Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores

International Journal of Heat and Mass Transfer 115 (2017) 566–579

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Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores Jinjie Wang a,⇑, Qingwang Yuan b, Mingzhe Dong c, Jianchao Cai d, Long Yu c a

Faculty of Earth Resources, China University of Geosciences, Wuhan 430074, China Faculty of Engineering and Applied Science, University of Regina, Regina, SK S4S0A2, Canada c Department of Chemical Engineering, University of Calgary, Calgary, AB T2N1N4, Canada d Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China b

a r t i c l e

i n f o

Article history: Received 10 January 2017 Received in revised form 15 August 2017 Accepted 17 August 2017

Keywords: Gas mass transport Surface diffusion coefficient Porous media Nanopores Shale gas reservoir

a b s t r a c t Understanding gas mass transport and determining diffusion coefficients are essential for investigating the gas flow mechanisms and evaluating porous media with nanopores. Multiple gas transport mechanisms coexist in porous media with complex pore size distribution, including viscous flow, Knudsen diffusion and surface diffusion. During pressure depletion of a reservoir, the adsorbed gas desorbs into pore space as additional ‘free gas’, and meanwhile, diffuses along the surface of nanopores in porous media. The surface diffusion itself increases the total gas transport capacity in pores and its effect cannot be neglected. The bulk gas transport (non-surface diffusion) data was excluded experimentally to intensively investigate the surface diffusion during gas mass transport based on the gas storage and flow mechanisms. Accordingly, a mathematical model is developed by incorporating the surface diffusion. The results show that the equilibrium time for gas transport process decreases quickly with temperature. Higher saturation pressure could accelerate the process and increase the amount of produced gas. Besides, the two-stage process of the gas mass transport can be identified by recording the decay of gas pressure, which implies that the surface diffusion dominates the late stage of the gas mass transport. The surface diffusion coefficient for shale is between 10
18 and 10
16 m2/s. This study provides a straightforward method to describe the gas mass transport in shale, simple but information–rich for the assessment of shale gas targets. Ó 2017 Published by Elsevier Ltd.

1. Introduction Shale, as one kind of porous media which is rich in nanopores, has been playing an increasingly important role in the energy industry and gradually contributes more to the energy supply of the world [1]. Although many ways have been used to improve the recovery of shale gas reservoir such as hydraulic fracturing [2], gas recovery is still less than 6% for shale wells in China [3]. Complex gas flow process happens in shale due to the nanopores introduced in this porous medium. One of the issues yet to be addressed is the mechanisms of the gas mass transport in porous media with nanopores which would be beneficial for enhancing the gas recovery to satisfy the domestic energy demand. Due to complex pore size distribution and different gas-storage processes, multiple gas transport mechanisms coexist in shale gas

⇑ Corresponding author at: China University of Geosciences, Wuhan, 430074, China. E-mail address: [email protected] (J. Wang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.08.057 0017-9310/Ó 2017 Published by Elsevier Ltd.

reservoirs, including Fick’s diffusion and Knudsen diffusion for free gas in inorganic pores, and surface diffusion for adsorbed gas in organic pores [4,5]. Different from conventional reservoirs, the pore size distribution for shale is wide (ranging from nanometers to milimeters) and many nanopores primarily exists in kerogen [6,7]. 15–70% of the total gas in shale reservoir is mainly adsorbed on the surface of nanopores and will influence the production dynamic tremendously [8]. With the development of science and technology, more and more attention has been paid to nanopores and the gas-solid interaction phenomena [9–11]. Consequently, clear gas transport mechanism, mass flow rate and diffusion coefficient will contribute a lot to the development of the shale gas reservoir. Worldwide research groups have addressed porous mediarelated (rich in nanopores) problems using different approaches, and the outputs of experimental results are steadily increasing [12–14]. Experimental methods for the measurement of the gas transportation in sub-micron pores includes variable-volume volumetric method (VVM) [12], constant-volume volumetric method

J. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 566–579

(CVM) [13], and pulse-decay method (PDM) [14]. The core of the VVM is to describe the dynamic gas storing/producing process when the temperature and the external pressure keep constant. Based on VVM and its mathematical model, the gas transport process can be divided into two stages and the apparent diffusion coefficients can be obtained. While for CVM method, it is mostly used for obtaining the adsorption isotherm curve when the system volume is constant while the pressure changes. Besides, by monitoring the pressure history, the gas transport in sub-micron pore is reflected and the transport stages can be obtained accordingly. PDM is a popular method for measuring the permeability of a core and the production process while the pressure keeps decreasing. Gas flow in porous media with micro/nano pores is considered as a combined result of free gas flow inside the pore and surface transport of the adsorbed gas along the solid wall [15]. Holt et al. [16] reported that in carbon nanotubes less than 2 nm in diameter, the measured CH4 flow rate exceeds predictions of the Knudsen diffusion model by more than an order of magnitude, which most likely resulted from surface diffusion. However, all the above mentioned methods cannot show any hint of investigating the surface diffusion while gas production. The driving force for surface diffusion is the concentration gradient. Since the adsorbed gas on organic pore walls has a large concentration gradient during the gas production process [17], the surface diffusion is a very important transport mechanism and cannot be neglected [10,18]. Some researchers stated that, compared with the bulk gas transport, surface diffusion in shale is more significant and even dominates the gas transport [19]. Darabi et al. [20] and Majumder et al. [21] concluded that the presence of surface diffusion can make the apparent diffusivity tens or even several orders of magnitude lower than that predicted by conventional hydrodynamic methods. Many experimental works have been conducted to measure the gas diffusion coefficients in mudstone, coal and shale. In these rocks, the measured diffusion coefficients range from 10
12 to 1 m2/s depending on different types of rock and testing methods [22,23]. Besides, the existed methods usually calculated the apparent coefficient which includes the process of Fick diffusion, Knudsen diffusion, and surface diffusion. However, the effects of surface diffusion and the roles of pressure and temperature on the gas mass transport have not been fully investigated. Specific method is still needed in order to measure the surface diffusion coefficient in shale. By referring to the organic rich porous media, mathematical simulation works are conducted to explore the gas flow behaviors through the nanopores [3,24–28]. The pore scale of the porous media determines the dominant mechanism of the gas flow process, Fick’s diffusion, Knudsen diffusion or surface diffusion, which is distinguished either by collision between molecules, by collision between molecules and pore walls, or by movement along the surface. Gas transport mechanism in conventional reservoir with large pores is dominated by viscous flow which can be described by Darcy’s law. However, in the tight gas reservoir gas transport is the combination of Fick’s and Knudsen diffusions [24]. In shale gas reservoir, except for the above mentioned mechanisms, surface diffusion along the adsorbed layer should also be considered [11,15,25]. Several mass transport models were accomplished to quantify gas transport considering the adsorbed gas existed in nanopores of porous media [26]. Carlson and Mercer [27] employed Langmuir isotherm theory to consider the effect of desorption behavior of shale gas and described the effect of diffusion by Fick’s law. Javadpour et al. [28] described gas flow in nanopores using a diffusive transport regime with a Knudsen diffusion coefficient and negligible viscous effects. Civan [7] coupled the mechanisms of viscous diffusion and Knudsen diffusion through a function of the Knudsen number (Kn) in the form of a product. Wu [3] calculated weighting coefficients of viscous flow and Knud-

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sen diffusion based on probabilities of gas molecules colliding with each other and with nanopore walls. Chen and Yang [29] used the kinetic method to derive the surface diffusion coefficient. They assume all the gas production comes from the adsorbed gas by surface diffusion after a certain period, which might overestimate the surface diffusion coefficient. Fractal models were employed to investigate the gas mass transport in porous media. Yu and Cheng [30] developed a fractal permeability model for bi-dispersed porous media. The model took into consideration of the non-uniform pore sizes and contained no empirical constants. Albaalbaki et al. [31] proposed a model which developed an interfacial boundary condition for diffusion, considering thermodynamic equilibrium, surface diffusion and interfacial exchange kinetics. It can be found that studies on the influence of the adsorbed layer on surface diffusion are rather scanty, and in the few currently published studies, not accurate calculation of surface diffusion coefficient limits those models for practical application with production prediction. The surface diffusion coefficient can be several orders of magnitude smaller than the apparent or effective diffusivity. It is a fundamental scientific problem that how surface diffusion is depicted and how much surface diffusion contributes during gas production, which still needs to be further explored. Therefore, in order to reasonably analyze gas mass transport and accurately forecast gas well deliverability, the surface diffusion must be considered in modelling the mass transport of shale reservoirs. In this paper, we investigate the gas mass transport behavior in porous shale. Experimental study on the gas mass transport and a model for determining the diffusion coefficient are presented which provide the fundament for numerical simulation and production forecast. First, the proposed experimental method enables the measurement of the dynamic gas mass transport when desorption massively occurs. Tests with helium (He) represent gas flow in porous media in the form of free gas, while tests with methane (CH4) represent the gas flow process including Fick’s diffusion and Knudsen diffusion for free gas, and surface diffusion for adsorbed gas. Second, we examine the effect of temperature and pressure on the gas flow behavior and the diffusion coefficient. Forty flow tests in total are shown under five pressures and four temperatures. Furthermore, a unified model is presented for depicting the gas transport mechanisms including the surface diffusion. By fitting the model with experimental results, the surface diffusion coefficient is obtained. Most importantly, the effect of pressure and temperature on the surface diffusion coefficient is described and analyzed. The experimental determination of the surface diffusion coefficient can depict the clear and accurate effect of parameters and provide more meaningful hint for the production of the shale gas reservoirs.

2. Experiment 2.1. Material Experiments were conducted to measure the characterization of shale sample and to investigate the gas mass transport through porous media. Since shale is a typical porous media with nanopores with large amount of adsorbed gas in it, shale is chosen as the test sample in this study. The shale samples are collected from Silurian age of Lower Jurassic Formation, Sichuan fold belt in China. The thickness of formation investigated is higher than 100 m, at depths between 585 and 643 m. The permeability of the sample is 3.8  10
6 mD with the average porosity 3.76%. According to the XRD test results and Rock Eval measurements, the dominant mineral for tested shale sample is quartz with an average of 48 wt%, followed by clay with an average of 41 wt%. The total organic carbon (TOC) is 1.58 wt%. The particles of the shale sample

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used in this study are assumed to be in spherical shape, with an average diameter of 1180 lm.

obtained with NMR represents the pore size distribution. This method is less time-consuming and nondestructive.

2.2. Characterization of the shale sample

2.3. Gas transport measurement

2.2.1. Adsorption isotherm measurement Adsorption is usually described through isotherms, that is, the amount of CH4 on the shale surface as a function of gas pressure at constant temperature. After accurately weighed, the shale samples are sealed into the sample tank, maintained their temperature within ±0.1 K of the designed temperature. At the same time, the leak test is conducted by injecting non-adsorbed gas (He is used for this study). Then the whole system with shale samples is evacuated for 12 h. Next, the void volumes of the tanks are measured with He before gas adsorption. After another evacuation, the CH4 in the buffer tank is piped into the sample tank and more than 24 h is needed for reaching equilibrium. The gas amount adsorbed by shale is then calculated and the corresponding pressure is recorded. Then other tests are conducted by repeating the above procedures to obtain the data of other pressures. After the measurement of the adsorption branch is finished, the desorption branch can be obtained afterwards by exporting a proper amount of gas from the tanks.

Fig. 1 illustrates the experimental set-up for measuring the gas transport behavior through porous media. It comprises a gas cylinder, buffer tank and sample tank, a gas flowmeter with an accuracy of 0.01 mL, and water bath unit. Gas is injected into the buffer tank first with different injection pressures. By releasing a proper amount of gas from the buffer tank and equilibrating the sample for 12 h, the shale particles can reach the equilibrium state under a saturation pressure. Before the measurement of gas production, gas regulator 4 is open for 10 s for releasing the gas between the particles. Record the value of produced gas volume and corresponding time. And the dynamic gas mass transport is obtained by multiplying the produced gas volume by gas density. A series of gas flow test with shale particle is measured at a temperature range of 299.15–328.15 K and pressure range of 0.69–3.45 MPa.

2.2.2. Scanning electron microscopy (SEM) Scanning electron microscopy (SEM) method is used to image the pore structure of the sample with argon ion-beam milling. Organic rich shale has a complex pore structure, with pore sizes ranging from nanometers to micrometers. Thus an SU 8000 microscope (Hitachi, Ltd., Tokyo, Japan) with a backscatter detector is utilized. Before imaging, the BIB-polished cross-sections were Au-coated first. From BIB cross-sections, large areas were selected to be imaged at magnifications of 10,000 and 80,000 to create a large representative map to study the pores and the distribution of kerogen. 2.2.3. Brunauer–Emmett–Teller (BET) Brunauer–Emmett–Teller (BET) theory aims to explain the physical adsorption of gas molecules on a solid surface and serves as the basis for an important analysis technique for the specific surface area of a material. BET equation [32] uses the adsorption data to determine the apparent surface area. The adsorption data is usually obtained by the measurement of nitrogen adsorption at 77 K. For a BET test, samples are first outgassed for 16 h to a residual pressure of P < 1 mPa. And then nitrogen adsorption is measured at isothermal condition. The physical parameters, including specific surface area (2.51 m2/g), pore volume (8.44  10
3 cm3/g) and average pore diameter (12.88 nm) were obtained for the tested sample in this study. 2.2.4. Nuclear Magnetic Resonance (NMR) Nuclear magnetic resonance (NMR) relaxation has been used to determine the physical properties of reservoirs, including porosity and its distribution, irreducible fluid volumes, and permeability [33]. NMR with short echo space is used in this study to identify the distribution of saturated pores which contribute to the conductivity of fluids. Theoretically, an NMR measurement consists of using an external magnetic field to align hydrogen magnetic moments, and creating a dipole moment in the hydrogenous fluid component of the sample. Because the amplitude of the dipole moment is proportional to the number of hydrogen atoms percent, it actually measures the pore volume which is filled by fluid. In NMR relaxation, water in smaller pores relaxes faster than that in large pores due to the fact that smaller pores experiences a greater surface relaxation. Therefore, a T2 (time) distribution

3. Gas mass transport model The total gas flux is the sum of fluxes by Fick’s diffusion, Knudsen diffusion and surface diffusion [4]. A model, which is for gas flow under constant external pressure, is adopted to study the effects of adsorption and surface diffusion through lowpermeability multi-scale porous media. Note that gas adsorption/ desorption is one kind of physical process which is the phase transition between free gas in the bulk gas phase and adsorbed gas on the surface, but the adsorbed layer affects bulk gas transport in the form of surface diffusion. Therefore, gas adsorption/desorption is not considered but the surface diffusion of adsorbed gas is intensively investigated while analyzing the contributions of gas mass transport from different mechanisms. Fig. 2 shows the physical model of free gas and adsorbed gas transport in a shale particle. It is assumed that single layer of adsorbed gas exists on a nanopore wall. Thus adsorbed gas thickness is less than the diameter of the CH4 molecule (0.38 nm), which is of relatively small size compared with the roughness depth of a nanopore wall. The adsorbed layer introduces the surface diffusion into the gas mass transport process in organic rich porous media but has little effect on the other gas flow regimes. The Fick’s law is still valid for modelling the transport for bulk gas through porous media. Based on Fick’s Second Law, the free gas transport in porous media is given as:

@C @ 2 C 2 @C ¼ Df þ @t @r 2 r @r

!

ð1Þ

where C is the concentration of free gas in the pores, mol/m3; Df is the free gas diffusion coefficient, m2/s; r is the distance to the particle center, m; t is time in s. The sample is assumed to be in spherical shape with radius r0. As indicated in Section 2.3, the external pressure of the spherical particle ce is kept constant during the tests. The internal boundary condition is an impermeable boundary condition. The initial concentration in the pore of the particle is ci. So the boundary conditions are:

cjr¼r0 ¼ ce

ð2Þ

 @c ¼0 @r r¼0

ð3Þ

And the initial condition is:

cjt¼0;06r
ð4Þ

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569

Fig. 1. Schematic diagram of the experimental apparatus.

radius for organic matter, ranging from 114.5 to 190.4 nm for Longmaxi Shale in Sichuan Basin in China [34]. Note that the surface diffusion affects total gas transport but not the free gas transport in pore spaces [3]. Therefore, the total gas production mt should be the summation of the free gas production mf and the adsorbed gas production ma, written as:

0 1 D n2 p2 t 2 2 1
f 2
Ds n 2p t 6 X 1@ r mt ¼ me
2 M f 1 e 0 þ M a1 e ra A p n¼1 n2

Fig. 2. Physical model for gas transport through a shale particle.

By transforming the variation of concentration into mass produced, the general solution for this model is:

0

1 6 X 1
mf ¼ Mf 1 @1
2 e p n¼1 n2

Df n2 p2 t r2 0

1 A

ð5Þ

where mf is the accumulative free gas mass production at a certain time, kg/m3; Mf1 is the total free gas production, kg/m3. Df is diffusion coefficient for free gas, m2/s. Yang et al. [34] stated that in micropores and fine mesopores, the surface transport rate would be triggered under concentration gradient when the bulk gas transport was seriously inhibited by the adsorption potential of the pore walls. Thus, the second Fick’s law is employed for describing the surface diffusion process and the mass transport equation can be written as:

ma ¼ Ma1 1


2 2 1 6 X 1
Ds nr2p t e a 2 2 p n¼1 n

! ð6Þ

where ma is the accumulative adsorbed gas mass production, kg/ m3; Ma1 is the total adsorbed gas production, kg/m3; Ds is the adsorbed gas surface diffusion coefficient, m2/s; ra is the apparent

ð7Þ

Here, we provide the solution of this model adapted to our notation as it will later be used for fitting purposes. me is the ultimate gas production, which can be obtained from the test results. The mass flow flux will be obtained by deriving of the mass production. Eq. (7) implies that the total gas mass transport considering the adsorption in porous media is related not only to the parameters that effect free gas flow, but also to the parameters that effect surface diffusion through the adsorbed layer. Thus the gas mass flow through micro- to nano- pores in shale can be derived accordingly. If the surface diffusion coefficient is zero, Eq. (7) reduces to the gas mass transport in porous media without nanopores (or organic matter). Different with the surface diffusion model developed by Yang et al. where no transport of adsorbed CH4 is considered before the calculated free gas produced [34], in the present study, both free gas and adsorbed gas produce simultaneously during the whole gas mass transport process. It is therefore more reasonable for the physical problems.

4. Results and discussions It is very complex and difficult to conduct experiments which consider both bulk gas transport and surface diffusion in nanopores at the same time. In this study, we try to explore this area by accurate measurement of mass transport of gas and mathematical model proposed in the last section. In this section, the results related to the characterization of tested samples are first presented and analyzed. Second, the comparison between mass transport of CH4 and He is made in order to distinguish the contribution of free gas and adsorbed gas to the total gas transport process. Besides, the effect of temperature and pressure on gas mass transport is analyzed. Eventually, the diffusion coefficient is calculated for different scenarios which allows more accurate prediction of the dynamic mass transport of gas in porous media with nanopores.

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4.1. Sample characterizing

BET NMR

0.016

4.1.2. Pore size distribution by BET and NMR Fig. 4 shows the pore size distribution with diameters ranging from 0.3 to 100 nm of the shale cuttings measured with BET and NMR method. The BET curve shows that the pore volume (PV) increases and then decreases sharply with respect to the pore diameter, so does the NMR curve. Both BET and NMR results show that the mesopores (between 2 and 50 nm) and micropores (<2 nm) occupy rather large pore volume, which provide a large number of adsorption sites for CH4. In contrast, the macropores (>50 nm) in shale matrix is fewer. The average pore diameter obtained by BET for the tested sample is 12.88  10
9 m. Integral of curves in Fig. 4 can obtain the total pore volume of different pore sizes, indicating that 92% of the pore volume is composed of pores with widths between 1 and 10 nm. The smaller the pore width, the more surface area it provides. Thus, nanopores for this shale sample provide very large surface area for CH4 adsorption and consequently the surface diffusion should be taken into account when discussing the transporting mechanism of CH4.

0.5

Pore Volume (cm3/g)

4.1.1. SEM imaging Fig. 3 shows the SEM images displaying the distribution of organic matter and the pore structure of shale. Clearly, considerable amount of organic material distributes in inorganic matter, making shale characterized by different types of pores in both the organic and inorganic materials. Many pores observed in shale are in the order of nanometers. According to the pore size, Haber [35] classified pores into the following three types: micropores with widths smaller than 2 nm, mesopores with widths between 2 nm and 50 nm, and macropores with width larger than 50 nm. The natural micro-fractures and many micropores exist primarily in the inorganic matter, while the nanometer pores exist primarily in organic kerogen. Gas stores in these pores in compressed free gas. The rest large amount of gas in shale is adsorbed onto the surface of kerogen. Fig. 3b is a magnified image of the observation for a kerogen area in Fig. 3a, showing the existence of nanopores in kerogen. SEM can qualitatively and intuitively describe the pores in shale and give us a rough order of the pore size. But it is crucial to obtain the exact value of pore size and the pore size distribution because different mechanisms occur in pores with different diameters, which can be indicated by BET (the liquid N2 adsorption).

0.6

0.012

0.4 0.3

0.008

0.2 0.004 0.1 0.000 0.1

1

10

Incremential Porosity (%)

570

0.0 100

Pore Diameter (nm) Fig. 4. Pore size distribution of tested shale sample.

4.1.3. Adsorption isotherm of CH4 and He Adsorption isotherms for CH4 and He were conducted under 308.15 K and the highest pressure was 19 MPa (Fig. 5). The total adsorbed CH4 amount for shale is 0.57 cm3/g, under 308.15 K and 19 MPa. It can be observed that the adsorbed CH4 quantity is much higher than the adsorbed He quantity. The total adsorbed He amount is much lower than that of CH4, no more than 0.02 cm3/g under the same test condition. As known, CH4 stores as free gas in natural fractures and pore space and as adsorbed gas in organic matter and on clay particle surface [36]. He stores as free gas, the amount of which is dominated by the porosity of shale particles, pressure and temperature and can be calculated with Equation of State. Since the adsorbed He gas amount is very low as is shown by Fig. 5, here we assume the amount of free gas inside the shale sample is equal to the amount of He gas. Compared with He, the extra amount of CH4 as adsorbed phase in shale not only increase the total gas production, but also play a significant role in influencing the dynamic process during gas production. Another point to discuss is the effect of adsorbed layer on the pore diameter and further on the gas transport process in porous media. Researchers believe that the adsorbed layer narrows the organic pores and thus impacts the gas flow during production process [11]. Besides the narrow-effect of adsorbed layer, surface diffusion of adsorbed gas also plays an important role. However, the effect of adsorbed gas or adsorbed layer is complicated during

Fig. 3. SEM images displaying (a) organic matter in inorganic matter and (b) micropores, mesopores and macropores in shale.

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Quantity Adsorbed (cm3/g)

0.5

Table 1 Numerical parameters used for calculating the Knudsen number.

T=308.15 K

CH4-adsorption CH4-desorption He-adsorption He-desorption

0.6

0.4 0.3 0.2 0.1 0.0

0

5

10

15

20

Fig. 5. Adsorption isotherm for CH4 and He at 308.15 K.

gas mass transport in porous media with nanopores. On one hand, the thickness of the adsorbed layer varies along the pass of gas transport because of the pressure change. On the other, the pressure for a specific point changes over time when unsteady state measurement is conducted. In addition, since gas adsorption/desorption is a physical process which is the phase transition between free gas in the pore volume and adsorbed gas on the surface, additional ‘free gas’ is generated from desorption during gas mass transport. What’s more, accurately obtaining the surface diffusion coefficient is not available currently. Related research needs to be continued for future work. Fig. 6 shows the schematic diagram of a nanopore before and after adsorption. The diameter for CH4 is 0.38 nm. When we suppose the adsorbed gas occupying all the adsorption sites, the pore diameter is narrowed by 0.76 nm. For the porous media studied in this study, the average pore diameter is 12.88 nm from BET test. After reaching the adsorption saturation, two layers of CH4 appear and the pore diameter is narrowed by 5.9%. 11.45% of the void space of a channel is occupied by the adsorbed layers with a thickness of 0.38 nm. Gas flow mechanisms largely determined by the pore diameter, which could be determined by the Knudsen number (Kn). To quantitatively analyze the effect of adsorbed layer on the gas mass flow mechanism in porous media, values Kn for nanopores with and without the adsorbed layer are shown by Table 1 for two test conditions. The mean free path of molecules and the Kn for the real gas in nanopores are presented below [10]. The parameters for the model are summarized in Table 1.

rffiffiffiffiffiffiffiffiffiffiffiffi

l pzRT P

Kn ¼

Value

Unit

Description

d dCH4 da

12.88 0.38 12.12

nm nm nm

M R P1 P2 T1 T2

z1 z2 Kn1

16 8.314 3 3.5 308.15 328.15 1.19  10
5 1.25  10
5 0.9435 0.9349 0.60

g/mol J/(mol K) MPa MPa K K Pa s Pa s Dimensionless Dimensionless Dimensionless

Kn1a

0.64

Dimensionless

Kn2

0.56

Dimensionless

Kn2a

0.59

Dimensionless

Average pore diameter Diameter of CH4 molecule Average pore diameter considering the adsorption layer CH4 molar mass Gas universal constant Pressure Pressure Temperature Temperature CH4 viscosity, 308.15 K, 3 MPa CH4 viscosity, 328.15 K, 3.5 MPa Gas deviation factor Gas deviation factor Knudsen number of the pore without adsorbed layer, condition 1* Knudsen number considering the adsorbed layer, condition 1* Knudsen number of the pore without adsorbed layer, condition 2* Knudsen number considering the adsorbed layer, condition 2*

l1 l2

P (MPa)

k¼

Parameter

ð8Þ

2M k l ¼ 2r 2rP

rffiffiffiffiffiffiffiffiffiffiffiffi pzRT 2M

ð9Þ

1*-tests conducted under 308.15 K and 3 MPa. 2*-tests conducted under 328.15 K and 3.5 MPa.

In Table 1 for the four calculations, the Knudsen number ranges from 0.56 to 0.64. Besides, the existence of the adsorbed layer increases the Knudsen number by 0.04 under the same test condition. When 0.1 < Kn < 10, transition flow happens in the porous media [3]. Therefore, the adsorbed layer does increase the Knudsen number, but the flow mechanisms in the porous media are still the same. In other words, the narrow effect of the adsorbed layer on the gas flow mechanism is very limited. For transition flow, the gas molecule velocity of a pore wall is no longer zero. Consequently, the gas flux is increased. Considering this fact, the most obvious effect of the adsorbed layer is introducing surface diffusion which enhancing the dynamic gas mass transport in the porous media.

4.2. Comparison of CH4 and He mass transport Gas adsorption increases the gas amount stored in porous media, and alters the mechanisms of the gas transport process. Unlike previous studies where an equilibrium amount of gas adsorbed or stored in shale is measured, a time-dependent gas transport process is investigated here. The illumination of our experimental results on the gas mass transport is discussed based on the two ways by which the adsorbed layer affects the shale gas flow in shale matrix: reducing the pore diameter, and changing the

Fig. 6. Schematic diagram of the effects of adsorbed layer on gas transport. (a) Without adsorbed layer and (b) with adsorbed layer.

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interaction between gas and solid walls. The thickness of the adsorbed layer on the pore diameter and the Knudsen number is less obvious as discussed in Section 4.1. Therefore, we only consider the second effect of the adsorbed layer, namely the surface diffusion. Since He flow in shale could be treated as a hint for describing the free gas flow during depressuring, production curves for CH4 and He under the same test condition (temperature and pressure) are displayed and analyzed in Fig. 7. CH4 needs more time for reaching equilibrium than He. Besides, the curve for CH4 is much higher than that of He. Under the same test condition, the accumulating gas produced for CH4 is 3–5 times more than that of He. Higher stored gas amount of CH4 in comparison to that of He due to the fact that the former have considerable amount of gas stored in the adsorbed phase [37]. That means, the adsorbed gas amount is 2–4 times more than the free gas amount. Therefore, adsorption effect cannot be neglected when discussing the shale gas transport. The adsorbed gas is expected to be more important in determining the transport behavior in porous media with nanopores. Fig. 8 shows the mass flux of CH4 and He for two conditions: one is under the same test condition and the other is reserving roughly the equal total gas. The same test condition denotes tests conducted with the same saturation condition (temperature and pressure) and produced under the same production condition

(temperature and pressure). Equal total gas denotes tests which reserve the equal amount of gas in pores, that is, tests with roughly equal total gas have the equal ability of producing the same amount of gas. Noting that having exactly the same amount of total gas reserved in the porous medium is difficult to achieve for two separate tests. The best we can do is to reach approximately equal total gas, which are presented in this study. Here the tests with roughly equal total gas are presented to demonstrate the effect of total gas on gas transport in porous media with nanopores. As shown by Fig. 8, with the same test condition, CH4 produces at a higher mass flux compared with He. Since the narrowed pore by adsorbed layer tends to slow down the gas transport, there must be another mechanism leading to the higher mass flux for CH4. While, compared with roughly equal total gas of tests of CH4 and He, CH4 still has a higher flux compared with He. Because only free gas exists for He test, the gas flux in pores is equal to the free gas transport value. That means, if all CH4 stored in pores as free gas (i.e., the adsorbed CH4 is transformed into free CH4, see curves for equivalent He), the mass flux is still lower than that of CH4 in shale when large amount of CH4 is reserved as adsorbed gas. Thus we conclude that, rather than delay the gas transport, the adsorbed layer of CH4 actually promote the gas flow behavior. Here we are not saying the thickness of adsorbed layer doesn’t work. We hold the opinion that besides the effect of the thickness of adsorbed

Fig. 7. Comparison of the gas mass transport between CH4 and He.

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573

Fig. 8. The mass flux of CH4 and He under the same test condition and roughly equal total gas. (a) At 318.15 K and (b) at 299.15 K.

layer on the pore diameter, there must be another mechanism, plays more positive role of enhancing the gas flow. Test results of two cases at different temperatures share the same observation. Based on the above discussion and description in Section 4.1, it can be found that the adsorbed layer has two main effects on the gas transport, one is the adsorption/desorption or the solid–gas interactions, the other is the surface diffusion. Since adsorption/ desorption is a physical process and completed instantaneously, adsorbed gas plays a role of changing the gas density profile by compensating the gas produced during the reservoir exploitation. However, the surface diffusion, with ultra-low diffusion coefficient, is considered as the primary transport mechanism of adsorbed gas and can enhance the dynamic gas mass transport obviously, which has also been mentioned in previous literature [19,29]. It is found that for a porous medium with nanopores, CH4 molecules are always affected by the molecule-wall interactions and behavior of the adsorbed molecules should be considered rather than the motion of free gas molecules.

4.3. Effect of temperature on CH4 mass transport Temperature can influence the capacity of physisorption of CH4 on nanopores as well as the gas flow rate significantly. Therefore, the CH4 mass transport process varies a lot under different test temperatures. The effect of temperature on the dynamic gas production process for the sample at 5 cases is shown by Fig. 9. For each case, tests were conducted under 299.15, 308.15, 318.15 and 328.15 K. It is shown by Fig. 9 that me (the equilibrium value of mt) decreases significantly when the temperature increases from 299.15 to 328.15 K. All 5 comparisons under different saturation pressures share the same observation. Taking Fig. 9a for examples, me decreases from 11 to 6  10
5 kg/m3 when the temperature rises from 299.15 to 328.15 K. This observation corresponds well with those of shale samples from other reservoirs [38]. Besides, it costs more time for test under lower temperature to obtain the equilibrium. That is, both the equilibrium gas content me and the equilibration time decrease with temperature. Because higher temperature can enhance the thermal motion, the velocity of the molecule will be increased under higher temperature. Thus, larger production rate of the gas can be expect for tests under higher temperature. However, higher temperature leads to short residence time for the adsorbed molecule and the desorption process is largely promoted, which means less adsorbed gas amount will be reserved. In other words, the potential gas amount produced from shale under higher temperature is much less than that of the lower

temperature. Therefore, less gas production and greater production rate lead to a shorter process duration. As described in Section 2.3 of the testing procedures, CH4 in the sample cell would experience an expansion process into the collector of the outlet, which is reflected by the real-time pressure decay data. Take the four tests in case b as examples (Fig. 9b). The curves in Fig. 10 display the four pressure decay processes over time. These 4 tests share the same observation and take the first one for example. In Fig. 10a, the pressure drops dramatically for the first 18 min, and for the next hundreds of min, the pressure keep dropping extremely slow till zero. Similar observation of the pressure profile is reported by Fathi et al. [15]. Based on the real gas Equation of State (EOS), the pressure drop reflects the gas amount in the pore space with the fixed void space. Since the pore pressure drops slowly for most of the time in Fig. 10a, the free gas amount in the pore changes slowly. However, the accumulating gas mass production mt continues increasing during this period. In other words, the produced gas amount for the last hundreds of min mainly comes from the desorption of the adsorbed gas. Besides, considerable percentage of the surface area is associated with micropores and mesopores in shale sample whereas macropores is less, indicating the surface diffusion probably dominates the late stage of the gas transport. According to the multi-scale gas flow mechanisms and the experimental observation, the gas flow would follow an order of free gas in natural fractures and macropores with bulk transport, and then the adsorbed gas in organic matter with micropores [19,34]. Therefore, a two-stage process is divided for the gas mass transport in porous media with nanopores, which is in conformity with the conclusion of previous study [37,39]. The early stage is gas expansion in larger pores and late stage is dominated by the surface diffusion. Holt et al. [16] reported that in carbon nanotubes with diameter less than 2 nm, the measured CH4 flow rate exceeds predictions of the Knudsen diffusion model by more than an order of magnitude, which most likely resulted from surface diffusion again. As main purpose of this work is to investigate the surface diffusion process, calculating the surface diffusion coefficient of adsorbed gas is of primary interest. 4.4. Effect of pressure on CH4 mass transport Fig. 11 presents 20 laboratory tests which were conducted to investigate the effect of pressure on the CH4 mass transport in porous media with nanopores. The curves show the total mass production (mt) over time (t) under different 5 saturation pressures. The saturation pressure varied from 0.69 to 3.25 MPa. As can be

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Fig. 9. Effect of temperature on the CH4 mass transport through the shale sample (a) 0.69 MPa; (b) 1.38 MPa; (c) 2.07 MPa; (d) 2.76 MPa; and (e) 3.45 MPa.

observed for all curves in Fig. 11, mt increases sharply with time at first and then the increasing rate decreases. As discussed in Section 4.3, for the early short period, gas production mainly comes

from the transport of free gas, while for the late long period, the production of the adsorbed gas dominates the process. For the higher pressure, it costs more time for obtaining equilibrium. Tak-

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Fig. 10. Curves for real-time pressure decay in shale (left coordinate) and cumulative mass production (right coordinate) under four temperatures and the saturation pressure of 1.38 MPa.

ing the test results in Fig. 11a for example, the accumulating mass production drops from 60  10
5 kg/m3 at 3.45 MPa to 10  10
5 kg/m3 at 0.69 MPa. The consumed time before equilibrium is much more at higher pressure than that of the lower pressure. At 3.45 MPa, a total of 700 min is needed while only 240 min for test under 0.69 MPa. The difference in time is probably attributed to the variations in mt and the production rate. Based on EOS for the real gas and the adsorption isothermal curve, for the fixed pore space of a porous media, both adsorbed gas amount and free gas amount increase with rising pressure. Besides, it is elucidated by Li et al. [40] that the molecule velocities in both the adsorbed layer and in free gas region can be enhanced by increasing pressure, which means that the gas flow rate at higher pressure is faster. However, under the test conditions, the effect of the total gas that can produced during the mass transport process exceeds that of the velocity for the gas transport. Thus it leads to more time consumed for obtaining the equilibrium under higher pressure. 4.5. Calculation of diffusion coefficient Before calculating the diffusion coefficient, comparison between models with and without considering surface diffusion is made in Fig. 12. For the model without considering the surface diffusion, it fits the experimental data in the early short period of time. However, the modelling curve deviates with the experimental data for the rest long period. It can be found in Fig. 12 that the matching results are much better for the model considering the surface diffusion, which indicates that this model is reliable for

the gas mass transport process through porous media with nanopores. Jin and Abbas [41] concluded that the surface diffusion could be regarded as the primary transport mechanism in small nanopores of porous media like shale [19]. The diffusion coefficient for free gas mass transport (Df) and the surface diffusion coefficient for adsorbed gas (Ds) in porous media with nanopores can be obtained with the aforementioned experimental results and mathematical model. First, diffusion coefficient Df can be calculated through Eq. (6) with He as free gas in porous media. Following this, surface diffusion coefficient Ds is to be calculated since tests conducted with CH4 simulates not only the free gas transport but also the surface diffusion of the adsorbed gas in porous media. Accordingly, Df and Ds are calculated and shown in Fig. 13 by fitting the experimental data discussed in Fig. 7 at various pressures and temperatures. It can be observed that the presented model can perfectly fit the experimental data. The good agreement in Fig. 12 again verifies the reasonability of previous two-stage assumption [37]. Under this assumption, surface diffusion dominates the late stage of the CH4 mass transport, indicated by the last term in Eq. (7). Since in most nanopores in shale, the pore size approaches the molecular mean free path, it’s therefore not accurate to model gas flow as a continuum process within the full range of flow regimes. Discrepancy may be caused using continuum approach and further work needs to be conducted once capability of fully characterizing the heterogeneity at pore scale is possible. Song et al. [10] share similar opinion and define a parameter j to analyze the probability of surface diffusion in pores and compare the

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Fig. 11. Effect of pressure on the gas mass transport.

Fig. 12. Comparison between mathematical modelling with and without considering the surface diffusion with test results. (a) 299.15 K, 0.69 MPa and (b) 328.15 K, 1.38 MPa.

result with micro-scale lattice Boltzmann model calculation result. The diffusion coefficient for free gas transport is 2.70  10
8 m2/s, while the surface diffusion coefficient Ds is about 9 orders of mag-

nitude lower, namely, 2.16  10
17 m2/s from Fig. 12. This is a reasonable estimation based on the published surface diffusion coefficient ranging from 10
14 to 10
20 m2/s [23,34]. Although

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Fig. 13. Fitting curves for the gas mass transport and the calculated diffusion coefficients.

Fig. 14. Sensitive of Df and Ds to pressure and temperature.

the diffusion coefficient is small, surface diffusion is able to transport the adsorbed gas to big flow channels, considering the short diffusing distance and large surface area of porous media with nanopores. Consequently, it greatly promotes the contribution of the adsorbed gas to the total gas mass production. Some simulation results demonstrated that the smaller the pore size, the larger the contribution of surface diffusion is [41]. Adsorbed gas on organic pore walls has a large concentration gradient in shale with a great specific surface area; therefore, the

surface diffusion is a very important transport mechanism. Some experimental studies show that, compared with the bulk gas transport, surface diffusion is even more important and can dominate the gas transport in shale when a pore network is not welldeveloped [19]. Hence it is a fundamental scientific problem of how surface diffusion is depicted and how the reservoir conditions influence the surface diffusion coefficient. To more clearly understand the effects of the pressure and temperature on Df and Ds, the diffusion coefficient and surface diffusion coefficient are calcu-

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lated with 20 groups of experimental data respectively, as shown in Fig. 14. Fig. 14a indicates that, Df increases with temperature while decrease with pressure. At higher temperatures, gas molecule with higher energy moves faster. Since the free gas flow is influenced by both Fick’s diffusion and Knudsen diffusion, the share for different type of diffusion depends on the ratio of the pore diameter and the molecular mean free path. As mentioned by Lunati et al. [42] and Yang et al. [34], the Fick’s diffusion coefficient decreases with pressure and the Knudsen diffusion coefficient rarely changes with pressure. Thus the free gas diffusion coefficient Df discussed in this paper is expected to decrease with rising pressure, which is consistent with the results of Fig. 14a. Under these test conditions, the measured surface diffusion coefficient Ds is in the range of 10
18–10
16 m2/s. Similar with the free gas diffusion, Ds. increases with the temperature. However, it increases when the gas pressure rises. This observation is in accord with the surface diffusion theories and agree with the results of Yang et al. [34]. This is because as the pressure increases, more adsorption sites are ‘activated’ and more CH4 molecule adsorbed on the surface of the porous medium. Besides, the thermal motion of the gas molecule is greatly enhanced due to the fact that higher pressure provides molecules with more energy. Gas molecule is easier to escape from the surface with higher energy but will soon attached to the surface again because of the large number of activated adsorption sites. What’s more, concentration gradient is the driving force of surface diffusion and higher pressure induces higher concentration gradient for the gas mass transport. During this escape-attach period, gas molecule moves towards the direction of the outlet or lowconcentration area, contributing to the total gas mass transport in the porous medium. 5. Conclusions A coupled experimental and numerical study of gas mass transport and diffusion coefficient for porous media with nanopores has been performed. Gas mass transport experimental results are presented and a novel mathematical model for calculating the surface diffusion coefficient of adsorbed gas is established. Effects of the temperature and the saturation pressure are explored in detail. The main conclusions of the present study are as follows: (1) Adsorbed layer affects the gas mass transport in nanopores of porous media through the following two ways: increasing the capacity of the gas production and changing the velocity of gas mass transport. The adsorbed gas increases the gas mass transport and enhances the capacity of the total gas production by more than 3 times compared with the free gas transport. With adsorbed layer existed in nanopores of the porous media, it is proved experimentally that the mass flux of the gas transport can be enhanced through the porous media because of the surface diffusion mechanism. (2) Effects of the adsorbed layer on the mass transport can be accounted for the surface diffusion in nanopores of porous media. Mathematical calculation reveals that the surface diffusion can enormously promote the gas transport process, even the diffusion coefficient is very low compared with the free gas diffusion coefficient. The presented model including the effect of surface diffusion can fit the experiments very well. In porous media with nanopores like shale plays, the surface diffusion is of great significance in the mass transport of adsorbed gas. Accurate calculation of its diffusion coefficient is crucial for the reservoir evaluation and the field production forecast. Such effect needs further study.

(3) Lower temperature and higher pressure can increase the total gas production, while they have adverse effect on the dynamic gas mass transport. For lower temperature, it makes the diffusion coefficient and surface diffusion coefficient smaller, thus leading to slower gas mass transport. However, under higher pressures, molecules gain more energy and move faster, resulting in a larger surface diffusion coefficient but smaller free gas diffusion coefficient. (4) This study is quite useful for deeper understanding of the behaviors of dynamic gas mass transport in porous media with nanopores. It would also be significant when more parameters of the reservoir like the diffusion coefficient/surface diffusion coefficient need to be accurately determined. Thus, it is beneficial to gain insight into the behavior of gas movement in natural systems with nanopores.

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