General Gas Permeability Model for Porous Media: Bridging the Gaps

Article pubs.acs.org/EF

General Gas Permeability Model for Porous Media: Bridging the Gaps Between Conventional and Unconventional Natural Gas Reservoirs Peng Cao,*,† Jishan Liu,†,‡ and Yee-Kwong Leong† †

School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia ‡ State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, PR China ABSTRACT: Many field observations have indicated that permeabilities of both conventional and unconventional gas reservoirs are not constant when gas pressure drops. For conventional reservoirs, permeability will decrease while for unconventional gas rocks, the apparent permeability may increase as gas pressure decreases to a lower magnitude. Evolution trends of permeability for different natural gas reservoirs are distinct. These differences are observed by laboratory experiments of sandstones, coals, or shales. In this study, we present a general permeability model to bridge the gaps between conventional and unconventional gas reservoirs. This model coupled three critical factors namely effective stress, adsorption, and flow regimes to reflect dynamic performances of permeability. On the basis of specific reservoirs properties, the model degenerates into four reduced types. The first reduced model is applicable for reservoirs with lower adsorption capacity. The second reduced model is adopted by unconventional reservoirs like coal seams when the intrinsic permeability is big and adsorption capacity is high. For the third reduced model, effective stress is the dominating factor for permeability evolution, which means that it is applicable for conventional reservoirs like sandstones. Unconventional gas reservoirs with low adsorption capacity like gas shales can apply the fourth reduced model because the flow regimes dominate the evolution. These reduced models are verified against the experimental data. Results show that effective stress is the main reason for the change of permeability for conventional gas reservoirs. Both effective stress and flow regimes together determine the apparent permeability of unconventional gas reservoirs. The impact of adsorption on permeability is relatively small. Permeability evolution trends can be classified into different zones for conventional and unconventional gas reservoirs. When the gas is depleted from reservoirs, the gas permeability has two bounds. For the upper bound, permeability is only affected by flow regimes and the apparent permeability will increase when gas pressure drops. For the lower bound, permeability is only affected by effective stress and the apparent permeability will decrease when the gas is depleted from the reservoirs. observations support this idea.8−11 These models have analyzed the importance of stress on permeability for tight rock especially during the gas depletion process as shown in Figure 1. According to the poroelasticity theory, when the effective stress increases, the porosity will be reduced because of the compaction of pores and grains. Variation of effective stress can affect the average pore radius and intrinsic permeability for gas reservoirs. However, for gas permeability of shales, a lot of experiments show that permeability will undergo an increasing trend when the gas pressure drops. For the same conditions, the gas permeability of conventional gas reservoirs should decrease. Thus, there is a gap between the dynamic behaviors for gas permeability between conventional gas rocks and unconventional gas rocks. For unconventional gas reservoirs, the evolution of permeability is more complex. The models above did not consider the effects of flow regime and adsorption on permeability, which will underestimate the real permeability of ultralow permeability sediments. Flow regimes will play an important role in the apparent permeability for unconventional

1. INTRODUCTION Permeability is a critical property of natural gas production from both conventional and unconventional gas reservoirs. Unconventional natural gas reservoirs are fine grained compacted sediments with very low permeability,1 and they are porous media with permeability below 1 millidarcy (mD). Most of the pores in unconventional gas reservoirs like shales and coals have diameters between 4 and 200 nm. The permeabilities of conventional gas reservoirs like sandstones are greater than 1 mD. Some scholars have pointed out that it is the matrix properties that dominate the gas production performance over longer periods of time.2 Thus, the evolution of permeability for conventional and unconventional reservoirs is very significant. In recent decades, many scholars have investigated the evolution of permeability for natural gas reservoirs when gas depletes from sandstones or shales. Some researchers observed that the permeability of the core sample would be changed when gas pressure or confinement pressure changed in the experiments.2−4 Generally, the gas pressure will decrease after production and this will change the effective stress. Many scholars have conducted experiments to study the effects of stress on permeability.5−7 It is well understood that gas permeability depends on effective stress and many experimental © XXXX American Chemical Society

Received: March 23, 2016 Revised: May 24, 2016

A

DOI: 10.1021/acs.energyfuels.6b00683 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels

Figure 1. Schematic of impacts of stress, flow regime, and adsorption on the apparent permeability.

gas reservoirs. In terms of the flow regime, it is related to one dimensionless parameter Knudsen number. For the traditional Darcy law, it is applicable that permeability is big and the Knudsen number is less than 0.01. But this condition is impossible for unconventional natural gas reservoirs. For example, for gas shale reservoirs, the pores are concentrated within several nanometers to hundreds nanometers, which means the Knudsen number is over 0.01 and the flow regime belongs to the slip flow. It is acceptable that permeability of gas flow is bigger than that of liquid flow in the same porous media especially for tight porous media.12−14 The apparent permeability is bigger than the intrinsic permeability for porous media because of slippage effect (flow regimes). There is a close relationship between intrinsic permeability and apparent permeability.15,16 Many scholars established apparent permeability models to reflect this effect by adding slippage factor to Darcy’s equation, which called Klinkenberg-corrected permeability model.3,17−20 Besides, many researchers thought flow regime like Knudsen diffusion will play a more important role in apparent permeability and adopted Knudsen number to represent the flow regime. Based on this theory, a lot of Knudsen numbercorrected permeability models have been developed.16,21 Additionally, some scholars have proposed several apparent permeability models based on the idea that slip flow and Knudsen diffusion together lead to the apparent permeability for shales, which means slip flow can appear independently without considering the diffusion.22−24 These models have good agreements with experiments for porous media with nanopores or microchannels under different fluids conditions like helium, methane, water, and decane. They focus on the variation of apparent permeability when using different fluids under the same confining pressure conditions.25 Based on these studies, it is easy to understand the impacts of flow regimes on the apparent permeability compared to the intrinsic or absolute permeability. However, for these permeability models, there is a common assumption or experimental condition that the stress for shale solid is

unchanged, which is impossible in the real process of gas production. This assumption is another gap between conventional gas reservoirs and unconventional gas reservoirs. On the other hand, adsorption phenomenon has two effects on gas reservoirs. First, the adsorption induced swelling can change the effective stress. When adsorbed gas injects into porous media, it will induce the swelling of the matrix and both the volumes of pores and grains will be changed. If it is under the free swelling condition, the effective stress is a constant and intrinsic permeability is unchangeable. However, if the boundary is fixed, the swelling will increase the stress and compact the pores, which leads to increase of effective stress and decrease of intrinsic permeability. Some scholars presented different models to reflect the impact of adsorption on permeability.26−28 Second, the adsorption will occupy the space of the pores, which can decrease the permeability. The adsorption on the surface of pores can reduce the porosity within the connected pores network, which means that the average pore radius should deduct the adsorption layer thickness29 as shown in Figure 2. When the pores of gas reservoirs are very small (less than 50 nm), the thickness of the adsorption layer should be taken into account. Overall, there are three main factors namely effective stress, adsorption and flow regime that affecting permeability of gas reservoirs as shown in Figure 1. Until now, all of these physical

Figure 2. Difference between average pore radius and effective pore radius from adsorption effect. B


Article

Energy & Fuels ⎞ ⎛ p σ ̅ − p = −K ⎜εv + − εs⎟ Ks ⎠ ⎝

processes have been investigated individually, but little work has been done to study the dynamic interactions of apparent permeability with respect to effective stress, adsorption, and flow regime. But for one specific gas reservoir, the importance for permeability from these factors is different. We present a general permeability model to contain all the three factors, and this model will bridge the gaps between conventional and unconventional gas reservoirs. In summary, the effective stress can change the intrinsic permeability and average pore radius, and the flow regimes can change the apparent permeability. The adsorption effect can change the intrinsic permeability and porosity.4,15,30−32

The porosity is calculated as a function of shale mechanical properties such as Young’s modulus and pore pressure. The following formulation is widely applied to describe the intrinsic permeability change with respect to porosity change in gas reservoirs like shale and coal.27

⎛ ϕ ⎞3 k∞ = ⎜⎜ ⎟⎟ k∞ 0 ⎝ ϕ0 ⎠

⎛ 1 ε 1 1 ⎞ α ⎟σ δ + pδij + s δij σij − ⎜ − kk ij ⎝ 6G 2G 9K ⎠ 3K 3

⎞ ⎛⎛ 1 1 ⎞⎟ ⎟ r = r0exp⎜⎜⎜⎜ − [ σ − σ − ( p − p )] 0̅ ̅ 0 ⎟ K p ⎟⎠ ⎠ ⎝⎝ K

(1)

When dads is the equivalent thickness of the adsorption layer, which occupies the porosity and reduces the effective pore radius; Dgas is the diameter of the adsorption gas molecule. Thus, the effective pore radius for gas flow will be re = r − dads

εLp PL + p

According to the study of other researchers, the effective intrinsic permeability can be described as

(2)

k∞ e =

re 2 ϕ 8 τ

(10)

2.3. Knudsen Number and Flow Regimes. The microporous fabric with pores or pore-throats in the nanometer size range of shale causes coexisting gas flow regimes (continuum flow, slip flow, transitional flow and Knudsen flow). In most cases, the gas permeability is higher than that of liquid permeability. The effect of gas slippage is more pronounced for gas reservoirs when the Knudsen number is over 0.1. The Knudsen number, a dimensionless parameter, is commonly used to classify flow regime in small pores within porous media. It is defined as the ratio of molecular mean free path, λ (nm) with a characteristic length (effective pore radius) re and is given as

(3)

where εL is a constant representing the maximum volumetric strain for porous media and PL is the Langmuir pressure constant for porous media. According to the poroelasticity theory,28,30,33 the dynamic porosity for porous media can be expressed as ⎛⎛ ⎞ ϕ 1 1 ⎞⎟ ⎟ = exp⎜⎜⎜⎜ − σ − σ − − [ ( p p )] 0̅ ⎟ ̅ 0 ⎟ ϕ0 K K ⎝ ⎠ p ⎝ ⎠

(9) 3

where εν = ε11 + ε22 + ε33 denotes the volumetric strain of porous media and σ̅ = −σkk/3 represents the mean compressive stress. The compressive pressure σij in this domain is negative. The sorption-induced volumetric strain εs is fitted onto Langmuir-type curve and has been verified through experiments.34 A Langmuir-type equation is used to calculate this volumetric strain, defined as εs =

(7)

The thickness of the adsorption layer can be defined as p dads = Dgas p + PL (8)

where σij represents the component of the total stress tensor and εij denotes the component of total strain tensor, G = E/2(1 + ν), K = E/3(1 − 2ν), α = 1 − K/Ks, and σkk = σ11 + σ22 + σ33, K is the bulk modulus of shale, Ks is the bulk modulus of shale grains, G is the shear modulus of shale, E is the Young’s modulus of shale rock, ν denotes the Possion’s ratio, α is the Biot coefficient, δij is the Kronecker delta, and p is the gas pressure within the matrix. From eq 1, we obtain 1 εv = − (σ ̅ − αp) + εs K

(6)

where k∞ denotes the intrinsic permeability and the subscript 0 refers to the initial value. It is noticeable that there are many other intrinsic permeability and porosity relations with different forms. But this model is more applicable for gas reservoirs. 2.2. Formulation for Effects of Adsorption. The effect of adsorption on the intrinsic permeability is reflected on the constitutive equation of the matrix deformation. Also, there is another impact that adsorption layer thickness will reduce the average pore radius and intrinsic permeability. To some extent, the average pore radius r should be calculated based on the geometry relationship. Based on eq 4, it yields that

2. GENERAL GAS PERMEABILITY MODEL 2.1. Formulation for Intrinsic Permeability. All the following formulations are derived from the assumptions: (a) Gas contained within the pores of porous media is ideal. (b) Porous media is saturated by gas. (c) Porous media is a homogeneous, isotropic, and elastic continuum. For a homogeneous and isotropic porous medium, the gas sorption-induced strain εs is presumed to result in volumetric strain only. The effects for the three normal components of strain are the same. According to poroelasticity theory14,33 and by making an analogy between thermal contraction and matrix shrinkage, the constitutive relation for porous media becomes εij =

(5)

Kn =

λ re

(11)

The mean-free-path of molecules λ is expressed as

(4)

λ=

where the subscript 0 refers to the initial state, and Kp denotes bulk modulus of pores. According to eq 2, we obtain C

KBT 2 πσ ̃ 2p

(12) DOI: 10.1021/acs.energyfuels.6b00683 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels where KB is the Boltzmann constant, T is the temperature (K) of gas, and σ̃ is collision diameter (m). From eq 10, the characteristic length re can be obtained as k∞ eτh ϕ

re = 2 2

Substituting eqs 14 and 16 into eq 15, the rigorous mathematical formulation for the general gas permeability model for porous media is given as kapp =

(13)

Kn =

ϕ

kapp =

2 πσ ̃ 2p 2 2k∞ eτh

(14)

Flow regimes can be classified into four types based on the Knudsen number. The four types are viscous (continuum) flow (Kn < 0.01), slip flow (0.01 < Kn < 0.1), transition flow (0.1 < Kn < 10), and Knudsen’s (free molecular) flow (Kn > 10). 2.4. Apparent Permeability for Porous Media. Because of complex flow mechanisms, the apparent permeability of porous media deviates significantly from the intrinsic permeability. Therefore, we developed an intrinsic permeability model as eq 10 for porous media. The value of the intrinsic permeability k∞ does not depend on the type of fluid used or the flow conditions in the experiments. It is just a property of the porous medium. However, the permeabilities measured by using gas such as He, CH4, and CO2 through ultralow permeability rock samples are different from permeabilities measured by liquids like water because of the effect of flow regime. The apparent permeability is a function of both intrinsic permeability and flow regime. The apparent permeability model for gas reservoirs is given by the following expression. We call it the BK (Beskok−Karniadakis) model, which is based on a unified Hagen−Poiseuille-type formulation.3,18,21 kapp = k∞ ef (Kn)

(16)

According to previous studies, the slip coefficient b is an empirical parameter. Beskok and Karniadakis suggested its value as b = −1 and that it is independent of the type of gas. The value of the dimensionless rarefaction coefficient ζ varies: 0 < ζ < ζ0 for 0 < Kn < ∞ where ζ0 is an asymptotic limit value.21 A correlation is presented by Civan et al.3

ζ0 1+

A Kn B

(17)

where A = 0.17, B = 0.4348, and ζ0 = 1.358. As the bulk modulus K is commonly several orders of magnitude larger than the pore volume modulus Kp, we obtain 1 1 1 1 − K ≈ − K , then define the compressibility ct = K , which K p

p

p

is a real property for rocks through experimental data. However, it is accepted that the compressibility is not constant and McKee’s model 35 is adopted in this work to presentct = ct0

1 − e−α0(Δσ̅ −Δp) , α0(Δσ ̅ − Δp)

(19)

3. MODEL ANALYSES AND VERIFICATIONS 3.1. Four Types of Reduced Models. The general apparent permeability model for porous media contains the three critical factors namely effective stress, adsorption and flow regime. This model is applicable for different testing conditions like changing the fluids or changing the fluid pressure. It is difficult to verify this model when considering the three factors together because the interactions between the factors are complex and the parameters are hard to measure in the experiments. However, in most cases, this complex model can be degenerated to simple models when adopting the specific conditions. Then we can use these reduced models to analyze the experimental and field data, which can verify the general model. For example, when we use nonadsorbed gas to measure permeability of porous media, the effect of the adsorption can be neglected and the model can remove the adsorption term. Then we can compare the experimental data with the results of this reduced model. Thus, based on reservoirs properties, the model can be reduced into four types. (1) First reduced model: when we adopt nonadsorbed gas like He to analyze the evolution of permeability or when porous media do not contain the adsorbed materials, the model just considers the effects of effective stress and flow regime. (2) Second reduced model: when the average pore radius is big and the Knudsen number is less than 0.01, the flow regime belongs to conventional flow and the model ignores the flow regime term. (3) Third reduced model: when the Knudsen number is small and the nonadsorbed gas is used to measure permeability, the flow regime and adsorption can be neglected together and the final model just is the function of the effective stress. (4) Fourth reduced model: in most cases the effective stress has great impact on the evolution of permeability but for the ultralow permeability porous media and the flow regime plays a more significant role in the variation of permeability. The change of intrinsic permeability is very small and we only consider the effect of flow regime. 3.2. Verification for the First Reduced Model. As discussed above, in some cases the adsorption effect is small and adsorption will be ignored in the general model. There two main cases for this simplified model. One is that the

where f(Kn) is slippage incremental factor as a function of the Knudsen number Kn, the dimensionless rarefaction coefficient ζ, and the slip coefficient b. It is defined as

ζ=

(r0exp( −ct[σ ̅ − σ0̅ − (p − p0 )]) − dads)2 8 ϕ0exp( −ct[σ ̅ − σ0̅ − (p − p0 )]) (1 + ζKn) τ ⎛ 4Kn ⎞ ⎜1 + ⎟ ⎝ 1 + Kn ⎠

This formulation contains the effects of mechanical deformations (effective stress), adsorption, and flow regime on the gas flow within reservoirs. This general gas permeability model can be applicable for both conventional and unconventional natural gas reservoirs.

(15)

⎛ 4Kn ⎞⎟ f (Kn) = (1 + ζKn)⎜1 + ⎝ 1 − bKn ⎠

(18)

where subscript 0 refers to the initial value for shale rocks. Then eq 18 becomes

where τh is the tortuosity of porous media. Therefore, eq 11 becomes KBT

⎛ (r − dads)2 ϕ 4Kn ⎞ ⎟ (1 + ζKn)⎜1 + ⎝ τ 8 1 + Kn ⎠

and this dynamic model has been

verified by many researchers in recent years.27 D


Article

Energy & Fuels

data fitting for the model, we could obtain the initial compressibility ct0 of sample 31A is 0.055 MPa−1 and the decline rate α0 for sample 31A is 0.02 MPa−1. The decane permeability is the intrinsic permeability for the shale sample, which ignores the effect of flow regimes. For the same shale rock 31A, helium permeabilities were measured at the gas pressure of 6.89 MPa and the results showed that permeability were nearly 1 order of magnitude higher than permeability of liquid oil under the same stress condition. The reason for the difference between oil permeability and gas permeability is the effect of flow regime, which indicates the coupled impact of effective stress and flow regimes on the unconventional gas reservoirs. The helium permeability declined from 3500 to 780 nanodarcy (nD) when the effective stress increased from 12 to 60 MPa as shown in Figure 3. There are strong mechanical, chemical and physical interactions between the fluid molecules and pores surface. In summary, both mechanical deformation and flow regime have significant impacts on the apparent permeability and the reduced model has been well validated by Cui’s data.4 This reduced model is applicable for gas reservoirs without the adsorption phenomenon. 3.3. Verification for Second Reduced Model. For some gas reservoirs, the pores are big and the corresponding Knudsen number is less than 0.01 and traditional Darcy law is applicable. The impact of the flow regime can be removed from the general model and this reduced model is acceptable for conventional gas reservoirs like sandstones. The second reduced model without the effect of the flow regime can be written as

nonadsorbed gases like He and N2 are used to analyze the evolution of permeability. Another case is that porous media has small capacity to adsorb gas like sandstones without the clay and kerogen. It is noted that many unconventional gas reservoirs have big capacity of the gas like coals, which means this reduced model is not applicable for these reservoirs. The first reduced model without the effect of the adsorption is reorganized as The stress−strain constitutive relationship (eq 1) becomes εij =

⎛ 1 1 1 ⎞ α ⎟σ δ + pδij σij − ⎜ − kk ij ⎝ 6G 2G 9K ⎠ 3K

(21)

Equation 2 becomes 1 εv = − (σ ̅ − αp) K

(22)

Then eq 5 can be rewritten as ⎛ p⎞ σ ̅ − p = −K ⎜εv + ⎟ Ks ⎠ ⎝

(23)

We verify this reduced model based on the experimental data of Cui.4 In the experiments, permeability of shale rocks was measured with helium. The same pressure was used for the axial and radial confining in the core cell.36 The axial confining pressure equaled to the radial confining pressure pr* = pz* = pc*. Therefore, the apparent permeability model for porous media can be simplified as * ) − (p − p )])(1 + ζKn) kapp = k∞ 0exp( − 3ct[(pc* − pc0 0 ⎛ 4Kn ⎞ ⎜1 + ⎟ ⎝ 1 + Kn ⎠

(r exp(−c [σ̅ − σ̅ − (p − p )]) − D = 0

k∞

(24)

where p*c is the mean confining pressure and ct denotes variable compressibility. Based on eq 23, the effective stress can be considered as Δσ̅ − Δp = (pc* − pc0 * ) − (p − p0). The detailed validation process can refer to the previous study.12,30 The data for shale sample no. 31A4 is selected to verify the model. As shown in Figure 3, permeability calculated using the new model shows a decline trend when the effective stress changes from 3.4 to 45 MPa. Based on experimental data, the decane permeability changed from 2 × 10−3 to 4.5 × 10−5 mD when the effective stress increased from 3.4 to 45 MPa. Based on the

t

0

0

p CH4 p + P

8 ϕ0exp( −ct[σ ̅ − σ0̅ − (p − p0 )]) τ

L

2

)

(25)

For the matrix deformation equation, it is the same as the general model and eqs 1, 2, and 11 are applicable in this simplified model. In the past decades, there are different models to describe the evolution of permeability of coal without the consideration of flow regime.26,37,38 Cui and Bustin26 presented the dynamic permeability model including the effect sorption induced volumetric strain as the following. ⎛ ⎛ ⎞ 1 1 ⎞⎟ k∞ = k∞ 0exp⎜⎜3⎜⎜ − [σ ̅ − σ0̅ − (p − p0 )]⎟⎟ ⎟ Kp ⎠ ⎝ ⎝K ⎠

(26)

Compared the reduced model to the CB model (Cui−Bustin model),26 there are the same structure of formulations. If we remove the thickness of the adsorption layer dads and the initial r2 ϕ

intrinsic permeability is defined as k∞ 0 = 80 τ0 , the two expressions will be the same. However, it is noted that the mathematic methods to derive the models are different and the general model comes from the poroelasticity theory. It is pointed that the CB model did not consider the effect of the flow regime. According to the study of Cui and Bustin, the CB model can degenerate to the same structure as PM model (Palmer−Mansoori model) and SD model (Shi−Durucan model)37,38 under the conditions of uniaxial strain and constant loading zone. The degenerated CB model for the coal reservoir under these conditions is

Figure 3. Comparison of the model with test data for permeability of shale with decane and helium. E


Article

Energy & Fuels ⎧ ⎡ (1 + v) 2E k∞ = k∞ 0exp⎨3ct⎢ (p − p0 ) − 9(1 − v) ⎩ ⎣ 3(1 − v) ⎤⎫ (εs − εs0)⎥⎬ ⎦⎭

(27)

The SD model for the coal reservoir under the same conditions is ⎧ 3 ⎡ v E k∞ = k∞ 0exp⎨ ⎢ (p − p0 ) − ⎣ − − v) K 1 v 3(1 ⎩ p ⎪ ⎪

⎤⎫ (εs − εs0)⎥⎬ ⎦⎭ ⎪ ⎪

(28)

The PM model for the coal reservoir is ϕ = ϕ0 +

Figure 4. Comparison of field data for coal seam with the prediction of the reduced model.

(1 − 2v)(1 + v) 2 ⎛ 1 − 2v ⎞⎟ (p − p0 ) − ⎜ E(1 − v) 3⎝ 1 − v ⎠

(εs − εs0)

(29)

adsorption together. Specifically, when porous media has high permeability (over 1 mD) and the mineral compositions do not include the clay and other adsorbed materials like kerogen, the model is applicable for describing the dynamic performance of permeability. Conventional gas reservoirs can adopt this assumption because of high intrinsic permeability and mainly composed of sandstones. Under this condition, the effective stress is the main reason for the variation of permeability and permeability is the function of the effectives stress. The third reduced model without the effects of flow regime and adsorption can be formulated as

This reduced model is similar to all these models and it can be verified by experimental data of coal permeability. The coal is a typical unconventional gas reservoir and permeability is about several millidarcy, which means the effect of flow regime can be ignored. In this study, we select the filed data of coal seams38 to verify this reduced model. Also, we adopt the variable compressibility in the reduced model. 35 The parameters for this reduced model based on data fitting are listed as Table 1. Table 1. Main Parameters for the Second Reduced Model Based on Experimental Data parameters

value

Possion’s ratio Young’s modulus (MPa) Langmuir pressure (MPa) maximum adsorption induced volumetric strain initial compressibility (MPa−1) decline rate of the compressibility (MPa−1) initial reservoir pressure (MPa)

0.25 3585 8.27 0.022 0.12 0.007 5.516

k∞ = k∞ 0exp( −3ct[σ ̅ − σ0̅ − (p − p0 )])

(30)

The stress−strain constitutive relationship becomes εij =

⎛ 1 1 1 ⎞ α ⎟σ δ + pδij σij − ⎜ − kk ij ⎝ ⎠ 2G 6G 9K 3K

Equation 2 becomes 1 εv = − (σ ̅ − αp) K Then eq 5 can be rewritten as

(32)

⎛ p⎞ σ ̅ − p = −K ⎜εv + ⎟ Ks ⎠ ⎝

The predicted results for the coal seam have good agreements with the field data as shown in Figure 4. Both the effective stress and adsorption have impacts on the evolution of permeability of the reservoir. When the gas produced from the well, the reservoir pressure will decrease and the effective stress will increase. If we do not consider the adsorption effect, permeability of the reservoir will decrease because of the matrix deformation. However, the filed data shows that permeability undergoes an increasing trend over the whole period of production. Thus, it is seen that desorption of the gas will induce the matrix shrinkage and permeability will become bigger when the gas pressure declines. In summary, for this special case, the effective stress and adsorption are the key factors that affecting the variation of permeability. The effect of the flow regimes can be ignored because of the small Knudsen number. 3.4. Verification for the Third Reduced Model. In comparison with the first and second reduced models, the third reduced model ignores the effects of flow regime and

(31)

(33)

It is acceptable that when the testing fluid pressure changes or the confinement pressure changes, permeability of porous media will be changed. In the laboratory, the effective stress can be obtained by the confining pressure and the gas pressure.27 Then eq 15 can be rewritten as * ) − (p − p )]) k∞ = k∞ 0exp( −3ct[(pc* − pc0 0

(34)

where pc* is the mean confining pressure and ct denotes compressibility. In eq 15, the effective stress can be considered as Δσ̅ − Δp = (p*c − p*c0) − (p − p0). Also, we adopt the dynamic compressibility model in this reduced model. We select the experimental data of core sample for the coal permeability39 to verify this reduced model. In the experiments, the N2 was used to measure permeability and the effect of adsorption can be neglected because of the very small adsorption capacity for N2. The gas pressure for the Anderson F


Article

Energy & Fuels ⎛ 4Kn ⎞ ⎟ kapp = k∞ 0(1 + ζKn)⎜1 + ⎝ 1 + Kn ⎠

01 is a constant (0.69 MPa), and model parameters are listed in Table 2.

For this reduced model, we assume the intrinsic permeability for porous media is a constant when the effective stress is unchangeable. The Knudsen number will be changed when the gas pressure changes. The apparent permeability is the function of the Knudsen number and we select the experimental data for the matrix permeability of a siltstone sample of the Horseshoe Canyon Formation in south central Alberta.40 The permeability of the sample was measured by helium and methane. We use permeability data of helium to validate the reduced model, which can ignore the effect of adsorption. The results of measurement and prediction of the model are shown in Figure 6. The predicted results of the degenerated model have good

Table 2. Main Parameters for the Third Reduced Model parameters

value

initial compressibility (MPa−1) decline rate of the compressibility (MPa−1) testing gas pressure (MPa)

0.0145 0.36 0.69

(35)

The predicted results of the third reduced model have good agreements with the experimental data39 as shown in Figure 5.

Figure 5. Comparison of experimental data with the prediction results of model for core sample of coal. Figure 6. Comparison of experimental data with the prediction results of model for sample one.

The effective stress is the most important factor for the evolution of permeability. When the gas pressure is a constant and the confining pressure increases, the effective stress for the core sample will increase. Under this condition, the intrinsic permeability will undergo a declining trend as shown in Figure 5. The effects of flow regime and adsorption are ignored because of the small Knudsen number and the low adsorption of N2. In summary, for this special case, the effective stress is the only key factor that affecting the variation of permeability. This reduced model is applicable for gas reservoirs with high permeability and low adsorption capacity like sandstones. 3.5. Verification for the Fourth Reduced Model. As discussed above, the flow regime will play an important role in the evolution of the apparent permeability of unconventional gas reservoirs. In this study, we adopted BK model21 to reflect the impact of the flow regime based on the Knudsen number. The Knudsen number can determine the impact degree on the apparent permeability and it is the function of the gas pressure and average pore radius. The intrinsic permeability is affected by the adsorption and effective stress. They are not constants during the production period. However, for the ultralow permeability reservoirs when the compressibility of the pores is small and the adsorption is weak, the flow regime is the significant factor that changing the apparent permeability. Under this condition, we can assume that the intrinsic permeability is a constant. Permeability is just a function of the gas pressure.40 The fourth reduced model without the effects of effective stress and adsorption can be expressed as

agreements with the experimental data. It is seen that, when the gas pressure drops, the apparent permeability for the sample one will increase because of the slippage effect. From the model, it is obtained that the intrinsic permeability for the siltstone is 380 nD and the Knudsen number is 0.044 (slip flow regime). The Knudsen number increases from 0.044 at 6.4 MPa to almost 0.12 at 1 MPa. In summary, the apparent permeability is bigger than the intrinsic permeability. The flow regime is the most important factor for this variation. Under this condition, the effect of effective stress and adsorption are weak and we can neglect them.

4. DISCUSSION 4.1. Governing Equations for Porous Media. The Navier-type equation for solid deformation can be expressed as G Gui , kk + uk , ki − αp , i − Kεs , i + fi = 0 (36) 1 − 2v where, f i denotes the component of the body force, α is the Biot coefficient, εs is the adsorption induced volumetric strain as shown in eq 3, ui is the component of the displacement, and K is the bulk modulus of porous media. Equation 36 is the governing equation for solid matrix deformation, where the gas pressure p can be solved from the gas flow equation. The equation for mass conservation of the gas is formulated as G


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∂m + ∇·(ρg qg ) = Q s ∂t

model ignores the effect of the adsorption and focus on the evolution of permeability with impacts of effective stress and flow regimes. We study three cases with different properties as follows. The first case adopts permeability model as shown in eq 24. The second case adopts the general model in the simulation as shown in eq 19 with very small adsorption capacity. The Langmuir volume for the second case is 0.0005 m3/kg, and the maximum adsorption induced strain is 0.001. The third case adopts the general model with high adsorption capacity. The Langmuir volume and maximum strain of this case is 0.005 m3/kg and 0.02, respectively. The initial intrinsic permeabilities for all three cases are the same and the value is 0.056 mD. The initial gas pressure is 10 MPa, and the right boundary pressure for gas is 0.1 MPa. Under these conditions, the gas will flow out from the sample and the gas pressure in porous media will decrease. If the gas is nonadsorbed gas like helium or porous media has no adsorbed materials, the adsorption term can be removed. The evolution trends for the three cases are shown in Figure 8. It is seen that when the adsorption capacity

(37)

where ρg is the gas density, qg is the Darcy velocity vector, Qs is the gas source or sink, t is the time, and m, the mass content including free gas and adsorbed gas, is defined as m = ρg ϕ + ρga ρs

VLp p + PL

(38)

where ρga is the gas density at standard conditions, ρs is porous media density, ϕ is the porosity, VL is the Langmuir volume constant, and PL represents the Langmuir pressure constant. Assuming the effect of gravity is relatively small and can be neglected, the Darcy velocity qg is given by

qg = −

kapp μ

∇p

(39)

where kapp is the apparent permeability of gas reservoirs and μ is the viscosity of the gas. Substituting eqs 38 and 39 into eq 37, we obtain ⎡ ⎛ kapp ⎞ ρ p VLPL ⎤ ∂p ∂ϕ ⎥ ⎢ϕ + s a − ∇·⎜ p∇ p⎟ = Q s +p 2 ∂t (p + PL) ⎦ ∂t ⎣ ⎝ μ ⎠ (40)

where pa is 1 atm of pressure. In eq 40, the apparent permeability model is shown in eq 19. The governing eqs 36 and 40 are a set of nonlinear partial differential equations (PDEs), which can reflect the dynamic evolution of gas pressure and permeability. The dynamic compressibility for pores is used to reflect the physical property of porous media. A finite element method is adopted to solve PDEs and study evolution trends of the apparent permeability. 4.2. Comparison between First Reduced Model and General Model. According to the conditions of the first reduced model, we present one simulation model for gas reservoirs as shown in Figure 7 and the model geometry is 0.1 Figure 8. Evolution of apparent permeability for point A under different conditions.

is very tiny, the general model has the same trend as the reduced model and this evolution also is verified by experimental data in the above section. When we increase the capacity of adsorption, the evolution curve will deviate that of the reduced model. The first reduced model is one kind of degenerated models for the porous medium with low adsorption capacity. The main factors for gas permeability evolution are effective stress and flow regimes. For this kind of model, when the gas pressure drops, the apparent permeability will undergo the decreasing trend. But the gas permeability will be rebounded to a higher value while the gas pressure decreases to a lower magnitude as shown in Figures 8 and 9. 4.3. Comparison between CB Model of Coals and General Model. Coal seam gas is one kind of unconventional natural gas over the world. The gas methane is generated and stored in the coals matrix. Permeability of coal plays a critical role in the production of coal seam gas from coal reservoirs. CB model is a popular permeability model with considerations of stress conditions and adsorption swelling for coal, which has been verified by many experiments of coal matrix. We adopt four cases of simulation to study the evolution behaviors with different permeability models. The first case with the second

Figure 7. Simulation model and conditions for the first and fourth reduced models.

m × 0.1 m. The point A is located in the center of the geometry. The upper boundary and the right boundary are constant stress. The displacements at the left and bottom sides are constrained in the horizontal and vertical directions, respectively. A distributed overburden load of 10 MPa is applied and remains unchanged during the whole process. The initial gas pore pressure is 10 MPa and gas pressure on the right side remains 0.1 MPa as shown in Figure 7. Zero fluxes are specified on the other boundaries. In this section, the reduced H


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The third case adopts the general model with ultralow permeability to analyze the evolution of permeability. As discussed above, when the intrinsic permeability of porous media is high and the corresponding Knudsen number is less than 0.01, the impact of the flow regimes can be neglected. It is clearly seen that the second reduced model experienced the same trend as the general model with high initial intrinsic permeability. Under these conditions, the impact of flow regime can be neglected and the apparent permeability will decrease with the gas depletion from the matrix as shown in Figure 11. It

Figure 9. Evolution of permeability ratio for point A under different conditions.

reduced model is adopted in the simulation while case two and case three with the general model are used. Because this reduced model just considers the effect of adsorption and effective stress on the gas permeability, we add the fourth case of CB model as shown eq 26 to study the difference between the general model and popular coal model. For the first case, the initial intrinsic permeability of the reduced model is about 3.75 mD. In terms of the second case, the initial intrinsic permeability is 3.75 mD. However, for the third case, we adopt a low intrinsic permeability of 0.056 mD to present the impact of flow regimes on the evolution of permeability. For the fourth case, we still adopt the initial intrinsic permeability of 3.75 mD. The other parameters such as Possion’s ratio, Young’s modulus, and compressibility in the CB model are the same as the general model. According to the conditions of second reduced model, we use the same simulation geometry as type one. The upper boundary and the right boundary are constant stress. The displacements at the left and bottom sides are constrained in the horizontal and vertical directions, respectively. A distributed overburden load of 30 MPa is applied and remains unchanged during the whole process. The initial gas pore pressure is 30 MPa, and gas pressure on the right side remains 4 MPa as shown in Figure 10. Point A is located in the center of the geometry. Zero fluxes are specified on the other boundaries. The second case adopts the general model with high permeability, which means that the slippage effect is weak.


is noted that the CB model has a good agreement with the reduced model, which means that the dominated factors are adsorption and effective stress. This reduced model is applicable for coals with high initial intrinsic permeability. When the initial permeability decreases like case three, the apparent permeability will rebound because of the slippage effect as shown in Figure 11. Through these simulation results, we can see that the impact of flow regime has a close relationship with the initial intrinsic permeability. For example, for the shale gas reservoirs, permeability is ultralow and the flow regime cannot be ignored. But for the conventional sandstones, the slippage effect is weak and permeability evolution is controlled mainly by effective stress. 4.4. Comparison between David Model of Sandstones and General Model. As discussed above, the effective stress is the main factor for the evolution of gas permeability for conventional gas reservoirs like sandstones because of the high initial intrinsic permeability and very low adsorption capacity. David et al. proposed a permeability model for sandstones based on experimental measurements. The David model9 can be expressed as k∞ = k∞ 0exp( −γ(σe − σe0))

(41)

where σe is the effective pressure, γ is the pressure sensitivity coefficient, and σe0 is the initial effective pressure at reference state. It is noted that this model has the same structure as the third reduced model as shown in eq 30. If we define that the pressure sensitivity coefficient γ = 3ct and the effective pressure σe = σ̅ − p based on effective stress theory, the reduced model is totally same as the David model. We investigate the difference between the David model and the general model. The simulation model geometry and

Figure 10. Simulation model and conditions for the second and third reduced models. I


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model is a function of effective stress. If we adopt the dynamic pressure sensitivity coefficient as the reduced model, the trends for David model and reduced model will be the same. Through these simulation results, we can see that the impact of effective stress has a close relationship with the gas pressure. When the intrinsic permeability is high, the effect of flow regimes is lower. For example, permeability of conventional gas reservoirs is higher than that of shale reservoir, which means that the effect of flow regime can be ignored. The adsorption capacity for conventional gas reservoirs is also lower compared with unconventional gas reservoirs like coals and shales. The main dominated factor for conventional gas reservoirs is the effective stress. Overall, this reduced model is applicable for conventional gas reservoirs like David model. 4.5. Comparison between Javadpour Model of Shales and Fourth Reduced Model. In recent years, many scholars emphasize the importance of the flow regimes for unconventional gas reservoirs and many experiments show that lower gas pressure will lead to big apparent permeability. The impact of effective stress on the gas permeability is low when the variation of gas pressure is small. Javadpour13 proposed an apparent permeability model for gas shales and this model has been used by many scholars. The Javadpour model can be expressed as

boundary conditions are the same as second type as shown in Figure 10. Point A is located in the center of the geometry. There are four cases for the simulations. The first case is the simulation with the third reduced permeability model. The second case adopts the general model for the porous medium with high permeability and low adsorption capacity. But the third case uses the general model with ultralow permeability model and high adsorption parameters. The fourth case adopts David model of sandstones. Both the David model and the reduced model focus on the evolution of permeability from effective stress. Under this condition, the effective stress is the main factor that changes the intrinsic permeability. According to the results, we can know the reduced model is useful for the conventional gas reservoirs like sandstones. But for the unconventional gas reservoirs, the evolution curve is different from that of the reduced model. For the first case, the initial intrinsic permeability is about 3.75 mD when we assume the average pore radius for the porous medium is 1000 nm. In terms of the second case with general model, the Langmuir volume is 0.0005 m3/kg and the maximum strain is just 0.001. Also, the initial intrinsic permeability of the second case is approximately 3.75 mD. For the third case, the simulation adopts that Langmuir volume is 0.005 m3/kg and the maximum adsorbed strain is 0.02. The intrinsic permeability is smaller than that of the above two cases and the value is about 0.056 mD. For the fourth case, high initial intrinsic permeability 3.75 mD is used and the other parameters are the same as the other cases. It is clearly seen that the third reduced model experiences the same trend as the general model with high initial intrinsic permeability and low adsorption capacity. Under these conditions, the impacts from flow regime and adsorption can be neglected and the apparent permeability will decrease when the effective stress increases as shown in Figure 12. The David

kapp =

2r ⎛⎜ 8RT ⎞⎟ 3 ⎝ πM ⎠

0.5

⎞⎞ r 2 ⎛ 8πRT ⎞0.5 μ ⎛ 2 μ ⎛ ⎟ + ⎜1 + ⎜ ⎜ − 1⎟⎟ ⎝ M ⎠ pr ⎝ β p ⎝ ⎠⎠ 8 (42)

where M is gas molar mass, R denotes the gas constant, T is the absolute temperature in Kelvin, r is the average pore radius, μ denotes the gas viscosity, and β is the tangential momentum accommodation coefficient varied from 0 to 1. Javadpour model takes into account the effect of Knudsen diffusion and slip flow on the gas flux. The intrinsic permeability for Javadpour model can be assumed to be a constant. As discussed above, the general permeability model for unconventional gas reservoirs like shales can be reduced into the fourth type because the variation of effective stress is small. We adopt the gas helium to do these simulations, which can ignore the effect of adsorption. The simulation model geometry and boundary conditions are the same as the first reduced model as shown in Figure 7. Point A is located in the center of the geometry. To some extent, the decrease of gas pressure means the increase of the Knudsen number and the apparent permeability will undergo an increasing trend in the gas depletion process. In this section, we propose two cases for numerical simulation with the same geometry and boundary conditions. The first case adopts the reduced model without considering the variable intrinsic permeability and the adsorption capacity. The second case adopts the Javadpour model to study the evolution of gas permeability for shales. The initial intrinsic permeability for the first case is 300 nD. The second case is based on the Javadpour model with constant intrinsic permeability 300 nD. The initial gas pressure is 10 MPa and the boundary gas pressure is just 0.1 MPa. It is clearly seen that the fourth reduced model experiences the same trend as the Javadpour model of shales with ultralow intrinsic permeability. Under these conditions, the impacts of effective stress and adsorption on the gas permeability can be neglected and the apparent permeability will increase with the gas depletion from the reservoirs as shown in Figure 13. When the gas pressure decreases from 10 MPa to only 0.1 MPa, the apparent permeability increases more than 60 times of the


model shows a good agreement with the reduced model at the early stage of gas depletion. Permeability ratio from reduced model reaches 0.83 when the gas pressure drops to only 4 MPa while the ratio from David model is almost 0.8 at the same pressure. Permeability from the David model is a little lower than that of the reduced model at the lower pressure magnitude. The reason is that the pressure sensitivity coefficient γ is a constant but the compressibility in the general J


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Generally, the increase of the effective stress can decrease the intrinsic permeability. The flow regimes can improve the flow rate for tight porous media when the intrinsic permeability is small. Desorption can lead to the matrix shrinkage and reduce the effective stress. Also, adsorption layer thickness can decrease the effective average pore radius as shown in Figure 2. In order to accurately predict the trend of permeability for porous media including conventional and unconventional gas reservoirs, we presented eight cases of simulation with the general permeability model to investigate the complex evolution. For the conventional or unconventional gas reservoirs, there are different properties for adsorption capacity, compressibility and initial intrinsic permeability as shown in Table 3. Based on these properties, we propose eight cases to investigate the evolution of apparent permeability for possible gas reservoirs. The results show that for cases of the same boundary conditions and geometries, when the properties of porous media such as adsorption parameters changes, the evolution trend will be different. These cases correspond to the possible gas reservoirs including conventional gas reservoirs like sandstones and unconventional gas reservoirs like shales and coals. This general permeability model bridges the gaps between conventional and unconventional gas reservoirs. Specifically, the permeabilities have decreasing trends for case two and case six as show in Figure 15. The reason is that initial intrinsic permeability is high and flow regime has small effect on the apparent permeability. For these two cases, the compressibility is high, which means the increase of effective stress can reduce permeability more efficiently. The apparent permeabilities undergo big increasing trends for case three, case five and case seven. The reason is that the initial intrinsic permeability is very low (0.006 mD), which means the flow regimes cannot be ignored. When the gas pressure drops to less than 1 MPa, the Knudsen number will become big and the slippage effect will significantly contribute to the apparent permeability. For other cases, the evolutions experience different trends. Generally, at the early stage, the gas pressure drops from 25 MPa to just 4 MPa, the effective stress will increase, which leads to the decrease of permeability. When the gas pressure is less than 4 MPa, the flow regime will lead to big apparent permeability because of high Knudsen number. Overall, the evolution of the apparent permeability is complex. The trend will be different when the properties change. Based on specific conditions, each case can correspond to either conventional gas reservoir like sandstones or unconventional gas reservoirs such as shales and coals as listed in Table 3. We also present a classification zone for possible gas reservoirs based on reservoirs properties as shown in Figure 16. For example, the conventional gas reservoirs like sandstones have little or no adsorption capacity for natural gas and the intrinsic permeability is usually big, which means the slippage effect will be lower. Permeability will experience a decreasing trend when the gas pressure drops as shown in Figure 16. For the unconventional gas reservoirs like shales, the intrinsic permeability is lower, which means slippage effect is important. The apparent permeability may undergo an increasing trend when the gas is depleted from the reservoirs. For the coals, permeability will decrease at the early stage of gas depletion and rebound to increase at the later time of production. The reason is that at the early stage, the effective stress dominates the variation of permeability and the flow regimes control permeability evolution when the gas pressure is lower as shown in Figure 16.


initial permeability (300 nD) for both the reduced model and Javadpour model. The reason for this increase is the existence of slippage effect and Knudsen diffusion in nanopores. Overall, this kind of model is different with the previous three reduced models because the flow regime is the only main factor for the gas permeability evolution. This kind of reduced model is applicable for unconventional gas reservoirs with low adsorption capacity like shale reservoirs and tight gas reservoirs. 4.6. Complex Evolution and Classifications for General Model. It is noted that the evolution of gas permeability for gas reservoirs is a complex process based on the simulation results above. There are some gaps for the dynamic behaviors of gas permeability between conventional gas reservoirs and unconventional gas reservoirs. In this study, we propose the general gas permeability for both conventional and unconventional gas reservoirs. The evolution trend for permeability depends on the interactions between stress, adsorption and flow regimes. In some cases, the effective stress plays the dominated role in permeability variation as discussed in section 4.4. In other cases, the flow regimes become the main factor for permeability evolution as discussed in section 4.5. In this section, we do more simulations as shown in Figure 14 to study this complex process. Under different conditions, the dominated factors for the variation of permeability will be changed. It is interesting to see that the evolution trends of permeability are not straight lines. Permeability may undergo increase or decrease trends when the gas pressure decreases.

Figure 14. Simulation model and conditions for the general apparent permeability of gas reservoirs. K


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Energy & Fuels Table 3. Properties of Porous Media for Eight Cases with the General Model adsorption capacity case one case two case three case four case five case six case seven case eight

3

Langmuir volume 0.005 m /kg maximum adsorption strain 0.02 Langmuir volume 0.005 m3/kg maximum adsorption strain 0.02 Langmuir volume 0.005 m3/kg maximum adsorption strain 0.02 Langmuir volume 0.005 m3/kg maximum adsorption strain 0.02 Langmuir volume 0.0005 m3/kg maximum adsorption strain 0.001 Langmuir volume 0.0005 m3/kg maximum adsorption strain 0.001 Langmuir volume 0.0005 m3/kg maximum adsorption strain 0.001 Langmuir volume 0.0005 m3/kg maximum adsorption strain 0.001

compressibility (MPa−1)

initial intrinsic permeability (mD)

0.02

0.006

unconventional, coals

0.02

1


0.001

0.006

unconventional, shales

0.001

1


0.02

0.006

unconventional, shales

0.02

1

0.001

0.006

0.001

1

possible gas reservoirs

conventional, sandstones unconventional, shales conventional, sandstones

for the gas permeability evolution laws between conventional and unconventional reservoirs. In this study, we propose a general model of gas permeability to bridge these gaps. This model is developed based on poroelasticity theory and flow regimes theory. Based on experimental data and simulation results, three main conclusions can be drawn. (1) For natural gas reservoirs, there are three main factors that affecting the evolution of gas permeability namely effective stress, adsorption and flow regimes. However, in terms of the specific reservoirs properties, the dominating factors are different. For the conventional gas reservoirs like sandstones, the dominating factor is effective stress. But for the unconventional gas reservoirs such as shales, the impact of flow regimes cannot be neglected. (2) Based on properties of gas reservoirs, the general model can degenerate into four kinds of reduced models. When the adsorption capacity is low, the adsorption term of the general model can be removed and the model becomes the first reduced model. When the initial intrinsic permeability of the porous medium is very high, the impact of flow regime can be neglected. Then the general model becomes the second reduced model. When the adsorption capacity is low and initial intrinsic permeability is high, the effective stress is the most important factor for gas permeability. The general model can be written as the third reduced model. Flow regimes play the dominating role in the fourth reduced model when the intrinsic permeability is lower. Each reduced model has been verified based on experimental data or field data. These reduced models can correspond to different kinds of gas reservoirs. (3) The evolution of permeability for gas reservoirs is a complex and complicated process. When the properties of a reservoir change, the evolution trend is different. Based on the possible properties of gas reservoirs, there are two bounds for permeability evolution zones. The upper bound is the gas reservoirs only affected by the flow regimes, and the lower bound is the reservoirs only affected by the effective stress. Overall, the general model is applicable for both conventional and unconventional gas reservoirs.


Figure 16. Permeability evolution zones from general model for conventional and unconventional natural gas reservoirs.

5. CONCLUSIONS Permeability is a critical property for both conventional and unconventional natural gas reservoirs. But there are some gaps L


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(17) Li, J.; Liu, D.; Yao, Y.; Cai, Y.; Xu, L.; Huang, S. Control of CO2 permeability change in different rank coals during pressure depletion: an experimental study. Energy Fuels 2014, 28 (2), 987−996. (18) Florence, F. A.; Rushing, J.; Newsham, K. E.; Blasingame, T. A. Improved permeability prediction relations for low permeability sands. Presented at Rocky Mountain Oil & Gas Technology Symposium, Denver, CO, USA, April 16−18; Society of Petroleum Engineers, 2007. (19) Klinkenberg, L. The permeability of porous media to liquids and gases. In Drilling and Production Practice; American Petroleum Institute, 1941. (20) Yao, J.; Sun, H.; Fan, D.; Wang, C.; Sun, Z. Numerical simulation of gas transport mechanisms in tight shale gas reservoirs. Pet. Sci. 2013, 10 (4), 528−537. (21) Beskok, A.; Karniadakis, G. E. Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 1999, 3 (1), 43−77. (22) Darabi, H.; Ettehad, A.; Javadpour, F.; Sepehrnoori, K. Gas flow in ultra-tight shale strata. J. Fluid Mech. 2012, 710, 641−658. (23) Li, J.; Liu, D.; Lu, S.; Yao, Y.; Xue, H. Evaluation and modeling of the CO2 permeability variation by coupling effective pore size evolution in anthracite coal. Energy Fuels 2015, 29 (2), 717−723. (24) Naraghi, M. E.; Javadpour, F. A stochastic permeability model for the shale-gas systems. Int. J. Coal Geol. 2015, 140, 111−124. (25) Tanikawa, W.; Shimamoto, T. Comparison of Klinkenbergcorrected gas permeability and water permeability in sedimentary rocks. Int. J. Rock Mech. Min. Sci. 2009, 46 (2), 229−238. (26) Cui, X.; Bustin, R. M. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. AAPG Bull. 2005, 89 (9), 1181−1202. (27) Pan, Z. J.; Connell, L. D. Modelling permeability for coal reservoirs: A review of analytical models and testing data. Int. J. Coal Geol. 2012, 92, 1−44. (28) Zhang, H. B.; Liu, J. S.; Elsworth, D. How sorption-induced matrix deformation affects gas flow in coal seams: A new FE model. Int. J. Rock Mech. Min. Sci. 2008, 45 (8), 1226−1236. (29) Sakhaee-Pour, A.; Bryant, S. Gas permeability of shale. SPE Reservoir Eval. Eng. 2012, 15 (4), 401−409. (30) Cao, P.; Liu, J.; Leong, Y. K. A fully coupled multiscale shale deformation-gas transport model for the evaluation of shale gas extraction. Fuel 2016, 178, 103−117. (31) Ghanizadeh, A.; Clarkson, C.; Aquino, S.; Ardakani, O.; Sanei, H. Petrophysical and geomechanical characteristics of Canadian tight oil and liquid-rich gas reservoirs: I. Pore network and permeability characterization. Fuel 2015, 153, 664−681. (32) Moghaddam, R. N.; Aghabozorgi, S.; Foroozesh, J. Numerical simulation of gas production from tight, ultratight and shale gas reservoirs: flow regimes and geomechanical effects. Presented at EUROPEC 2015, Madrid, Spain, June 1−4, 2015. (33) Detournay, E.; Cheng, A. H. D. Fundamentals of poroelasticity. In Comprehensive Rock Engineering: Principles, Practice and Projects, Fairhurst, C, Ed.; Pergamon Press, 1993; Vol. 2, Chapter 5, pp 113− 171. (34) Yuan, W. N.; Pan, Z. J.; Li, X.; Yang, Y. X.; Zhao, C. X.; Connell, L. D.; Li, S. D.; He, J. M. Experimental study and modelling of methane adsorption and diffusion in shale. Fuel 2014, 117, 509−519. (35) McKee, C.; Bumb, A.; Koenig, R. Stress-dependent permeability and porosity of coal and other geologic formations. SPE Form. Eval. 1988, 3 (1), 81−91. (36) Connell, L. D.; Lu, M.; Pan, Z. J. An analytical coal permeability model for tri-axial strain and stress conditions. Int. J. Coal Geol. 2010, 84 (2), 103−114. (37) Palmer, I.; Mansoori, J. How permeability depends on stress and pore pressure in coalbeds: a new model. Presented at SPE Annual Technical Conference and Exhibition, Denver, CO, October 6−9; Society of Petroleum Engineers, 1996. (38) Shi, J. Q.; Durucan, S. Drawdown induced changes in permeability of coalbeds: a new interpretation of the reservoir

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by Australian International Postgraduate Research Scholarship (IPRS), Australian Postgraduate Award (APA), and partially funded by National Natural Science Foundation of China (51474204). This support is gratefully acknowledged.

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REFERENCES

(1) Loucks, R. G.; Reed, R. M.; Ruppel, S. C.; Jarvie, D. M. Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett Shale. J. Sediment. Res. 2009, 79 (11−12), 848−861. (2) Heller, R.; Vermylen, J.; Zoback, M. Experimental investigation of matrix permeability of gas shales. AAPG Bull. 2014, 98 (5), 975−995. (3) Civan, F.; Rai, C. S.; Sondergeld, C. H. Shale-gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Transp. Porous Media 2011, 86 (3), 925− 944. (4) Cui, A.; Wust, R.; Nassichuk, B.; Glover, K.; Brezovski, R.; Twemlow, C. In a nearly complete characterization of permeability to hydrocarbon gas and liquid for unconventional reservoirs: a challenge to conventional thinking. Presented at the Unconventional Resources Technology Conference, Denver, CO, USA, August 12−14, 2013. (5) Kwon, O.; Kronenberg, A. K.; Gangi, A. F.; Johnson, B.; Herbert, B. E. Permeability of Illite-bearing shale: 1. Anisotropy and effects of clay content and loading. J. Geophys. Res.: Solid Earth 2004, 109 (B10), B10205. (6) Soeder, D. J. Porosity and permeability of eastern Devonian gas shale. SPE Form. Eval. 1988, 3 (1), 116−124. (7) Li, S.; Tang, D.; Xu, H.; Yang, Z.; Guo, L. Porosity and permeability models for coals using low-field nuclear magnetic resonance. Energy Fuels 2012, 26 (8), 5005−5014. (8) Bhandari, A. R.; Flemings, P. B.; Polito, P. J.; Cronin, M. B.; Bryant, S. L. Anisotropy and stress dependence of permeability in the Barnett shale. Transp. Porous Media 2015, 108 (2), 393−411. (9) David, C.; Wong, T.-F.; Zhu, W.; Zhang, J. Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust. Pure Appl. Geophys. 1994, 143 (1−3), 425−456. (10) Liu, J.; Chen, Z.; Elsworth, D.; Qu, H.; Chen, D. Interactions of multiple processes during CBM extraction: a critical review. Int. J. Coal Geol. 2011, 87 (3), 175−189. (11) Raghavan, R.; Chin, L. Productivity changes in reservoirs with stress-dependent permeability. SPE Reservoir Eval. Eng. 2004, 7 (4), 308−315. (12) Cao, P.; Liu, J.; Leong, Y. K. Combined impact of flow regimes and effective stress on the evolution of shale apparent permeability. J. Unconv. Oil Gas Resour. 2016, 14, 32−43. (13) Javadpour, F. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 2009, 48 (8), 16−21. (14) Ziarani, A. S.; Aguilera, R. Knudsen’s permeability correction for tight porous media. Transp. Porous Media 2012, 91 (1), 239−260. (15) Garrouch, A. A.; Smaoui, N. Application of artificial neural network for estimating tight gas sand intrinsic permeability. Energy Fuels 1996, 10, 1053−1059. (16) Zhang, L. J.; Li, D. L.; Lu, D. T.; Zhang, T. A new formulation of apparent permeability for gas transport in shale. J. Nat. Gas Sci. Eng. 2015, 23, 221−226. M


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Energy & Fuels response to primary recovery. Transp. Porous Media 2004, 56 (1), 1− 16. (39) Robertson, E. P.; Christiansen, R. L. Modeling laboratory permeability in coal using sorption-induced strain data. SPE Reservoir Eval. Eng. 2007, 10 (3), 260−269. (40) Letham, E. A. Matrix permeability measurements of gas shales: gas slippage and adsorption as sources of systematic error. Ph.D. Thesis. University of British Columbia, Vancouver, 2011.

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General Gas Permeability Model for Porous Media: Bridging the Gaps

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