Experimental and Numerical Investigation of Dynamic Gas Adsorption

Oct 19, 2016 - In contrast, the specific surface area of macropores (>50 nm) is very small. ... is monitored using thermocouples with a resolution of ...
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Experimental and Numerical Investigation of Dynamic Gas Adsorption/Desorption−Diffusion Process in Shale Jinjie Wang,† Zehao Yang,† Mingzhe Dong,*,‡ Houjian Gong,† Qian Sang,† and Yajun Li† †

College of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266580, China Department of Chemical Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada

‡

ABSTRACT: Shale gas is produced by gas transport under constant reservoir temperature and down hole pressure conditions. Therefore, it is of great importance to study the dynamic gas adsorption/desorption-diffusion process in shale, under isothermal and constant production pressure conditions. Accordingly, a new experimental method and apparatus has been designed and tested for studying shale gas transport behavior. The essence of the method includes accurately measuring the gas going into or coming out of a shale sample with respect to time. The accuracy and sensitivity of the method are confirmed by conducting experiments with methane and helium, and comparing the outcomes from adsorption isotherm obtained using the traditional constant-volume method. With this newly designed method, a two-stage transport process was observed by comparing the dynamic gas transport of N2 and CH4. Free gas transports first due to the pressure gradient, which is followed by the desorption and transportation of the adsorbed gas. Besides, tests under five pressures were conducted. It is found that for the same differential pressure, higher external pressure could accelerate the process while decrease the amount of transported gas. Finally, the dynamic adsorption−diffusion (DAD) mathematical model is presented to analyze the gas transport mechanisms in shale depicting the adsorption/desorption−diffusion process under isothermal and constant external pressure. By calculating the production rate for free gas and adsorbed gas, the two stages of the transport process can be identified. This study provides a straightforward method to experimentally determine the dynamic gas adsorption/desorption−diffusion process in shale, which is a relatively simple but information−rich technique for the assessment of shale gas targets. fracture network.8,23−25 Even with the wide application of hydraulic fracturing and horizontal well in production of shale gas reservoirs, gas recovery remains as low as 10−30% of original gas in place.26 More mechanism study on gas transport in a shale matrix needs to be done to make shale gas production predictable and economical.27 The issue yet to be addressed is to find a valid predictive tool for shale-gas production. During that transport process, adsorption or desorption mainly occurs in organic matter (kerogen), whereas diffusion is the main form of gas transport from inner and micropores to fractures.28−31 And the most fundamental step is to describe the dynamic adsorption/desorption-diffusion process in shale. Some researchers have recently studied the amount of gas stored in shale reservoirs, as well as the effect of influencing factors on gas adsorption and desorption, such as pore structure and organic and inorganic mineral content.32−37 Based on the storage capacity of a shale reservoir, the “sweet spot” can be determined and the development/exploitation plan can be made accordingly. The adsorption isotherm method is one of the popular ways of evaluating shale reservoirs.28 However, in applying the adsorption isotherm method, only the final amount of adsorbed gas under each equilibrium pressure is obtained, rather than the gas transport dynamics. Besides, the pressure in the cell of each adsorption isotherm test keeps changing, which has been depicted by Etminan et al. in detail.38 Those issues make it difficult to simulate the dynamic process

1. INTRODUCTION World production of unconventional gas has increased as the availability of light crudes and conventional gas declines. Due to its emergence as one of the backup energy resources and large geological reserves, shale gas with a high content of organic components has received renewed attention in recent years.1−5 As the reservoir bed and source rock for shale gas, shale reservoirs are complex and anisotropic geologic systems, which makes it difficult to successfully adapt the assessing methods for conventional gas reservoirs or coal bed.6−11 Javadpour et al. studied gas flow in nanoscale, in which the difference in gas evolution curves between coal methane and shale gas was stated.12 Considerable efforts are underway to make this resource economically and environmentally available. Gas storage and transport in shale differs significantly from conventional gas reservoirs, due to the ultralow permeability (as low as 10−10−10−6 μm2) and the rich organic components (total organic content could reach more than 30%).13−16 Shale gas may be stored as free gas in natural fractures and pore space, as adsorbed gas in organic matter and on clay particle surface, or in small amount as dissolved gas in oil and water.17,18 The amount of free gas could be calculated with the pore volume, and the dissolved gas is usually assumed to be negligible. The amount of adsorbed gas varies from 15 to 70% of the total gas stored in shale reservoir.4,19 The fact that gas adsorption or desorption plays a significant role, complicating the study of gas transport in shale reservoirs. It has been stated that a major component of the flux is due to diffusion in shale.20−22 Gas transport is controlled by rock petrophysical properties (e.g., matrix permeability), mineralogical features, and the © XXXX American Chemical Society

Received: June 14, 2016 Revised: September 13, 2016 Published: October 19, 2016 A

DOI: 10.1021/acs.energyfuels.6b01447 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels

PVVME is constant during the entire process (the horizontal straight line in Figure 1), simulating the constant external pressure. Different from PCVMI, PVVMI of an inner spot will experience continuous pressure drop, which mimics the actual production process in the field. A review on models of shale gas flow has been recently presented by Yang et al.40 Some systematic gas transport and storage models have also been proposed, which consider both the processes of free gas diffusion and gas adsorption.41,42 Javadpour et al., for the first time, systematically introduced Knudsen mechanism into the gas diffusion in shale when the diameter of the flow channel has the same order as the mean free path of the gas molecules.12 Freeman et al. and Yao et al. mixed the diffusion mechanisms through dusty gas model (DGM).42,43 Wu et al. present a study on the multiple transport mechanisms through nanopores of shale gas reservoirs with real gas effect−adsorption-mechanic coupling.26 However, the model presented in this article focuses on the effect of organic matter on the dynamic adsorption process. A generalized mathematical model for gas flow in shale is further improved to incorporate the process of gas adsorption/desorption. In this work, we focus on investigating the dynamic gas transport behavior in shale under isothermal and constant external pressure conditions by conducting experimental tests and numerical simulation. First, we develop a VVM and set up the apparatus correspondingly. The accuracy and sensitivity of VVM are verified by comparing the measured adsorbed gas content with that from CVM. Second, with VVM, a two-stage gas transport process is proposed based on the test results with He and CH4. Third, to verify the two-stage gas transport process and examine the effect of the external pressure on gas transport process, tests with shale particle under five different external pressures are conducted. Finally, mathematical study which can promote experimental results is presented for dynamic production prediction and shale gas reservoir evaluation.

of gas transport of shale reservoir under constant production pressure. Qin et al. showed an apparatus for measuring the dynamic gas adsorption process.39 In their study, the system volume is constant, which is similar to the usual adsorption isotherm method. Wang et al.33 experimentally investigated the dynamic gas adsorption-diffusion process in shale. They obtained the adsorption rate coefficients from the test results. To the knowledge of the authors, no descriptions of the isothermal and constant external pressure tests have been conducted for the production process of shale experimentally, leading to a lack of basic data and mechanism knowledge relevant to the dynamic process. To acquire a basic knowledge for designing and optimizing shale gas development, it is crucial to determine the dynamic gas transport behavior under isothermal and constant external pressure conditions. Different from constant-volume volumetric method (CVM), a new variable-volume volumetric method (VVM) has been designed in this article, for measuring the gas transport in shale under constant external pressure. This new method maintains constant external pressure which is satisfied by changing the system volume. The new method shows not only the amount of gas adsorbed at equilibrium pressure, but also depicts the whole dynamic process of gas transport in shale. Figure 1

2. EXPERIMENTAL SECTION 2.1. Material. Shale samples used in this paper were collected from Sichuan Basin, which is located on the upper Yangzi block in the eastern Sichuan fold belt in China. The samples are from Lower Jurassic Ziliujing Formation. The thickness for the studied formation is 104 m. Figure 2 shows the geologic map of Jiannan shale. Shale sample with total organic carbon (TOC) of 1.58 wt% was taken from Jiannan shale at depths of 585−643 m, and was grounded to 10−20 mesh. The particles of the shale sample used in this study are assumed to be in spherical shape. The permeability of the sample is between 0.6 to 1.68 mD. The average porosity for the sample is 3.76%. The physical parameters, including specific surface area (2.57 m2/g), pore volume (9.85 × 10−3 cm3/g), and average pore diameter (16.92 nm) were obtained by using liquid nitrogen adsorption/desorption method. Table 1 shows the geochemical characteristics of studied sample derived from Rock-Eval Pyrolysis. Since shale composition, including minerals and organic matter, strongly influences the pore structure, it is important to investigate the shale composition for better understanding their pore structure. According to the XRD test result, the dominant mineral for tested shale sample is quartz with an average of 48 wt%, and followed by clay with an average of 41 wt%. N2 physisorption experiment usually measures the adsorption and desorption branches (Figure 3a). Nitrogen isotherm curves are type IV according to the IUPAC classification. Previous studies indicate that type IV isotherms are common for shale as the presence of mesoporosity (2−50 nm). The left end of desorption branch is convex toward the upper left, indicating that the slit-shaped pores are predominant, followed by cylindrical and spherical pores. A significant amount of adsorption at low relative pressure (lower than 0.05) is observed. Figure 3b shows a broad pore size distribution with

Figure 1. Pressure dynamic history during the desorption process at one inner spot of the shale particle. PCVME: external pressure for CVM; PCVMI: inner pressure for CVM; PVVME: external pressure for VVM; PVVMI: inner pressure for VVM.

schematically shows the pressure history for the dynamic production process for both CVM and VVM technique. As previously mentioned, the external pressure was not regulated for CVM. Whereas a constant external pressure is maintained for each test with VVM. The superscripts “I” and “E” represent the inner pressure and the external pressure, respectively. Usually, CVM shows how much gas the reservoir could provide at a specific pressure−temperature condition, but gives very limited information regarding the process of field exploitation. Here, we take one inner spot of the particle as an example. Immediately after a pressure decrease, both PCVME and PCVMI change over time: PCVME increases over time because of desorbed and diffused gas from the matrix; PCVMI drops first, due to the pressure pulse from the reference cell, then increases because desorption supplements the amount of gas in the space near the inner spot. This pressure bounce-back will cause a change from desorption to adsorption at the regions near the surface of the sample. This is a complex process, and does not actually happen during the exploitation of actual reservoirs. For VVM at isothermal and constant external pressure condition, B

DOI: 10.1021/acs.energyfuels.6b01447 Energy Fuels XXXX, XXX, XXX−XXX

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Figure 2. Geologic map for Jiannan shale in Sichuan Basin. The location is in light cinnamon in the center. The samples tested in this article were collected from well HF-1.

Table 1. Geochemical Characteristics of Shale Sample TOC (wt%)

Tmax (°C)

kerogen type

S1 (mg/g)

S2 (mg/g)

S3 (mg/g)

S1/TOC (mg/g)

HI

OI

1.58

444

II

51.33

3.87

0.27

0.32

244.94

17.09

Figure 3. Nitrogen adsorption/desorption results for shale samples from two reservoirs: (a) N2 sorption isotherm; (b) pore size distribution. diameters ranging from 0.3 to 200 nm by BJH calculation.43 It is shown that mesopores (between 2 and 50 nm) and micropores (50 nm) is very small. The pore size distribution features are (1) the volume percentage of pores smaller than 20 nm is extremely high, occupying 79% of the total pore volume; (2) the pore volume percentage gradually increases by 13% as pore size increases from 20 to 50 nm; and (3) the pore volume percentage of pores larger than 50 nm is 8%. Integral of curve in Figure 5b indicates that 92% of the pore volume is composed of micropores and mesopores, providing large surface area for gas adsorption and pores with small diameter for gas transport. 2.2. Apparatus Design. To understand the gas transport process, an apparatus for VVM at isothermal and constant external pressure is designed and set up. The schematic diagram of the apparatus is shown in Figure 4. Some measures have been taken to ensure the accuracy of the temperature control, pressure record, and data acquisition, which will be discussed in detail in this section. The apparatus essentially consists of four parts: the cells (for gas and shale sample), the pressure

controlling system, the temperature controlling system, and the data acquisition system. A microdose pump is connected to the reference cell to keep the system pressure constant by changing the system volume. A reference cell is necessary for effectively keeping the external pressure constant. Specifically, large or small reference cell could all lead to the problem of pressure detecting. Since the gas transport rate is slow for shale and the gas amount transported is relatively small, thus the pressure vibration for large reference cell would be too small to be detected. On the contrary, too rapid and severe pressure change occurs if the small reference cell is employed. Based on the above discussion, the most straightforward solution is to (1) choose a proper volume of the reference cell; and (2) test as much sample powder as possible each time. Consequently, such a system is less sensitive to pressure fluctuations, and, most importantly, the change of gas volume due to gas transport is reliably recorded. By trial and error, the proper volume for reference cell is set to be 500 cm3, and the sample weight is 160 g. Under these experimental conditions, the pressure vibration is either visible, or not too sensitive to control. Besides, because the gas will easily diffuse or largely adsorb when a rubber O-ring is used, the cells employed the metal hard seal technique to guarantee the C

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the entire gas transport process. Prior to each experiment, all parts of the system were thoroughly cleaned to remove any solid and liquid contaminants. Toluene and acetone were used to clean and remove oily solvent, water, dust, and any contaminants from the cells. Then, to ensure no contaminants were left inside the system, cells and lines were evacuated and flushed with dry helium. Tests were conducted for observing the volume change at isothermal and constant external pressure conditions. The testing procedures have a total of five steps, as follows: (1) Sample weighing: according to the volume of the sample cell, crushed shale samples are accurately weighed and sealed into the sample cell (approximately 160 g). (2) Air-tightness checking: prior to each measurement, the entire setup is pressurized to 21 MPa with helium. If pressure is equilibrated for more than 4 h at a designed temperature, airtightness will be guaranteed. (3) Free-space volume determination: since the volumetric method is adopted, the accurate volume of each part is measured with nonadsorbed helium gas. An average value from three runs is used as the final result, to ensure a reliable measurement. Based on the free-space volume obtained, pressure change induced by the nonadsorbed gas expansion will be known. (4) Dynamic gas transport experiment: the temperature is set to the desired value and held there for more than 5 h. Methane is first pressurized, and then injected into the reference cell, keeping V2 and V5 open, and V1, V3, and V4 closed. Following that, V2 is closed and V4 is opened, allowing the gas to expand into the sample cell. The system pressure is recorded, and operations on microdose pump 8 is made: during the adsorption process, the microdose pump is screwed in, while it is drawn back for the desorption process. However, for both processes, the volume change needs to be recorded over time. Once pressure fluctuation does not happen anymore, V4 is shut down and the reference cell pressure is reset to a different level for the next measurement. After the end of all tests, the pressure in the system is quickly brought down to atmospheric, and the sample cell is allowed to degas completely. (5) Data processing: the free space volume is calculated with a real gas equation of state (EOS), according to the results of Step (3). After Step (4), the volume change due to gas transport in shale is recorded. Given a specific temperature, the gas amount induced by adsorption/desorption and diffusion under isothermal and constant external pressure could be calculated by eq 1:

Figure 4. Schematic diagram of apparatus for VVM under isothermal and constant external pressure. air-tightness. The high-performance metal seal overcomes the limitations of the traditional sealing materials, being suitable for pressures up to 35 MPa. Pressure is controlled with two pumps for this apparatus: the boost pump for increasing the gas pressure before injecting into the reference cell, and the microdose pump for maintaining constant pressure in the cells. Due to the low rate of gas transport in shale, the microdose pump was utilized to increase the test accuracy, which is capable of recording as small volume as 2.6 μL. Figure 5 shows the

ΔV =

PVz 0T0 P0zT

(1)

where ΔV denotes the changed gas volume due to gas transport into or out of shale at standard conditions (S cm3/g); P, V, T, and z stand for the pressure (MPa), volume (cm3), temperature (K), and compressibility factor under experimental conditions, respectively; P0, T0, and z0 are pressure (MPa), temperature (K), and compressibility factor under standard condition.

Figure 5. Pressure dynamic history for crushed sample at 308 K. curve of the pressure dynamic history through one test. During this 2 h history, the pressure was maintained around 13.93 MPa, with an error of less than 0.1%. The temperature control is another fundamental aspect to consider for the design of the apparatus. This part is a dual temperature control system with a water bath and an incubator. The temperature is monitored using thermocouples with a resolution of 0.01 K. The main components of the data acquisition system are the data gathering line connected to the pressure transducers and a computer with designed software. The software shows the real-time temperatures and pressures during each test. The real-time pressures give the sign of control action of the microdose pump. By adjusting the piston of the microdose pump, the system volume is changed to maintain constant external pressure. 2.3. Measurement of Dynamic Gas Transport Process. The basic principle of VVM is to keep the external pressure constant during

3. MATHEMATICAL MODEL A real challenge in describing the gas transport in shale is the characterization of the contribution of the kerogenic material imbedded into the rock. For the shale gas reservoir, the effect of organic matter on gas transport through nanopores cannot be ignored due to a large share of the production comes from the adsorbed gas stored in the organic matter. This section presents the dynamic adsorption-diffusion (DAD) model40 to describe gas transport in shale reservoirs, considering the effect of adsorption/desorption in organic matter. 3.1. Theory. Under a certain pressure and a temperature, one or more gas transport mechanisms coexist in shale rocks. In order to facilitate field applications, we need a comprehensive D

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Energy & Fuels constitutive equation to describe this gas transport process. The core issue is to characterize the share of each gas in the total transport, especially the extra amount of gas transport due to the existence of organic matter in shale. Therefore, based on the knowledge of free gas flow in pores and adsorbed gas adsorption/desorption from kerogen, we defined the apparent gas diffusion coefficient and the adsorption rate coefficient in this study as the physical parameters of free gas diffusion and adsorbed gas adsorption/desorption, respectively. Based on this consideration, a mathematical model derived and modified from Fick’s Law is developed. The control equation is given as ⎛ ∂ 2c ∂c f 2 ∂c f ⎞ ∂ca ⎟− = D⎜ 2f + r ∂r ⎠ ∂t ∂t ⎝ ∂r

The initial conditions:

c f |t = 0,0 ≤ r < r0 = c fi

(8)

ca|t = 0,0 ≤ r < r0 = cai

(9)

3.2. Analytical Solution. The physical parameters for the gas transport process are denoted as apparent gas diffusion coefficient and the adsorption rate coefficient, which might reveal the nature of the gas−solid interaction as a dynamic process. By solving the equations in Section 3.1, the analytical solutions for the free gas concentration (cf) in the shale particle and equivalent surface concentration (ca) of adsorbed gas at position r are equal to40

(2)

where cf is the concentration of free gas in the pore space of the shale particle (spherical) (mol/m3), D is the apparent gas diffusion coefficient (m2/s), r is the distance to the center of the shale particle (m), ca is equivalent surface concentration or adsorbed gas concentration (mol/m3), and t is time (s). The dynamic gas adsorption/desorption process before equilibrium could be described with the following equations:40 ∂ca = λc f − μca ∂t

(3)

The expressions of pn and mn in eqs 8 and 9 are as follows, respectively:

−1

where λ is the adsorption rate coefficient (s ), μ is the desorption rate coefficient (s−1). Because λ and μ have fixed relationship under equilibrium pressure, the μ can be expressed as below: −1

μ = λR

⎛ c eq ⎞−1 = λ⎜⎜ aeq ⎟⎟ ⎝ cf ⎠

mn =

(4)

where cfeq is the equilibrium concentration of free gas in the pore space (mol/m3), caeq is the equilibrium concentration of the adsorbed gas concentration on the surface (mol/m3), and R is the ratio of caeq and cfeq. Substituting eq 4 into eq 3, we can find

∂ca c eq = λc f − f eq λca ∂t ca

pn (ca eqλ + λc f eq + ca eqpn ) D(ca eqpn + λc f eq)

(13)

Integrating the total concentration from the surface of a particle to its center, the amount of gas transported in the particle at an arbitrary time can be obtained as

(5)

This equation reveals the interaction or the influence of the adsorbed gas on the dynamic gas transport process. It is assumed that the gas concentration in the external boundary (cfe) keeps constant during the experiment. The internal boundary condition is an impermeable boundary condition. The initial concentration in the pore of the particle is cfi and the initial equivalent surface concentration on the surface of the pore is cai. Gas transport out and into the shale sample are defined as the gas production process and gas storage process, respectively. For the gas production process, the external boundary concentration (cfe) is smaller than initial concentration (cfi) in the pore of the particle. While for the gas storage process, cfe is larger than cfi. r0 is the radius of the shale particle. So the boundary conditions and initial conditions could be stated as The boundary conditions: c f |r = r0 = c fe

(6)

∂c f ∂r

(7)

When the time approaches infinity, an equilibrium state will be achieved and the total amount of gas transport veq in the particle is Veq = lim V = t →∞

4πVmr0 3ϕ(c fi − c f0)(ca eq + c f eq) 3c f eq

(15)

It can be seen from eqs 14 and 15 that, when the time for gas transport is long enough, the total gas volume has nothing to do with the adsorption rate coefficient λ. That is, the adsorption rate coefficient is the parameter which reveals the dynamic process before the adsorption/desorption and diffusion equilibrium is reached. Adsorption/desorption process in kerogen causes the delay effect. The delay effect here means that actual gas transport process in organic-rich shale may takes more time than that of instantaneous adsorption or solo diffusion model. It is verified that the dynamic gas production processes calculated from the DAD model distinguishes from that of the instantaneous adsorption−diffusion (IAD) model or gas diffusion

=0 r=0

−

E

DOI: 10.1021/acs.energyfuels.6b01447 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels model (without adsorption). By fitting eq 14 with the experimental data based on the least-squares method, the adsorption rate coefficient and apparent diffusion coefficient value for each isothermal and constant external pressure condition can be obtained.

4. RESULTS AND DISCUSSION In this section, validation of the approach is carried out first by comparing the adsorption isotherm with CVM and VVM. Then tests with VVM are conducted to investigate the gas transport stage and the effect of pressure on this dynamic process. And finally numerical simulation was conducted to calculate the adsorption rate coefficient for test results with five different external pressures. 4.1. Validation of the VVM Approach. Experiments were conducted to test the sensitivity and accuracy of the VVM method. Figure 6 shows the test results for methane and helium

Figure 7. Contrast between the adsorbed gas content with two methods at 308 K for 40−50 mesh shale sample.

As can be seen, the trend for two curves is the same and Va for both methods coincide at the same pressure condition. The agreement of the two curves indicates that VVM is accurate in the measurement of dynamic adsorption−diffusion process. Take one point for example. When the pressure is 12 MPa, Va for CVM and VVM could be read from the tendency of the curves, i.e., 1.20 S cm3/g and 1.21 S cm3/g. Thus, it could be concluded that the data from VVM has a high degree of accuracy and sensitivity. 4.2. Gas Transport in Shale. N2 and CH4 Dynamic Transport Analysis. Tests for CH4 and He in shale particle were conducted to study the effect of adsorbed gas on the dynamic transport process. Organic matter (kerogen) in shale provides most of the adsorption sites for CH4. Unlike CH4, gas He is considered to be nonadsorption gas, or its adsorption amount is negligible in shale. Thus, for test with He, only free gas involves in the gas transport process. If the production of CH4 is much higher than that of He, both free gas and adsorbed gas should be considered during CH4 transport in shale. Figure 8 shows the test results of production process for CH4 and He with 10−20 mesh shale sample. The initial saturation pressure in the pore of the sample particle is 3.4 MPa and the external pressure is 0.1 MPa. The experimental temperature is 308 K. In Figure 8a, it is apparent that the total production volume of CH4 is much higher than that of He. The total volume V for CH4 and He are 2.60 S cm3/g and 1.33 S cm3/g, respectively. We assume that the amount of free gas (methane) inside the sample is nearly equal to the amount of helium gas. Thus, the produced volume of free gas for unit mass shale particles under standard condition is 1.33 S cm3/g. Therefore, the volume of produced adsorbed gas for unit mass shale sample under standard condition is equal to 1.27 S cm3/g (2.60 − 1.33 = 1.27). Since the produced gas volume is propotional to the gas concentration, the ratio R in eq 4 can be calculated to be 0.995. Another information concluded from Figure 8a is that the equilibrium time for CH4 and He is significantly different. The gas production time for CH4 (420 min) is much longer than that of He (45 min). Higher amount of production and longer period of the process of CH4 are due to the fact that large amount of CH4 in shale is stored as the adsorbed gas. To further investigate the effect of adsorbed gas on the dynamic gas production process, efforts have been made to distinguish different production stages. Figure 8b shows the semilog plot of the experimental data presented in Figure 8a. It depicts that the CH4 production process for the shale particle exists two distinct stages, i.e., the early stage dominated by free gas and the late

Figure 6. Test results with methane and helium at 308 K and 3.31 MPa for 40−50 mesh shale particles.

on 40−50 mesh shale particle. The external pressure was 3.31 MPa. The increment of ΔV for CH4 was more obvious at first, and then decreased to zero as time elapsed. The test with helium had a comparatively low equilibrium value of ΔV. The amount of He filling into the pore space is as low as 0.03 S cm3/g. However, the equilibrium volume for CH4 is 0.40 S cm3/g, 13 times more than that of He. Helium is considered to be none-adsorbing, or its adsorption amount is negligible in shale, whereas CH4 adsorbs substantially on the inner surfaces of shale.30 That difference in adsorption phenomena causes the gas transport dynamic to vary significantly between CH4 and He. As to the CH4-shale system, free gas and adsorbed gas coexist and interact, hence the cumulative volume of gas transported and the time for reaching equilibrium of CH4 are much greater than those of He-shale system. The difference could be detected and recorded with this approach. There is no experimental data available in the literature for us to validate VVM method. In order to confirm the accuracy of the results obtained from VVM, the 40−50 mesh shale sample was once again employed and tested at 308 K with both CVM and VVM. With CVM, the adsorption isotherm curve can be obtained. With VVM, in addition to the isotherm curve, the dynamic gas transport process is recorded. Since the adsorbed gas content can be obtained by both CVM and VVM, when the measurement of the adsorbed gas amount is accurate, the VVM can be proved to be accurate. The comparison of the adsorption isotherm is presented in Figure 7. Va denotes the adsorbed gas content per gram of the shale sample at standard condition. F

DOI: 10.1021/acs.energyfuels.6b01447 Energy Fuels XXXX, XXX, XXX−XXX

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Figure 8. Comparison of dynamic transport processes of CH4 and He in shale particle (10−20 mesh) with VVM at 3.4 MPa and 308 K. (a) Time in linear scale, and (b) time in log scale.

stage by adsorbed gas. This conclusion coincides with the observations from Wang et al.33 To the triangle-labeled curve for CH4, it increases linearly until about 11 min and then experiences a long period of growth with the production rate declining. One thing needs to be noted is that, the transported gas volume at 11 min for CH4 is 1.39 S cm3/g. This value is quite close to the total transport volume for He, 1.33 S cm3/g. This suggests that the first 11 min for CH4 transport in shale is dominated by free gas flow. Some but not very much contribution of the adsorbed gas during this period leads to the CH4 production slightly higher than that of He. The experimental results mentioned above could roughly determine the point between the free gas dominant stage and the adsorbed gas domiant stage. In order to further precisely determine the point mathematically, the DAD model in Section 3 is used to describe the production process of methane (Figure 8a). First, the experimental data (Figure 8a) obtained from VVM for methane production process is matched with eq 14 to obtain the physical parameters: adsorption rate coefficient (λ) and apparent diffusion coefficient (D). The fitting technique used in this article is multilevel single-linkage (MLSL) global optimization method.33,40 Because the dependent variable in the DAD model (Section 3) is concentration instead of pressure, the initial saturation pressure in the pores of the shale samples and the external pressure should be transformed to concentration through the equation of state of real gas: PV = znRT ⇒ c =

P zRT

Figure 9. Comparison of experimental results and fitting curve for the transport process of CH4 at 3.4 MPa and 308 K.

shale particle, the amount of free gas and adsorbed gas inside the particle can be obtained. Then, the relationship between the amount of produced gas (free gas and adsorbed gas) and time can be further given by subtracting the amount of gas remaining inside the particle from the initial gas (free gas and adsorbed gas). The analysis result is shown in Figure 10a. It depicts that at the beginning of the production process, the production rate of the free gas is much higher than that of adsorbed gas. Compared with the production of the adsorbed gas, the decay of the free gas is very fast and after a certain time point, the rate of adsorbed gas exceeds that of free gas. We further present the derivative curves (Figure 10b) of the accumulated free gas and adsorbed gas to show the production rates of free gas and desorbed gas as a function of time. The critical point denotes the point that free gas and desorbed gas produce at the same rate. Thus, after the critical point, the production of the adsorbed gas becomes dominant. As can be seen from Figure 10b that, there is a critical point at 11.3 min, at which point the production rates for free gas and adsorbed gas are the same. After this point, the gas production is mostly from the contribution of the adsorbed gas. This confirms the previous assumption and the aforementioned experimental results. That is, the free gas is produced very quickly at the early period (from the beginning to the ccritical point) of the gas production process, while the adsorbed gas is produced at a relatively low rate but lasts for a long period at the late stage (from the critical point to the end of the process). Based on the intersection point of the two curves in Figure 10b, the gas transport process can be divided into two stages: the early stage dominated by free gas production and the late stage dominated

(16)

where P is the gas pressure, Pa; V is the volume of gas, m3; z is the gas compressibility factor; T is the absolute temperature, K; and R is the gas constant, J/(mol·K). The gas compressibility factor can be obtained through fitting the Carnahan−Starling equation of state to the Standing and Katz Z-factor correlation using a simple matlab procedure.45 Comparison between the experimental data and the fitting curve is made in Figure 9. It is shown that the DAD model can match the experimental data very well, indicating that DAD model is reliable to describe the gas production process. The physical parameters λ and D obtained from DAD with MLSL method are 3.61 × 10−4 s−1 and 8.16 × 10−11 m2/s, respectively. After physical parameters (λ and D) and experimental conditions are determined, the distribution of free gas concentration cf (eq 10) and equivalent surface concentration ca (eq 11) of DAD model become just the function of time (t) and the distance to the center of the shale particle (r). Integrating cf and ca from the outside boundary to the center of the G

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Figure 10. Simulated results of (a) the cumulative productions and (b) the production rates of free gas and adsorbed gas for the transport process of CH4 at 3.4 MPa and 308 K.

not been investigated. To fill in this gap, our apparatus was designed to determine the dynamic production behavior under constant production pressure, related to the field gas production process. (2) Although the amount and transport process of free gas can be obtained by tests with helium, it is nearly impossible to simulate the exact dynamic process of desorption. (3) Once adsorbed gas is released into the free gas phase, it can no longer be differentiated from the original-free-gas, without knowing whether the adsorbed gas is produced or still in the reservoir. For shale gas content determination, the gas released includes the adsorbed gas, as well as free gas in the matrix.26 It can be concluded that the early stage gas production is mainly from the diffusion of original free gas. As time elapsed, the desorbed gas becomes the dominant contributor to the production stream. Effect of Pressure on Gas Transport Dynamic. Take the gas production process as an example to show the pressure distribution in shale particle as depicted in Figure 12. The shale

by the adsorbed gas production. It is also shown that, through the analysis of the production rate changes, the DAD model can be used to determine the point between the free gas dominant stage and the adsorbed gas dominant stage. There is an interplay between the free gas diffusion and the adsorbed gas desorption for the gas transport process in shale. The changes in free gas and adsorbed gas contribution to total gas production lead to two distinguished stages during the CH4 transport process. Figure 11 depicts the two stages of gas

Figure 11. Gas transport during gas production process in shale. (a) Initial state; (b) free gas transport; (c) free gas and desorbed gas transport; and (d) equilibrating state.

transport process from shale particles. In the initial state (Figure 11a), free gas and adsorbed gas coexist in shale. However, due to the pressure drop at the boundary, transport of free gas occurs near the edge of the particle first (Figure 11b). This is because the pressure drop near this area is more obvious. Thus, free gas flow dominates the production of the shale reservoir from this state. As a result of the pressure reduction due to the production of the original-free-gas, the originally adsorbed gas desorbs, supplementing free gas, and begins to contribute to the overall gas production stream. This increases the amount of “free gas” and, thus, enhances the diffusion process but with different production rate. As the pressure change passes on to the inner part of the matrix, more and more adsorbed gas desorbs and flows out of the matrix (Figure 11c). The process continues until the pressure inside the shale particle becomes equal to the external pressure, Pe (Figure 11d). It should be noted that (1) the amounts of adsorbed gas could be obtained from the adsorption isotherm thus the free gas and adsorbed gas for each gas transport process could be obtained. However, the dynamic process at constant external pressure has

Figure 12. Pressure distribution inside the shale sphere initially saturated with methane at Pi when the pressure at the sphere boundary is suddenly dropped to Pe.

sphere is initially saturated with methane at the initial pressure Pi. To start the gas production process, the cell pressure is suddenly dropped to a lower pressure Pe at t = to. When the pressure at the surface of the sphere or the external pressure Pe is kept constant, the gas starts to transport from inside the sphere to the external space. To keep the cell pressure or the external pressure constant, the released gas from the shale must be removed out of the cell. As time elapses, the pressure H

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of each single test (Δt) decreases with rising external pressure. One reason for this is that, gas molecular moves faster at higher pore pressure in shale pores.44 Another reason is that the total gas volume (ΔV) for higher pressure is comparatively lower, droping from 0.52 S cm3/g at 4.24 MPa to 0.11 S cm3/g at 17.09 MPa. Greater rate and lesser total volume make the process duration shorter, thus tests under higher external pressure tend to attain equilibrium sooner. 4.3. Model Analyses and Comparisons. The experimental results reveal that adsorbed gas contributes a significant portion of the total volume of gas transportation. eq 14 was used to determine the adsorption rate coefficient based on the multilevel single-linkage method (MLSL) global optimization method.33,40 Figure 14 presents the modeling curve for the experimental results as well as the values of the adsorption rate coefficient of each segment. The values in the legend are the external pressures, under which each part of the test was conducted. The values above each segment are the corresponding adsorption rate coefficient λ in 10−3 s−1. The adsorption rate coefficient was calculated under the average pressure (p)̅ of the initial pressure (pi) in the pore of the crushed sample and the external pressure (pe) of each part of the test. From Figure 14, it is clear that the calculated results from the mathematical model (eq 14) can match the experimental data very well. Table 3 shows the adsorption rate coefficient λ and gas diffusion coefficient D at different average pressure p̅. As is shown, the gas diffusion coefficient is in the order of magnitude of 10−12 and increases with average pressure. Existence of kerogen in shale could extend or delay the gas transport process.40 The delay effect here means that actual gas transport process in organic-rich shale may take more time than the time predicted from traditional mathematical model assuming instantaneous adsorption/desorption. This delay effect will be addressed by the adsorption rate coefficient (λ) in this article. It can be derived from the dynamic gas adsorption/desorption equation (eq 3) that smaller adsorption rate coefficient corresponds to smaller adsorption/desorption rate. And smaller adsorption/ desorption rate means that it takes longer time for the gas to attain adsorption/desorption equilibrium. From Table 3, it can be seen that when the external pressure decreases from 17.09 to 4.24 MPa, the adsorption rate coefficient decreases from 4.43 × 10−3 s−1 to 1.19 × 10−3 s−1. It means that delay effect becomes more obvious under lower pressures. Another observation is that, λ are approximately the same for gas storage process and for gas production process under the same average pressure and temperature. Previous experimental study showed that the size of the testing sample has no effect

change will spread gradually into the sphere and the pressure within the sphere varies between Pi and Pe as schematically shown in Figure 12 (from t1 to t3). During this test, the differential between the inner and external pressures decreases. Eventually, a uniform pressure distribution inside the sphere will be attained at time t = teq, and an equilibrium state at Pe is obtained. Dynamic gas transport measurements were conducted for 40−50 mesh shale sample, under different external pressures. The results are summarized in Table 2 and plotted in Figure 13. Table 2. Summary of Tests for 40−50 Mesh Shale Sample at 308 K P (MPa)

Veq (S cm3/g)

gas storage process 4.24 0.52 6.95 0.86 10.53 1.15 13.82 1.36 17.09 1.47 gas production process 0.14 0.02 3.49 0.55 7.00 0.91 10.35 1.18 13.74 1.40

Δt (min)

ΔV (S cm3/g)

B (%)

69 42 33 24 12

0.52 0.34 0.29 0.21 0.11

59.7 63.5 67.0 70.9 72.4

142 65 44 35 11

0.53 0.36 0.27 0.22 0.07

58.7 60.1 63.2 67.3 71.0

For the parameters in Table 2, P is the constant external pressure in MPa; Veq is the total amount of gas stored in shale at an equilibrium state in S cm3/g; Δt is the time consumed for obtaining the equilibrium state of each single test in min; ΔV is the change of gas volume caused by gas transport for each test in S cm3/g; and B is the contribution of the free gas for each process. Figure 13 shows test results for 40−50 mesh shale at five external pressure conditions, among which Figure 13a shows the gas storage process, V is the accumulating gas amount adsorbed by and diffused into the shale sample by time t in S cm3/g; and Figure 13b shows the gas production process. It can be seen that pressure influences the dynamic process, not only on the gas amount transported during the process, but also the time needed for obtaining the equilibrium. For the same differential pressure, when the external pressure increases, the change of gas volume caused by gas transport (ΔV) decreases. In this study, ΔV decreased from 0.52 at 4.42 MPa to 0.11 at 17.09 MPa. This could be explained by the fact that the increasing rate of adsorption isotherm decreases with rising pressure. Besides, the time consumed to achieve the equilibrium state

Figure 13. Curves for dynamic gas transport process in 40−50 mesh shale at 305 K. (a) Gas storage process; (b) gas production process. I

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Figure 14. Comparison curves between mathematical model fitting and VVM results. (a) Gas production process; (b) gas production process.

Table 3. Relationship between Average Pressure and Adsorption Rate Coefficient p̅ (MPa)

λ* (10−3s−1)a

D (10−12m2/s)

p̅ (MPa)

λ# (10−3s−1)a

D (10−12m2/s)

2.12 5.60 8.74 12.18 15.46

0.66 1.07 1.52 1.97 2.46

2.86 6.53 9.04 11.50 14.00

1.82 5.25 8.68 12.05 15.42

0.67 1.07 1.52 1.97 2.45

2.50 6.15 9.01 11.41 13.98

a

λ* is for the gas storage process and λ# for gas production process.

on the adsorption rate coefficient.33 This suggests that λ is a parameter governed by the average pressure and temperature, not by the dimension of the tested shale sample. Therefore, for a specific shale reservoir, the λ obtained from its shale sample can be applied to the large scale reservoir under the same pressure and temperature conditions to predict the gas transport process for reservoir evaluation and production prediction.

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• For the same differential pressure, the gas transport process in shale reaches equilibrium faster and transports less gas at a higher external pressure than at a lower external pressure. The contribution of free gas to the total gas transported in shale increases with external pressure. • The dynamic adsorption-diffusion (DAD) model considers the effect of organic matter on the dynamic gas transport process. A smaller adsorption rate coefficient (λ) means a greater delay effect on the gas transport process due to the adsorption/desorption-diffusion of gas in the organic matter in shale. • The mathematical model presented in this article can not only be applied in characterizing gas transport under laboratory conditions, but also can be applied to gas transport in shale gas reservoirs under reservoir conditions.

APPENDIX Derivation of correction factor for bulk gas transport in nanopores of shale gas reservoirs

5. CONCLUSIONS A coupled experimental and mathematical study for investigating the dynamic gas transport process has been performed. A new experimental methodVVMis designed and applied to measure the gas adsorption/desorption-diffusion process in shale. The main idea of VVM is measuring the gas volume transported into or out of shale sample by changing the system volume and keeping the external pressure of the shale particles constant. The sensitivity and accuracy of the approach were confirmed by comparing the test results with constant-volume volumetric method. The effect of pressure on the dynamic gas transport process has been investigated and the two stages of the gas transport process in shale have been identified. The major conclusions of this study are • The gas volume change over time under constant external pressure due to gas transport through pores in shale is recorded and analyzed. The information obtained from VVM tests includes: dynamic gas transport process, contributions of free gas and adsorbed gas to the total gas production, and the apparent diffusion coefficient and the adsorption rate coefficient. • Dynamic gas transport process obtained from VVM could be divided into two stages: the early stage dominated by the diffusion of free gas in the pores and the late stage dominated by the adsorption−diffusion for gas storage process or the desorption−diffusion for gas production process.

AUTHOR INFORMATION

Corresponding Author

*Tel: +1 403 210 7642; Fax: +1 403 284 4852; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from the National Basic Research Program of China (No. 2014CB239103) and the Natural Science Foundation of China (No. 51274225, 51204197) are greatly appreciated.

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NOMENCLATURE PCVME=External pressure for CVM, MPa PCVMI=Inner pressure for CVM, MPa PVVMI=Inner pressure for VVM, MPa PVVME=External pressure for VVM, MPa ΔV=Gas volume change under standard conditions, S cm3 P=External pressure, MPa V=Gas volume, cm3 T=Test temperature, K z=Compressibility factor, dimensionless P0=Equals 0.1 MPa T0=Equals 273 K z0=Equals 1 DOI: 10.1021/acs.energyfuels.6b01447 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels cf=Free gas concentration in the pore space, mol/cm3 D=Apparent diffusion coefficient, m2/s r=Distance to the particle center, cm ca=Equivalent surface concentration, mol/cm3 t=Time, s λ=Adsorption rate coefficient, s−1 cfeq=Equilibrium concentration of free methane gas in the pore space, mol/m3 caeq=Equilibrium concentration of the adsorbed gas concentration on the surface, mol/m3 cfe=Gas concentration in the external boundary, mol/m3 cfi=Initial gas concentration in the pore, mol/m3 r0=Radius of the shale particle, m Vm=Mole volume of gas, 22400 cm3/mol V(t)=Amount of gas transported at time t, cm3/g Veq=Total amount of gas transported, cm3/g Pi=Initial pressure, MPa Pe=External pressure, MPa Veq=Total amount of gas stored in shale at an equilibrium state, S cm3/g Δt=Time consumed for obtaining the equilibrium state of each single test, min B=Contribution of the free gas, %

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