the gas molecules and the pore surface. Among them, there are monolayer adsorption models, such as Langmuir model (Langmuir, 1918), and multimolecule layer adsorption models, such as BET model (Brunauer et al., 1938). In porous media, the mechanisms of gas transport mainly include viscous flow, Knudsen diffusion of free gas, and surface diffusion of adsorbed gas. When the average free path of the gas molecule is comparable to the pore size, it is necessary to consider the slippage effect of the gas (Klinkenberg, 1941). In order to depict shale gas transport in nanopores comprehensively, the researchers put forward the concept of apparent permeability, and simplify the complicated gas flow equation into the form similar to Darcy's law, which is convenient to solve. Based on the flow mechanism of shale gas in nanopores, scholars

Corresponding author. E-mail address: [email protected] (C. Guo).

https://doi.org/10.1016/j.petrol.2019.106291 Received 20 February 2019; Received in revised form 16 June 2019; Accepted 16 July 2019 Available online 19 July 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 182 (2019) 106291

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the unit is Pa. According to the research of many scholars (Aringhieri, 2004; Boyer et al., 2006; Jin and Firoozabadi, 2014), there is a layer of water film on the surface of minerals in inorganic shale due to the existence of clay minerals. The existence of water film can impose an impact on the adsorption and desorption of shale gas. Li et al. considered the existence of water film on the inorganic pore surface (Li et al., 2015), and proposed following equation:

have put forward different gas transport models. Javadpour studied the migration of shale gas by atomic mechanical microscopy and proposed an apparent permeability model considering the effect of Knudsen diffusion and gas slippage on the gas transport in nanopores of tight rock layers (Javadpour, 2009). Chen et al. considered multiple transport mechanisms and derived a simplified model of apparent permeability (Chen et al., 2015). Xia et al. considered the nonlinear and nonequilibrium gas adsorption and surface diffusion, and introduced the transport equation of shale gas in dual porosity media, which synthesizes all kinds of mechanisms (Xia et al., 2015). Su et al. constructed gas transport equations for inorganic and organic shale respectively (Su et al., 2016). Guo et al. considered the effect of gas desorption on gas viscosity, and derived the weak form of flow equation in dual porosity media (Guo et al., 2015a). Wu et al. built a mathematical model considering the surface diffusion of adsorbed shale gas, and discussed the effects of energy heterogeneity, isosteric heat of adsorption, and nonisothermal desorption on the diffusion of gas molecules (Wu et al., 2015). For shale reservoirs, the existence of water distribution in the organic pores is still a controversial topic, but most scholars have a consensus on the distribution of water in the inorganic pores (Hu et al., 2014). Haghighi et al. found that under natural conditions, the actual shale reservoirs often have a certain degree of water saturation (Haghighi and Ahmad, 2013). Roos et al. and Gasparika et al. tested a large number of shale samples in Canada, the United States and Europe, and found that shale adsorption capacity decreased under water-existing conditions (Ross and Bustin, 2009; Gasparika et al., 2014). Zhang et al. found that the thickness of the water film arranged on the surface of shale clay minerals is about 0.4 nm (Zhang et al., 2010). Spears et al. found that the volume of irreducible water in inorganic shale clay could reach 7.2% of shale sample volume (Spears et al., 2011). This shows that the water distribution in the pores of inorganic shale cannot be ignored. Through molecular simulation experiments, Jin et al. studied the interaction between methane and water molecules on the pore surface of clay (Jin and Firoozabadi, 2014). Based on thermodynamics, Li et al. established a quantitative model of water film thickness and discussed the distribution characteristics of water saturation (Li et al., 2016). Although they have studied the water distribution characteristics of inorganic shale clearly, there is still little discussion on the influence of water characteristics on gas transport in nanopores. On the basis of the previous research, we introduced the three-phase adsorption model in the inorganic shale (Li et al., 2015). Considering the effect of gas desorption process on the pore radius and porosity, the corresponding transport model of shale gas in nanopores has been constructed. By using the finite element method, we obtained the numerical results and discussed the influence of the water molecular layer on shale gas transport in nanopores and the pore size range which has significant effect.

V=β

(2) 3

In the equation, V is the gas adsorption volume, the unit is m /kg; VL is the solid-gas saturated adsorption volume (Langmuir saturated adsorption volume), the unit is m3/kg; PL is the solid-gas pressure constant (Langmuir pressure constant), the unit is Pa; VH is the liquidgas saturated adsorption capacity, and the unit is m3/kg; PH is the liquid-gas pressure constant, the unit is Pa. β is defined as the water molecular coverage ratio, dimensionless, which is the ratio of wetting area of water molecules to total pore area, dimensionless. The β can be calculated using following equation:

β=

AH Atotal

(3)

Furthermore, there is a relationship between the water molecule coverage ratio and irreducible water saturation. We combined the definition of water molecular coverage with the capillary irreducible water saturation equation (Li et al., 2016), and then obtained the equation that correlates irreducible water saturation with water molecular coverage ratio, which is shown as follows:

h 2 Sw = ⎡1 − ⎛1 − ⎞ ⎤⋅β ⎢ r⎠ ⎥ ⎝ ⎣ ⎦

(4)

In the equation, Sw is the irreducible water saturation in the capillary, dimensionless; r is the pore radius; h is the thickness of water molecular layer. The detailed discussion on how to calculate water molecular coverage ratio is given in Appendix A. Considering the three-phase adsorption model, the equation of the total gas adsorption in the presence of water film can be obtained, which can be given by the following equation (Guo et al., 2015b):

qa =

ρs MV ρM V P V P + (1 − β ) L ⎤ = s ⎡β H + + PL ⎥ Vm Vm ⎢ P P P H ⎦ ⎣

(5)

In the equation, ρs is the rock density, the unit is kg/m ; M the gas mole mass, the unit is kg/mol; Vm is the gas molar volume under the standard condition (0 °C, 1 atm), the unit is m3/mol; qa is gas adsorption capacity, the unit is kg/m3. Considering the surface coverage of the adsorbed gas, the surface coverage ratio can be obtained from the solid-gas adsorption model, which is as follows (Cui et al., 2003): 3

2. Three-phase adsorption and desorption mechanism

θ1 = In shale gas reservoirs, methane usually exists in shale pores in three forms: adsorbed gas on the shale matrix and pore surface, free gas in the pores and natural fractures, and dissolved gas in water and kerogen, as shown in Fig. 1. The adsorbed gas is the main component, which accounts for about 85% of the total gas content in shale gas reservoirs (Zhang et al., 2019). For the adsorption model of shale gas on the pore surface of shale, scholars generally adopt the Langmuir isotherm adsorption model (Guo and Hu, 2017; Xiong et al., 2012), which is given as follows:

P V1 = VL P + PL

VH P V P + (1 − β ) L P + PH P + PL

V1 P = VL P + PL

(6)

Based on the liquid-gas adsorption model, the surface coverage ratio can be obtained as follows:

θ2 =

V P = VH P + PH

(7)

In the three-phase adsorption model, the total surface coverage ratio of the adsorbed gas is as follows:

θ = β⋅ (1)

P P + (1 − β )⋅ P + PH P + PL

(8)

When the adsorbed gas desorbs from the surface of nanopores in the shale matrix, the gas molecules will fall off from the pore surface, which will increase the pore radius (Pruess et al., 1999). In inorganic clay shale, the gas desorption process considering the three-phase

In the equation, V1 is the gas adsorption volume, the unit is m3/kg; VL is the saturated adsorption volume of Langmuir, the unit is m3/kg; PL is the pressure constant of Langmuir when the gas adsorption is 50%, 2

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Fig. 1. The state of shale gas in the matrix: dissolved gas, adsorption gas and free gas.

pore radius and porosity. It can be found that these parameters are not only related to the pressure, but also related to the existence of water molecules on the water-wet inorganic pore surface. 3. The gas transport model in shale nanopores Under the initial reservoir condition, the shale gas is mainly adsorbed in the shale matrix. With the pressure of shale gas reservoir decreases, the shale gas gradually desorbs. The increasing gas concentration difference will be produced. In this process, the gas molecules will diffuse through the nanopores of the matrix, as shown in Fig. 3. By making reasonable assumptions and simplifying the complex underground geological conditions, we constructed a mathematical model to describe shale gas flow in nanopores. 3.1. Model assumptions (1) The gas adsorption obeys the three-phase adsorption model of methane, water film and shale clay, and it is the monomolecular adsorption. (2) The process of gas flow is isothermal. (3) The shale matrix is slightly compressible. (4) The effect of gravity can be ignored. (5) Only shale gas is involved in flow, the relative humidity of the gas is extremely low.

Fig. 2. Effect of three-phase desorption on pore radius. rmax is the pore radius without any fillers; rmin is the pore radius filled with water molecules and gas molecules; r is the equivalent pore radius.

adsorption model is shown in Fig. 2. Thus, the pore radius can be modified as follows:

re = rmax − dH ·β − dC ·θ = rmax − dH ·β − dC 3.2. Gas transport model in nanopores

P P ⎤ + (1 − β ) ⋅⎡β ⎢ P + P P + PL ⎥ H ⎦ ⎣

(9)

The equation of continuity of shale gas in nanopores is given as follows:

Where, re is the equivalent pore radius, dH is the water molecular layer thickness, dC is the methane molecular diameter, rmax is the pore radius without any adsorption gas in the inorganic pore, which is also called the ideal maximum pore radius. According to Li's research, the water molecular layer thickness is related to the relative humidity of the gas and pore radius (Li et al., 2015). In this paper, we consider the flow of shale gas with the relative humidity extremely low. In this case, the water molecular layer thickness is almost the thickness of one layer of water molecule. The change of pore radius can lead to the change of porosity. Thus, the porosity also needs to be modified, which is shown as follows:

ϕ = χ⋅ϕmax

re2 2 rmax

{r

max

= χ⋅ϕmax

P

P

− dH ·β − d⋅⎡β P + P + (1 − β ) P + P ⎤ H L⎦ ⎣ 2 rmax

}

∂ρtotal ∂t

+ ∇ (ρ⋅v) = q

(11)

Where, ρtotal is the total amount of gas in nanopores, the unit is kg/m3; v is the gas flow velocity (vector form), the unit is m/s; q is the source

2

(10) In the equation, ϕmax is the porosity under ideal maximum pore radius, which is called the ideal maximum porosity, ϕ is the intrinsic porosity, χ is the comprehensive coefficient, which is related to formation heterogeneity, and it is defined as the ratio of actual measured porosity to theoretical porosity at the same pore radius. Considering the three-phase adsorption model, we modified the

Fig. 3. Flow mechanisms of shale gas in nanopores. Red solid dots represent surface diffusion; yellow ones represent Knudsen diffusion; green ones represent viscous flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 3

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m/s; vd is velocity of viscous flow considering slippage effect (vector form), the unit is m/s. Because the thickness of the water film is very small, here we ignore the influence of the water film on the Knudsen diffusion and the viscous flow. In the equation, the Knudsen diffusion velocity is given as follows (Wu et al., 2015):

term. 3.2.1. Time-gradient changes of shale gas density There are both adsorbed gas and free gas in the matrix. In the presence of water film, the adsorbed gas is given by Eq. (4), while the free gas is given by the real gas equation of state, which is shown as follows:

ρ=

PM ZRT

vk = −

(12)

In the equation, ρ is the gas density and the unit kg/m ; R is the ideal gas constant, the value is 8.314 J/(mol·K); Z is the compressibility factor, dimensionless; T is the temperature, the unit is K. Then, the change of the total gas density with time can be expressed as follows: 3

∂ρtotal ∂t

=

∂ [(1 − ϕ) qa ] ∂ (ϕρ) + ∂t ∂t

v0 = −

∂q ∂ϕ ∂P = ⎡ (1 − ϕ) a − qa ⎤ ⎢ P⎥ P ∂ ∂ ⎦ ∂t ⎣

∂t

(15)

(16)

8πRT μ 2 r2 ⎛ − 1⎞ ⎤ ∇P ⎥ M Pr α 8 ‾ ⎝ ⎠⎦ μ

2μre M 3RTρ

8RT πM

8πRT μ ‾e M Pr

+ ⎡1 + ⎣ μ

∂ϕ

reason why the term (ρ − qa ) ∂P can be ignored is presented in Appendix B). Thus, the equation can be approximated as follows:

∂t

ρ M (1 − β )⋅VL⋅PL β⋅VH ⋅PH ⎤ ⎫ ∂P ϕ ≈ ρ ⎧ + (1 − ϕ) s ⎡ + 2 ⎨ (P + PH )2 ⎥ ρVm ⎢ P ⎣ (P + PL ) ⎦⎬ ⎭ ∂t ⎩

kapp =

ρ M (1 − β )⋅VL⋅PL β⋅VH ⋅PH ⎤ ϕ + (1 − ϕ) s ⎡ + 2 ( ) ( ρVm ⎢ P P P P + PH )2 ⎥ + L ⎣ ⎦

(18)

kapp kd

The unit of storage coefficient S is 1/Pa. Then Eq. (16) can be simplified as:

∂ρtotal ∂t

= ρS

∂P ∂t

2 α

)

−1 ⎤ ⎦

re2 8

∇P

(25)

2μre M 3RTρ

8RT + ⎡1 + ⎢ πM ⎣

r2 8πRT μ 2 ⎛ − 1⎞ ⎤ e ⎥ M Pr ‾ e ⎝α ⎠⎦ 8

(26)

=

16μM 3re RTρ

8RT +1+ πM

8πRT μ 2 ⎛ − 1⎞ M Pr ‾ e ⎝α ⎠

(27)

Then, the equation of gas motion after the introduction of apparent permeability can be simplified to the form similar to Darcy's law, which is as follows:

(19)

kapp → v =− ∇P μ

3.2.2. Gas velocity term in nanopores The mechanisms of gas transport in nanopores mainly includes Knudsen diffusion, slippage effect, surface diffusion, and etc. In this paper, we mainly consider the equation of gas motion under the effect of Knudsen diffusion and gas slippage effect, which is as follows:

v = vk + vd

(

In the equation, kapp is called the apparent permeability. It is not only related to the intrinsic properties of reservoirs, but also related to fluid properties, formation temperature, formation pressure and other factors. Further considering the difference between apparent permeability and reservoir intrinsic permeability, the ratio of apparent permeability to reservoir intrinsic permeability can be obtained:

(17)

In order to simplify the solution equation, the water storage coefficient S, is defined as the follows:

S=

(24)

It can be found that Eq. (25) is similar to Darcy flow equation. The apparent permeability kapp can be obtained:

∂ϕ

In the above equation, the term (ρ − qa ) ∂P can be ignored (The

∂ρtotal

(23)

Combine Eqs. (9), (20) and (23), we can obtain the following equation:

v=−

∂ϕ ⎫ ∂P ∂P ⎬ ⎭ ∂t

8πRT μ 2 ⎛ − 1⎞ M Pr ‾ ⎝α ⎠

vd = F ⋅v0 = ⎡1 + ⎢ ⎣

ρ M (1 − β )⋅VL⋅PL β⋅VH ⋅PH ⎤ ϕρ =⎧ + (1 − ϕ) s ⎡ + 2 ⎢ ⎨ ( ) ( V P P P + PH )2 ⎥ P + m ⎣ L ⎦ ⎩ + (ρ − qa )

(22)

In the equation, α is the tangential momentum correction coefficient, dimensionless; P is the average pressure, the unit is Pa. Then, the velocity of viscous flow considering the Klinkenberg effect is as follows:

By substituting Eqs. (14) and (15) into Eq. (13), the following equation can be obtained:

∂ρtotal

r2 ∇P 8μ

F=1+

(14)

Similarly, the time-gradient change of the adsorbed gas can be expressed as follows:

∂t

(21)

In the equation, v0 is the velocity of viscous flow without considering the effect of gas slippage, and μ is the gas viscosity, the unit is Pa·s. In the nanopores, the average free path of the gas molecule is comparable to the pore size, so the gas will slip on the pore surface (Klinkenberg, 1941). Brown et al. introduced the Klinkenberg effect coefficient F to modify the viscous flow in the capillary tube (Brown et al., 1946), and the equation is as follows:

It can be seen from the above formula that the change of the total gas density consists of two parts, the free gas and the adsorbed gas, which are discussed separately as follows. The time-gradient change of the free gas is further simplified as follows:

∂ [(1 − ϕ) qa ]

8RT ∇P πM

The motion of the viscous flow in the long straight circular capillary follows the Hagen-Poiseuille equation (Florence et al., 2007), and the following equation is obtained:

(13)

ϕρ ∂ϕ ∂P ∂ϕ ∂ρ ∂ (ϕρ) =⎛ +ρ ⎞ +ρ =ϕ ∂P ⎠ ∂t ∂t ∂t ∂t ⎝ P

2rM 3RTρ

(28)

3.2.3. The mathematical equation of shale gas in nanopores Combining Eqs. (11), (19) and (28), the flow equation of shale gas matrix can be obtained as follows:

(20)

ρS

In the equation, v is the velocity of gas flow (vector form), the unit is m/s; vk is the velocity of Knudsen diffusion (vector form), the unit is 4

ρ⋅kapp ∂P − ∇⋅⎛⎜ ∇P ⎞⎟ = q ∂t ⎝ μ ⎠

(29)

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In the equation, S is the water storage coefficient, kapp is the apparent gas permeability, ϕ is the porosity, and re is the effective pore radius. The four proposed parameters can be calculated as follows:

ρ M (1 − β )⋅VL⋅PL β⋅VH ⋅PH ⎤ ϕ S= + (1 − ϕ) s ⎡ + ⎢ (P + PL )2 ⎥ (P + PH )2 ⎦ ρVm ⎣ P kapp =

2μre M 3RTρ

{r

max

ϕ = χ ⋅ϕmax

(30)

r2 8πRT μ 2 ⎛ − 1⎞ ⎤ e ⎥ M Pr ‾ e ⎝α ⎠⎦ 8

8RT + ⎡1 + ⎢ πM ⎣

P

P

Table 2 Value of basic parameters.

− dH ·β − dC⋅⎡β P + P + (1 − β ) P + P ⎤ H L⎦ ⎣ 2 rmax

(31)

}

2

(32)

P P ⎤ + (1 − β ) re = rmax − dH ·β − dC⋅⎡β ⎢ + + P P P PL ⎥ H ⎦ ⎣

(33)

4. Model solution Generally speaking, shale gas transport equation in nanopores is a system of the non-linear partial differential equation with variable coefficients. The analytical solution is very difficult and sometimes impossible to obtain. Therefore, we often use numerical analysis method to discretize the equations and obtain the numerical solution. One of these numerical methods is called the finite element method. In this paper, we use the finite element method to solve Eq. (29) and analyze the effect of water film and the significant pore size range. The basic parameters required for solving Eq. (29) and values of some parameters are listed in Tables 1 and 2. We build a shale gas reservoir with the size of 100 m × 100 m. Point A is a point in the shale gas reservoir, the coordinate is (10, 6), which will be analyzed later. And the marked boundary is a horizontal well. The model is shown in Fig. 4. The initial and boundary conditions for the model are as follows:

∂Ω1, vy = 0,

∂P =0 ∂y

(34)

∂Ω2 , vx = 0,

∂P =0 ∂x

(35)

∂Ω3, vy = 0,

∂P =0 ∂y

(36)

∂Ω4 , P = P1

(37)

Ω, P (t = 0) = P2

(38)

Parameter name

Values and units

The meaning of parameters

P1 P2 μ rmax dC dH PH PL VL VH R M ρs α

3 MPa 50 MPa 2.22e-5Pa·s 3 nm 0.38 nm 0.4 nm 44.56 MPa 5 MPa 0.003 m3/kg 0.003 m3/kg 8.314 J/(mol·K) 0.039948 kg/mol 2600 kg/m3 0.8

ϕmax Vm

0.1 22.414 L/mol

Z T D χ

1 300 K 2m 1

Pressure in the horizontal well Initial pressure of shale gas reservoir Gas viscosity Ideal maximum pore radius Gas molecular mean diameter Water molecular layer thickness Gas-liquid adsorption pressure constant Langmuir pressure constant Langmuir saturated adsorption volume Gas-liquid saturated adsorption volume Ideal gas constant Gas molar mass Density of shale rock Slip effect tangential momentum correction factor Ideal maximum porosity Gas molar volume under standard condition Gas compressibility factor The temperature of the stratum Diameter of horizontal well Comprehensive coefficient

Table 1 Basic parameter expression. Parameter name

Expression

Three-phase adsorption volume Water molecular coverage ratio Gas adsorption capacity

V=β

Pore radius

β= qa =

ϕ=

Apparent permeability Ratio of apparent permeability to rock absolute permeability

+ (1 − β )

VL P P + PL

5. Model verification

AH Atotal ρs M V P ⎡β H Vm ⎣ P + PH

+ (1 − β )

Since there is no actual production data of oil and gas fields which is the same as the theoretical situation in this paper, it is difficult to verify the transport model of this paper with the actual production data. Therefore, we compare the numerical simulation results with the analytical results derived by previous scholars to verify the model in this paper. In 1998, Wu et al. deduced the analytical solution of one-dimensional steady state gas migration based on the Yucca Mountain limestone experimental study (Wu and Pruess, 1998). The equation for the one-dimensional steady-state gas transport solution is as follows:

VL P ⎤ P + PL ⎦

P P ⎤ re = rmax − dH ·β − dC⋅⎡β + (1 − β ) P + PL ⎦ ⎣ P + PH

Porosity

Water storage coefficient

VH P P + PH

Fig. 4. Geometrical model in this paper: a shale gas reservoir with the size of 100 m × 100 m.

S=

2 P P ⎤⎫ ⎧r + (1 − β ) β max − dH ·β − dC ⋅ ⎡ ⎢ P + PH ⎥⎬ ⎨ + P P L ⎣ ⎦ ⎭ χ ⋅ϕmax ⎩ 2 rmax

ϕ P

k app = kapp kd

=

+ (1 − ϕ) 2μre M 3RTρ

16μM 3re RTρ

ρs M (1 − β ) ⋅ VL ⋅ PL ⎡ ρVm ⎣ (P + PL )2

8RT πM

8RT πM

+

β ⋅ VH ⋅ PH ⎤ (P + PH )2 ⎦

( (

+ ⎡1 + ⎣

8πRT μ 2 M P re α

+1+

8πRT μ 2 M P re α

) − 1)

−1 ⎤ ⎦

re2 8

P (x ) = −b +

5

b2 + [P (x = L)]2 + 2bP (x = L) +

2qm μ (L − x ) k∞ β

(30)

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Fig. 6. Effect of water film on porosity variation with time. As the value of parameter β increases, the value of porosity in each time is gradually reduced.

Fig. 5. Comparison between the analytical solution of Wu and the numerical results in this paper.

varying with time at each β are shown in Figs. 7 and 8 respectively. The maximum pore radius rmax is 1 nm in this analysis. From Fig. 7, we can know that the pressure distribution is as follows: the closer to the edge of the horizontal well, the pressure changes more quickly. The farther away from the horizontal well, the less obvious the pressure change. Compared with the left and right figure, the pressure drop is faster when the parameter β is larger in the same space position. As shown in Fig. 8, each pressure curve decreases sharply at first, then decreases slowly. As the value of β increases, the value of each pressure decreases gradually. During the period of sharp change of pressure, pressure difference (Δp) is the main influencing factor and the influence caused by other factors can be ignored. In the period of slow change of pressure, pressure difference (Δp ) is no longer the main factor and other factors cannot be ignored. Water film affects apparent permeability of gas flow and also influences the change of pressure, which results in the difference of pressure at different β. At the same time, the value of each pressure curve decreases with the increase of β. Also, the change of pressure in turn will affect the changes of physical parameters. When the pressure changes rapidly, the physical quantity of flow also changes sharply.

The meaning and value of each parameter in the above equation have been detailed explained in the reference (Wu and Pruess, 1998). Therefore, we can verify the model by comparing the numerical results obtained from the simplified model with the analytical solution of Wu and Pruess (1998). As shown in Fig. 5, the results obtained by the model proposed in this paper are basically in agreement with the analytical solution provided by Wu and Pruess (1998). 6. Results and discussion The existence of water film will directly affect the pore radius and the amount of gas adsorption. Also, the pore radius change will affect the apparent permeability and porosity. Therefore, water film in shale has a certain influence on shale gas flow parameters directly and indirectly. In this paper, we proposed the term water molecular coverage ratio β to analyze the effect of water film on porosity, pressure, adsorption capacity etc. And then, the range of pore size with significant water film effect was discussed. 6.1. Effect of water film on other parameters 6.1.1. Porosity The effect of water film on porosity variation with time is shown in Fig. 6. It can be found that the value of porosity in each time is gradually reduced as the value of parameter β increases. In the period when the porosity changes slowly, the difference of porosity under various parameters β is increasing with time. At the t = 0 s, the difference of porosity under various parameters β is also very large. When the water molecular coverage rate reaches to 1, the decrease of porosity can be about 18.1%. When the ideal maximum pore radius is fixed, the pressure in the shale decreases gradually with time, which results in desorption of the adsorbed gas. Thus, the pore radius and porosity will increase. Since the porosity is proportional to the square of the radius, with the desorption of the gas, the difference of the pore radius at the same time with the presence of the water film will be larger and larger. At the initial time, when β = 1, the water film completely covers the pore surface, and the gas adsorption reaches its maximum at this time, which can be considered as the double layer adsorption. When β decreases, the adsorption gradually becomes the single layer solid-gas adsorption model.

6.1.3. Apparent permeability and permeability ratio kapp/kd The curves of apparent permeability and kapp/kd vs. time for each β parameter are shown in Figs. 9 and 10. As it can be seen from Fig. 9, the apparent permeability increases with time. With the increase of β, the apparent permeability curve decreases. And with time going, the apparent permeability difference becomes more and more obvious. As shown in Fig. 10, with the increase of β, the ratio of apparent permeability to intrinsic permeability increases and becomes more and more obvious with time going. When the water molecular coverage rate reaches to 1, the decrease of apparent permeability is 8.4%, and kapp/kd will be 9.96%. With the decrease of the reservoir pressure, the apparent permeability and kapp/kd will increase. The apparent permeability is also related to the pore radius: the larger the pore radius is, the larger the apparent permeability is. However, kapp/kd decreases with the increase of pore radius. Due to the existence of water film, the pore radius becomes small. And the larger the value of β is, the higher the water film coverage is, and the more obvious the decreasing of pore radius is. With time going, the influence of water film is becoming greater. However, it can be found that the pressure values of gas reservoirs under each β are almost the same and the difference is very small. So, the apparent permeability and kapp/kd are mainly related to the change of β. From the

6.1.2. Gas reservoir pressure The cloud diagram of gas reservoir pressure distribution with β equal to 0 and 1 at t=2 ×107 s and the curves of pressure 6

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Fig. 7. The pressure distribution map of the gas reservoir when t = 2 × 107s, the unit is Pa. The left: β = 0; the right: β = 1.

Fig. 10. Effect of water film on kapp/kd variation with time. As the value of parameter β increases, the value of kapp/kd curve in each time gradually increases.

Fig. 8. Effect of water film on pressure variation with time. As the value of parameter β increases, the value of pressure in each time is gradually reduced.

Fig. 9. Effect of water film on apparent permeability variation with time. As the value of parameter β increases, the value of apparent permeability curve in each time is gradually reduced.

Fig. 11. Effect of water film on gas adsorption capacity variation with time. As the value of parameter β increases, the value of gas adsorption capacity curve in each time is gradually reduced.

expression of apparent permeability and kapp/kd, we can know that kapp/ kd increases with the increase of β. With the passage of time, the difference becomes larger. While the apparent permeability decreases with the increase of β, and the difference becomes larger with time.

6.1.4. Gas adsorption capacity The gas adsorption capacity under each β is shown in Fig. 11. From Fig. 11, it can be found that each adsorption capacity curve decreases 7

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sharply at first, then decreases slowly to the minimum. With the increase of β, the adsorption amount of each adsorption capacity curve at each time gradually decreased. Also, it can be seen that the larger the β value is, the faster the adsorption capacity curve decreases overall in the period from 0 to 2.5 × 107 s. When the pressure and the maximum adsorption capacity are fixed, the larger the adsorption pressure constant is, the smaller the gas adsorption capacity is. This reflects the gas adsorption ability of different substances. By comparing the data, it is known that the adsorption ability of water molecules to gas is smaller than the adsorption ability of the pore surface to gas. Therefore, under the same pressure conditions, the larger the water film coverage is, the lower the adsorption ability of shale is, and the smaller the gas adsorption capacity is. At the same time, gas molecules are more easily desorbed from the surface of water membrane when the pressure drops rapidly, which makes the adsorption capacity decrease more rapidly. 6.1.5. Shale gas production rate For the shale gas production rate, consider the situation of perfect wells, we can calculate it by integral the area fraction of the flow velocity along the semi-cross-sectional area of the horizontal well. The diameter of the horizontal well is D. The definition is as follows:

qt =

∬

vdA =

∫

Dπ vdl = 2

∫

Dπ vdx 2

Fig. 13. Relationship between cumulative gas production and water molecular coverage ratio β.

approximately the same, so the percolation velocity is mainly related to the apparent permeability. When the value of β increases, the apparent permeability decreases gradually. Thus, the flow velocity decreases. So the integral with the same upper and lower limit of velocity will also be decreased. At the time t = 0 s, the apparent permeability decreases with the increase of β, so the initial gas production rate decreases with the increase of β. The change of gas production rate determines the change of cumulative gas production, which makes the cumulative gas production decrease with the increase of β. The results show that the existence of water film reduces the cumulative gas production.

(31)

In order to further discuss the effect of water film on shale gas production, we introduce the cumulative gas production parameter, which is defined as the integral of shale gas production rate to time. The definition is as follows:

Q=

vdxdt ∫ qt dt = ∬ Dπ 2

(32)

By means of numerical integration, we obtained Fig. 12 and Fig. 13. In Fig. 12, it can be found that the gas production rate curves decrease sharply initially and then decrease slowly. In the sub-figure of Fig. 12, with the increase of β, the value of gas production rate decreases gradually. We can find that the initial values of gas production rates decrease gradually with the increase of β. In total, the gas production rate is not sensitive to the change of water film parameter. However, in Fig. 13, we can see that with the increase of β, the cumulative gas production decreases obviously. And when the water molecular coverage rate reaches to 1, the accumulative production will decrease about 49.6%. The flow velocity is related to the apparent permeability and the pressure-gradient. The pressure-gradient at each β value is

6.2. Discussion on the range of pore size with significant water film effect In this section, we studied the effect of water film on permeability ratio under the conditions of different rmax and β. After scanning different rmax and β, Fig. 14 can be obtained. In Fig. 14, it can be found that the smaller pore size is, the more significant the water film effect is. As rmax increases, the change of kapp/kd with β is not obvious. In the range of rmax less than or equal to 11 nm, the change of kapp/kd with β is obvious. But the range is only the perceptual understanding. So, we need to carry the quantitative study for the range of pore size with the significant water film effect. The apparent permeability k, which is affected by the water film, is called a response variable. Variable k can be regarded as a function of

Fig. 12. Effect of water film on gas production rate variation with time. As the value of parameter β increases, the value of gas production rate in each time is gradually reduced.

Fig. 14. The effect of water film on permeability ratio under different pore radius and water molecular coverage ratio. The effect is not obvious with the increase of pore radius. 8

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water molecular coverage ratio β, ideal maximum pore radius rmax, and time t, that is:

k = k (β , rmax , t )

(33)

Here we use kd and ϕ to characterize k, which is as follows:

16μM k = kd ⎜⎛ ⎝ 3re RTρ

8RT +1+ πM

r2 8πRT μ 2 ⎛ − 1⎞⎟⎞ and ϕ = ϕmax 2e M Pr rmax ‾ e⎝α ⎠⎠ (34)

For the given rmax and t, the set of parameter β, {β0, β1, β2, β3⋅⋅⋅βn} (β0 specifically refers to β = 0), and the corresponding set of variable k, {k 0, k1, k2, k3⋅⋅⋅kn} , give the following definition: 1

δn (β , rmax , t ) = | n

n

∑i = 1 ki − k 0 1 n

n

∑i = 1 ki

n

|=|

∑i = 1 ki − nk 0 n

∑i = 1 ki

| (35)

In the equation, k 0 refers specifically to the value of the response variable at β = 0. For the given rmax and t, δn (β , rmax , t ) is called the absolute offset ratio for the given rmax and t. δn is used to quantitatively characterize the effect of water film. Similarly, given rmax and t, when there are only two parameters β, it can also be characterized by relative offset ratio, as follows:

δ 2 = δ 2 (β , rmax , t ) = |

k (β1) − k (β2) | k (β1)

Fig. 16. Relative offset ratio of porosity variation with time.

water film. When the pore radius is beyond 29 nm, the effect of water film on reservoir physical property can be ignored. At the same time, it reflects the complexity of gas flow mechanism in nanopores.

(35)

7. Conclusions

Moreover, the relative offset ratio can be used to determine the maximum pore radius range. When we discuss the change of function δn or δ 2 to rmax, when β is certain, for each given rmax, δn or δ 2 is a univariate function of t. In addition, for a given critical value ε , if

δnmax ≤ ε or δ 2max ≤ ε

(1) By introducing the mechanism of three-phase adsorption, the equation of effective pore radius and effective porosity were obtained. Then, a new transport model of shale gas which has universality with considering three-phase adsorption was constructed; (2) The existence of water film on shale pore surface has a negative effect on porosity, apparent permeability and gas adsorption. That is to say, the existence of water film reduces the value of each parameter and makes the reservoir physical property and gas flow property of shale gas reservoir become worse; (3) The existence of shale gas water film also has effect on the actual gas production rate and cumulative gas production. The simulation results in this paper show that water film can reduce gas production and even reduces the economic benefits of the gas reservoir in actual production; (4) For the small pore radius, the effect of water film is significant. And the smaller pore size is, the more significant the effect is. Based on the critical value ε given in this paper, for gas flow in nanopores, when rmax is below 11 nm, the effect of pore water film is obvious. And for reservoir physical property, it can be ignored when rmax is beyond 29 nm. The influence of water film can be neglected in large pores.

(36)

Then, it is considered that under this parameter rmax, the water film parameter β has no effect on k. We choose β2 = 0 and β1 = 1 to construct the relative offset ratio δ 2 , and analyze the change of the relative offset ratio of permeability ratio and porosity as shown in Fig. 15 and Fig. 16. In order to obtain higher accuracy, we choose the critical value ε = 0.02 . As shown in Fig. 15, we can see that the smaller pore size is, the larger the ratio is, and we can also know that the smaller pore size is, the more significant the effect is. When rmax = 11 nm, δ 2 max < 0.02 , it can be considered that the water film has no effect on kapp/kd at this time. And in Fig. 16, when rmax = 29 nm, it can be considered that the water film has no effect on ϕ at this time. It shows that the effect of water film on gas flow can be ignored for the pore radius above 11 nm because of the weak effect of

Acknowledgement The authors want to thank the supports from National Natural Science Foundation of China NSFC (No. 51704265 PI: Dr. Chaohua Guo), the Outstanding Talent Development Project of China University of Geosciences (Wuhan) (CUG20170614), and the Fundamental Research Founds for National University, China University of Geosciences (Wuhan) (1810491A07). The authors also want to acknowledge the anonymous reviewers of JPSE. Appendix A. Process to calculate the water molecular coverage ratio In the main body of the paper, we obtain the equation that correlates irreducible water saturation with water molecular coverage ratio, which is as follows: Fig. 15. Relative offset ratio of kapp/kd variation with time. 9

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h 2 Sw = ⎡1 − ⎛1 − ⎞ ⎤⋅β ⎢ r⎠ ⎥ ⎝ ⎣ ⎦

(A.1)

In the equation, the water film thickness, h, can be obtained using following equation (Li et al., 2015):

γ r A RT Pv ⎛ H + ⎞= ln r − h ⎝ h3 r⎠ Vm P0

(A.2)

The meanings of the parameters in Eq. (A.2) have been given in reference (Li et al., 2015). However, Eq. (A.1) only applies to a single capillary. At the capillary scale, it is difficult to obtain irreducible saturation data in a single capillary. Besides, in a shale gas reservoir, there are lots of capillaries. The workload of calculating each water molecular coverage ratio in the single capillary one by one is very huge. So the equation on reservoir scale needs to be derived. We make some adjustments to Eq. (A.1):

Sw

β=

(

⎡1 − 1 − ⎣

h 2 ⎤ r ⎦

)

Sw

=

(2 − )⋅ h r

h r

(A.3)

Meanwhile, we define the relative thickness of the water film, ζ , which is expressed as follows:

h r

ζ=

(A.4)

Then, Eq. (A.3) can be transformed as follows:

β=

Sw Sw + 2⋅ζ 2⋅(2 − ζ )

(A.5)

When the spatial distribution of oil field data in the development area is not taken into account, we make the cumulative addition operation to Eq. (A.5): n

∑ βi = i

n

1 2

∑ i

Swi 1 + ζi 2

n

S

∑ 2 −wiζ i

(A.6)

i

In the equation, n is number of experimental data. Through the definition of the arithmetic average, we obtain:

β‾ =

1 2n

n

Swi 1 + ζi 2n

∑ i

n

S

∑ 2 −wiζ i

(A.7)

i

For the heterogeneous shale gas reservoir, it is inaccurate to use arithmetic average under some cases, so it is necessary to introduce weighted average: n

∑i βi⋅Wi

βw =

n

∑i Wi

(A.8)

In the equation, Wi is the weight coefficient, whose value is determined by the heterogeneity of the gas reservoir. The equation for calculating water molecular coverage ratio using weighted averages is as follows: n Swi Wi ζi

∑i

βw =

n Swi Wi 2 − ζi

+ ∑i n

2⋅ ∑i Wi

(A.9)

When the spatial distribution of oil field data in the development area should be taken into account, we can obtain the following equation:

βS =

1 2SΩ

∫ Sζw((SS)) dS + 2S1Ω ∫ 2 S−w (ζS()S ) dS Ω

(A.10)

Ω

In the equation, SΩ is total area of shale gas reservoir; S is the area that can be seen as homogeneous. Similarly, we can obtain the arithmetic average equation:

βSw =

(S ) dS ∫Ω Sw (Sζ)(⋅SW) (S ) dS + ∫Ω Sw2(S−) ζ⋅ W (S )

2 ∫Ω W (S ) dS

(A.11)

On the scale of shale gas reservoir, we can use equations above to calculate water molecular coverage ratio. Appendix B. Rationality of ignoring the derivative of porosity to pressure In the main body of the paper, we ignore the derivative of porosity to pressure. Here, we want to explain its rationality. If considering the derivative of porosity to pressure, the true water storage coefficient is as follows:

S′ =

q ∂ϕ ρ M (1 − β )⋅VL⋅PL β⋅VH ⋅PH ⎤ ⎛ ϕ + (1 − ϕ) s ⎡ + + ⎜1 − a ⎞⎟ ⎢ (P + PL )2 ⎥ ⎝ (P + PH )2 ⎦ ρVm ⎣ ρ ⎠ ∂P P

(B.1)

(1 − β ) PL ⎤ ⎛⎜1 − qa ⎟⎞ ∂ϕ = ⎜⎛ qa − 1⎟⎞ 2χdC ϕmax re ⎡ βPH + 2 2 ⎢ (P + PL )2 ⎥ ρ ⎠ ∂P rmax ⎣ (P + PH ) ⎦ ⎝ρ ⎠ ⎝

(B.2) 10

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The approximate water storage coefficient is as follows:

S=

ρ M (1 − β )⋅VL⋅PL ϕ β⋅VH ⋅PH ⎤ + (1 − ϕ) s ⎡ + 2 P ρVm ⎢ (P + PH )2 ⎥ ⎣ (P + PL ) ⎦

(B.3)

We respectively substitute the approximate water storage coefficient and the true water storage coefficient into Eq. (29) in the main body of the paper, and compare the difference between the two results. Here we use the relative error to describe the difference between the two, and get Fig. B1. From Fig. B1, we can know that the relative error caused by ignoring Eq. (B.2) is not more than 0.2%. So, it is reasonable to ignore the derivative of porosity to pressure.

Fig. B1. The relative error caused by ignoring Eq. (B.2) variation with time. The maximum relative error is no more than 0.2%.

Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2019.106291.

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