TY - JOUR
AU - Tao,, Zheng-Wu
AB - Abstract The multiscale pore size and specific gas storage mechanism in organic-rich shale gas reservoirs make gas transport in such reservoirs complicated. Therefore, a model that fully incorporates all transport mechanisms and employs an accurate numerical method is urgently needed to simulate the gas production process. In this paper, a unified model of apparent permeability was first developed, which took into account multiple influential factors including slip flow, Knudsen diffusion (KD), surface diffusion, effects of the adsorbed layer, permeability stress sensitivity, and ad-/desorption phenomena. Subsequently, a comprehensive mathematical model, which included the model of apparent permeability, was derived to describe gas production behaviors. Thereafter, on the basis of unstructured perpendicular bisection grids and finite volume method, a fully implicit numerical simulator was developed using Matlab software. The validation and application of the new model were confirmed using a field case reported in the literature. Finally, the impacts of related influencing factors on gas production were analyzed. The results showed that KD resulted in a negligible impact on gas production in the proposed model. The smaller the pore size was, the more obvious the effects of the adsorbed layer on the well production rate would be. Permeability stress sensitivity had a slight effect on well cumulative production in shale gas reservoirs. Adsorbed gas made a major contribution to the later flow period of the well; the greater the adsorbed gas content, the greater the well production rate would be. This paper can improve the understanding of gas production in shale gas reservoirs for petroleum engineers. shale gas, apparent permeability, production rate, PEBI grid, fractured horizontal well Nomenclature Aij area of the interface between block i and block j, m2 C molarity of real gas, mol m–3 cμ molarity of adsorbed gas, mol m–3 Cg gas compressible coefficient, 1 Pa–1 dm molecular diameter of gas, m Di distances from the nodes of block i to the interface shared by blocks i and j, m Dj distances from the nodes of block j to the interface shared by blocks i and j, m Dk Knudsen diffusion coefficient, m2 s–1 Ds surface diffusion coefficient, m2 s–1 F total mass flux of gas for real gas, kg m–2 s–1 Fa total mass flux of adsorbed gas for real gas, kg m–2 s–1 Fb total mass flux of bulk gas for real gas, kg m–2 s–1 Fk mass flux by Knudsen diffusion for real gas, kg m–2 s–1 Fs mass flux of gas by surface diffusion for real gas, kg m–2 s–1 Fv mass flux by slip flow for real gas, kg m–2 s–1 Ga amount of adsorbed gas, m3 kg–1 i block i j block adjacent to block i Jik element of row i and column k in sparse Jacobian matrix J⃗ J⃗ sparse Jacobian matrix k permeability of shale matrix, m2 k0 permeability at zero effective stress, m2 kapp apparent permeability of shale gas reservoirs, m2 kv permeability of slip flow for real gas, m2 k′k gas flow conductance of Knudsen diffusion, m2 k′s gas flow conductance of surface diffusion, m2 k′v gas flow conductance of slip flow, m2 Kn Knudsen number for real gas in nanopores m associated with geometry of the conductive pore space M gas molecular mass, kg mol–1 Mi total mass density of the gas in block i, kg m–3 n current time level n + 1 next time level p pressure of shale gas reservoir, Pa p1 effective stress when pores are closed completely, Pa pc confining pressure, Pa pi pressure of block i, Pa pj pressure of block j, Pa pL Langmuir pressure, Pa pwf bottom hole pressure, Pa qg gas production or injection rate, m3 s–1 Q gas production/injection rate, kg s–1 r pore radius of shale matrix, m r0 initial pore radius of shale matrix, m rw well radius, m reff effective pore radius of shale matrix, m reff_stress effective pore radius of shale matrix considering geomechanicals effects, m R general gas constant, J K–1 mol–1 Sg initial gas saturation t elapsed time, s Tij transmissibility of the interface between block i and block j, kg Pa–1 s–1 tol calculating tolerance, Pa T reservoir temperature, K VL Langmuir volume in reservoir conditions, m3 kg–1 Vstd molar volume in standard conditions (273.15 K and 1.01325 MPa), m3 mol–1 Z gas deviation factor Δt time step, s ΔVi volume of block i, m3 α effective stress coefficient φ porosity of shale matrix φ0 initial porosity of shale matrix φeff effective porosity of shale matrix ρr shale density, m3 kg–1 σeff effective stress in shale matrix, Pa σeff_0 initial effective stress in shale matrix, Pa εv weighting factor of slip flow for real gas εk weighting factor of Knudsen diffusion for real gas θ gas coverage of real gas τ tortuosity of shale matrix λ mean free path of molecules for real gas, m μg gas viscosity, Pa s ρg gas density, kg m–3 1. Introduction With the dramatic decline in the production rate and reserves of conventional natural gas resources, shale gas has gradually provided an increasing contribution to the world’s natural gas supply in recent years. The development of shale gas reservoirs has attracted much interest and has played an important role in the energy industry of North America. However, in comparison to conventional gas reservoirs, shale gas reservoirs are characterized by multiple gas transport mechanisms during the development process. Hence, an accurate model for simulating and forecasting well production performance is urgently needed. Typical organic-rich shale gas reservoirs generally have ultralow porosity and permeability. Specifically, the porosity of shale formations is usually less than 10% and their permeability varies from the nanodarcy to the microdarcy range (Ghanizadeh et al2014). Nanoscale pores are dominant in shale gas reservoirs and pores with a pore radius of less than 10 nm account for 42% of the total pore space (Akkutlu and Fathi 2012). Organic pores usually have an even smaller radius (Adesida et al2011). In general, shale gas is stored as a bulk gas (free gas) in the matrix pore spaces and an adsorbed gas on the surface of organic material (Jenkins and Boyer 2008, Sun et al2013). Therefore, the specific pore sizes and forms of gas storage in shale gas reservoirs mean that gas transport in such reservoirs comprises a combination of several transport mechanisms such as viscous flow, slip flow (SF), Knudsen diffusion (KD), and surface diffusion (SD), as well as the adsorption/desorption of adsorbed gas with depressurization during production. In the past, a large number of papers have been published that concentrated on correcting the apparent permeability for gas transport in shale gas reservoirs. Beskok and Karniadakis (1999) developed a unified Hagen–Poiseuille-type equation, which has been extensively used to address the fundamental mechanisms of fluid transport in tight porous media, including continuum flow, SF, transition flow, and free molecular flow. Javadpour et al (2007) described gas flow in micropores in terms of Darcy flow and in nanopores in terms of KD, and a later paper (Javadpour 2009) presented an apparent model based on a linear superposition of SF and KD for gas flow in shale gas formations. Civan (2010) developed a permeability model for gas flow in tight porous media, which considered rarefaction effects and slippage effects on the basis of the Beskok–Karniadakis model. Civan et al (2011) presented an improved model to accurately describe the apparent permeability and diffusivity of shale gas. Azom and Javadpour (2012) incorporated SF and KD into a model of apparent permeability to simulate dual-continuum shale gas reservoirs. Xiong et al (2012) analyzed the impacts of non-Darcy flow effects and SD of an ideal gas on apparent permeability. Singh et al (2014) proposed that a combination of Darcy flow and Knudsen flow (without considering the Klinkenberg effect) can describe gas flow for a range of Knudsen flow conditions applicable to a shale gas system. Wu et al (2015a, 2015b, 2015c) combined viscous flow and KD based on weighting coefficients to study real gas transport in nanopores. Wang et al (2015a) discussed the influence of multilayer adsorption on SD and the contribution of SD to apparent permeability. Wu et al (2016) developed a comprehensive model of apparent permeability by coupling multiple mechanisms, including non-Darcy effects, real gas effects, geomechanical effects, and the effects of adsorption layers on gas transport. However, the impacts of geomechanical effects on effective pore radius and porosity were treated separately in this paper. Song et al (2016) developed a unified model of apparent permeability for both organic matrices and inorganic matrices by coupling multiple mechanisms separately. Furthermore, both gas adsorption/desorption phenomena (Sigal 2013, Yao et al2013a, 2013b, Negara and Sun 2016, Negara et al2016) and geomechanical effects (Yu and Sepehrnoori 2014, Cao et al2016, Song et al2016) can significantly lead to dynamic changes in the effective pore radius of the shale matrix during the depressurization process in shale gas reservoirs, which can further affect apparent permeability. Real gas effects can also distinctly affect gas transport at relatively high gas pressures (Wang et al2008, Ma et al2014, Zhao et al2015). Therefore, the influencing factors mentioned above must be considered in forecasts of numerical gas flow and productivity. None of the studies mentioned above comprehensively considers the impacts of factors that influence gas transport on the apparent permeability and production behaviors of shale gas, including gas transport mechanisms (viscous flow, SF, KD, and SD), adsorption/desorption phenomena of adsorbed gas, real gas effects, and geomechanical effects. Furthermore, the influence that these factors exert on the productivity performance of shale gas wells has rarely been quantitatively analyzed. In this paper, a more comprehensive model of apparent permeability that considers multiple influencing factors was first developed. In this model, the impacts of both the adsorbed layer and geomechanical effects on the effective pore radius, real gas effects, and different gas transport mechanisms are considered simultaneously in modeling the apparent permeability. Next, the governing mass balance equation was derived and a fully implicit numerical discretization scheme was presented based on the finite volume method with perpendicular bisection (PEBI) grids. Finally, sensitive analyses were performed to study the effects of the related influencing factors on the well production behaviors of multiple fractured horizontal wells in shale gas reservoirs. 2. Gas transport mechanisms in organic-rich shale gas reservoirs During shale gas exploitation, shale gas transport mechanisms mainly consist of viscous flow, SF, and KD for bulk gas and SD for adsorbed gas, as well as gas adsorption and desorption phenomena in the depressurization process (figure 1). Figure 1. View largeDownload slide Schematic diagram of gas transport in nanoscale shale gas reservoirs. Figure 1. View largeDownload slide Schematic diagram of gas transport in nanoscale shale gas reservoirs. 2.1. Effect of adsorption/desorption on effective pore radius Adsorbed gas on the surface of an organic matrix usually obeys a Langmuir isotherm. The relationship between the adsorbed gas volume and pressure for a real gas at a constant temperature is expressed as follows (Civan et al2013): Ga=VLp/ZpL+p/Z. 1 The coverage can be defined as the ratio of the adsorbed gas volume to the Langmuir volume. The coverage for a real gas can be expressed as follows: θ=p/ZpL+p/Z. 2 Hence, considering the pore space occupied by adsorbed gas molecules owing to adsorption/desorption phenomena, the effective pore radius of a shale matrix can be expressed as follows: reff=r0-θdm. 3 2.2. Influence of geomechanical effects on effective pore radius The influence of geomechanical effects in a shale matrix on permeability can be expressed as follows (Wasaki and Akkutlu 2015, Song et al2016, Cai et al2017): k=k01-σeffp1m3, 4 σeff=pc-αp. 5 The relationship between the permeability, porosity, pore radius, and tortuosity of a shale matrix can be expressed as follows: k=φr28τ. 6 When the effective stress varies from σeff_0 to σeff, the corresponding permeability varies from k0 to k. Simultaneously, the tortuosity is regarded as constant. By equations (4) and (6), the ratio between k and k0 can be expressed as follows: φr2φ0r02=1-σeffp1m31-σeff_0p1m3. 7 The porosity is directly proportional to the second power of the pore radius (Florence et al2007, Xiong et al2012, Cai et al2017): φeff=φ0reffr02. 8 Equation (7) can be simplified as follows: r4r04=1-σeffp1m31-σeff_0p1m3. 9 The effective pore radius considering geomechanical effects can be expressed as follows: reff_stress=r01-σeffp1m1-σeff_0p1m34. 10 Hence, considering the pore space occupied by adsorbed gas molecules and geomechanical effects, the effective pore radius of a shale matrix can be expressed as follows: reff=reff_stress-θdm. 11 2.3. Effect of gas properties Shale gas reservoirs are generally under relatively high-pressure conditions. The influence of the volume of gas molecules and gas intermolecular interactions cannot be neglected. Hence, gas under shale reservoir conditions cannot be regarded as an ideal gas. In this paper, a gas deviation factor is used as calculated in the literature (Dranchuk et al1973). The gas viscosity is as calculated in the literature (Lee et al1966). 2.4. Bulk gas transport The mean free path of molecules for a real gas can be expressed as follows (Civan 2010): λ=πZRT2Mμgp. 12 The Knudsen number for a real gas in nanopores can be expressed as follows: Kn=λreff. 13 2.4.1. Slip flow When the Knudsen number Kn is less than 0.1, the pore size of the shale matrix is larger than the mean free path of gas molecules. Gas transport is dominated by collision between gas molecules. This regime is referred to as SF. Thus, the mass flux of a real gas considering SF can be expressed as follows (Civan et al2011): Fv=-ρgkvμg∇p, 14 where kv=k0(1+α(Kn)Kn)1+4Kn1+Kn, 15 α(Kn)=12815π2tan-1(4Kn0.4). 16 By combining equations (6) and (11), equation (15) can be rearranged as follows: kv=φeffreff28τ(1+α(Kn)Kn)1+4Kn1+Kn. 17 2.4.2. Knudsen diffusion When the Knudsen number Kn is larger than 1, the pore size of the shale matrix is smaller than the mean free path of gas molecules. Gas transport is dominated by collisions between gas molecules and matrix pore walls. This regime is referred to as KD. Thus, the mass flux of a real gas considering diffusion can be expressed as follows (Javadpour 2009): Fk=-MDk∇c, 18 where Dk=φeffτ2reff38ZRTπM. 19 In terms of the molarity of a real gas, c can be expressed as follows: c=pZRT. 20 By substituting equation (20) into (18), we get Fk=-MDk∇pZRT=-DkρgCg∇p, 21 where Cg=1p-1Z∂Z∂p. 22 2.4.3. Total bulk gas flux Considering the typical conditions in shale reservoirs, the bulk gas transport mechanisms in shale matrix pores include continuum flow, SF, and transition flow (Civan et al2013). On the basis of two transport mechanisms, namely, SF and KD, respectively, Wu et al (2015b) sets the ratios of the frequency of gas intermolecular collisions and the frequency of collisions between gas molecules and pore walls to the total frequency of collisions as the weighting factors for SF and KD, respectively. Then, the total mass flux of the bulk gas for a real gas can be expressed as follows: Fb=εvFv+εkFk, 23 where εv=11+Kn, 24 εk=11+1/Kn. 25 2.5. Transport of adsorbed gas Under the initial conditions in shale gas reservoirs, gas is stored in nanopores in the form of bulk gas and free gas. During the depressurization process, adsorption/desorption phenomena accompany the production of gas. At the same time, the residual adsorbed gas on the surface of organic particles is transported via SD. 2.5.1. Surface diffusion The concentration gradient of the adsorbed gas is generally regarded as the driving force of SD (Zhao et al2016a). Thus, the mass flux of SD for a real gas can be expressed as follows (Wu et al2015a): Fs=-MDs∇cμ, 26 where cμ=ρrVstdGa. 27 By substituting equation (27) into (26), we get Fs=-MDsρrVLVstd∂θ∂p∇p, 28 where ∂θ∂p=p/(pLZ)(1+p/(pLZ))2Cg. 29 2.5.2. Total flux of adsorbed gas The total mass flux of adsorbed gas for a real gas can be expressed as follows: Fa=Fs. 30 2.6. Model of apparent permeability The total mass flux for a real gas can be expressed as the sum of the total mass flux of the bulk gas and the total mass flux of the adsorbed gas: F=Fb+Fa. 31 By substituting equations (23) and (30) into (31), we get F=-εvρgkvμg∇p+εk(DkρgCg∇p)+MDsρrVLVstd∂θ∂p∇p. 32 Equation (32) can be rearranged in the form of Darcy’s law: F=-ρgkappμg∇p, 33 where kapp is the apparent permeability, which can be expressed as follows: kapp=εvkv+εkDkμgCg+MDsμgρgρrVLVstd∂θ∂p. 34 According to equation (34), with considering weighting factors, the gas flow conductance of SF, KD, and SD can be expressed as follows: kv′=εvkv, 35 kk′=εkDkμgCg, 36 ks′=MDsμgρgρrVLVstd∂θ∂p. 37 2.7. Development of the continuity equation On the basis of the newly developed model of shale gas apparent permeability, which can accurately describe gas transport in shale pores by coupling multiple mechanisms, the continuity equation can be expressed as follows: ∇⋅ρgkappμg∇p+ρgQ=∂∂t(φeffρg)+∂∂t(Mcμ). 38 3. Numerical solution 3.1. Mesh generation The traditional method of modeling fractured shale gas reservoirs using regular Cartesian grids could have some disadvantages: (1) inflexibility in representing complex geologies; (2) inadequacy in capturing the elliptical flow geometries expected around the fracture tips; (3) problems caused by grid orientation effects (Olorode 2011). PEBI grids, which are also known as Voronoi grids, can overcome these disadvantages. Owing to the ultralow permeability of shale gas reservoirs, the multiple fractured horizontal well has been proved to be a common way to achieve commercial value in the process of shale gas extraction (Zhao et al2013, 2014, 2016b). In this paper, 2D PEBI grids are used to simulate the productivity behaviors of fractured shale gas reservoirs. Figure 2 shows PEBI grids of a fractured horizontal well with 28 fractures. Figure 3(a) represents a logarithmic radial refinement of the grid around the fracture tips and figure 3(b) represents a logarithmic refinement of the grid along the direction of the horizontal well. Figure 2. View largeDownload slide Plane view of the mesh used for application of the model. Figure 2. View largeDownload slide Plane view of the mesh used for application of the model. Figure 3. View largeDownload slide Plane view of the refinements of the grids. Figure 3. View largeDownload slide Plane view of the refinements of the grids. 3.2. Discretization of continuity equation Using the finite volume method with an implicit scheme, equation (38) and the supplemental equations can be numerically discretized as follows: -∑j(Tij(pi-pj))n+1+(ρgqg)in+1-(Min+1-Min)ΔViΔt=0 39Mi is the total mass density of the gas in block i, which can be expressed as follows: Mi=(φeffρg+Mcμ)i 40Tij is the transmissibility of the interface between block i and block j. On the basis of an upstream weighting scheme, Tij can be expressed as follows: Tij=AijDi+Djρgμgkappipi≥pjAijDi+Djρgμgkappjpi<pj. 41 Because of the strong nonlinearity of some parameters and variables with pressure in equations (39)–(41), the Newton–Raphson method is used to solve these equations. Equation (39) can be simplified as follows: -∑jF¯ijn+1(pin+1,pjn+1)+Q¯in+1(pin+1)-(M¯in+1(pin+1)-M¯in(pin))=0, 42 where F¯ijn+1=(Tij(pi-pj))n+1; Q¯in+1=(ρgqg)in+1; M¯in+1=ΔViΔtMin+1; M¯in=ΔViΔtMin. Thus, the residual vector of the Newton–Raphson method R⃗i can be expressed as follows: R⃗i=-∑jF¯ijn+1(pin+1,pjn+1)+Q¯in+1(pin+1)-(M¯in+1(pin+1)-M¯in(pin)). 43 When pi ≥ pj, the elements of the sparse Jacobian matrix J⃗ can be expressed as follows: Jik=-∑j∂F¯ijn+1∂pin+1+∂Q¯in+1∂pin+1-∂M¯in+1∂pin+1,k=i0,other. 44 When pi < pj, the elements of the sparse Jacobian matrix J⃗ can be expressed as follows: Jik=∂Q¯in+1∂pin+1-∂M¯in+1∂pin+1,k=i-∂F¯ijn+1∂pjn+1,k=j0,other. 45 After calculating the sparse Jacobian matrix J⃗, the sparse linear system equation (46) can be solved for each iteration l: J⃗⋅δp⃗ln+1=-R⃗. 46 When the solution is convergent on condition that ∥R⃗∥<tol, p⃗ln+1 can be calculated by equation (47) p⃗ln+1=p⃗l-1n+1+δp⃗ln+1. 47 4. Model verification To ensure the reliability and validity of the simulation, verification was performed by history matching of production data from the Barnett Shale gas reservoir. The parameters used in the simulation are shown in table 1. The reservoir parameters were obtained from the literature (Al-Ahmadi and Wattenbarger 2011, Wasaki and Akkutlu 2015, Wang et al2015b). The parameters of the hydraulic fractures were obtained by matching production data. The results of matching historic production data are shown in figure 4. It is obvious that the results of the simulation show good agreement with historic production data for the Barnett Shale. Table 1. Parameters used for simulation of Barnett Shale. Parameter Value Units Model dimensions 916 × 290 × 90 m × m × m Grid block number 16158 — Initial reservoir pressure, pi 21 MPa Bottom hole pressure, pwf 3.5 MPa Well radius, rw 0.1 m Reservoir temperature, T 340 K Reservoir porosity, φ0 0.06 — Initial gas saturation, Sg 0.7 — Initial permeability, k0 1.4 × 10-19 m2 Tortuosity, τ 1.5 — Number of hydraulic fractures 28 — Hydraulic fracture half-length 40 m Hydraulic fracture spacing 30 m Hydraulic fracture height 90 m Hydraulic fracture width 0.003 m Hydraulic fracture conductivity 2 × 10-16 m3 Langmuir volume, VL 2.72 × 10-3 m3 kg-1 Langmuir pressure, pL 4.48 MPa Surface diffusion coefficient, Ds 1 × 10-7 m2 s-1 Gas molecular mass, M 0.016 kg mol-1 Molar volume in standard conditions, Vstd 0.0224 m3 mol-1 Shale density, ρr 2580 kg m-3 Gas molecular diameter, dm 3.8 × 10-10 m p1 179 MPa Confining pressure, pc 103 MPa m 0.5 — α 0.5 — Parameter Value Units Model dimensions 916 × 290 × 90 m × m × m Grid block number 16158 — Initial reservoir pressure, pi 21 MPa Bottom hole pressure, pwf 3.5 MPa Well radius, rw 0.1 m Reservoir temperature, T 340 K Reservoir porosity, φ0 0.06 — Initial gas saturation, Sg 0.7 — Initial permeability, k0 1.4 × 10-19 m2 Tortuosity, τ 1.5 — Number of hydraulic fractures 28 — Hydraulic fracture half-length 40 m Hydraulic fracture spacing 30 m Hydraulic fracture height 90 m Hydraulic fracture width 0.003 m Hydraulic fracture conductivity 2 × 10-16 m3 Langmuir volume, VL 2.72 × 10-3 m3 kg-1 Langmuir pressure, pL 4.48 MPa Surface diffusion coefficient, Ds 1 × 10-7 m2 s-1 Gas molecular mass, M 0.016 kg mol-1 Molar volume in standard conditions, Vstd 0.0224 m3 mol-1 Shale density, ρr 2580 kg m-3 Gas molecular diameter, dm 3.8 × 10-10 m p1 179 MPa Confining pressure, pc 103 MPa m 0.5 — α 0.5 — View Large Table 1. Parameters used for simulation of Barnett Shale. Parameter Value Units Model dimensions 916 × 290 × 90 m × m × m Grid block number 16158 — Initial reservoir pressure, pi 21 MPa Bottom hole pressure, pwf 3.5 MPa Well radius, rw 0.1 m Reservoir temperature, T 340 K Reservoir porosity, φ0 0.06 — Initial gas saturation, Sg 0.7 — Initial permeability, k0 1.4 × 10-19 m2 Tortuosity, τ 1.5 — Number of hydraulic fractures 28 — Hydraulic fracture half-length 40 m Hydraulic fracture spacing 30 m Hydraulic fracture height 90 m Hydraulic fracture width 0.003 m Hydraulic fracture conductivity 2 × 10-16 m3 Langmuir volume, VL 2.72 × 10-3 m3 kg-1 Langmuir pressure, pL 4.48 MPa Surface diffusion coefficient, Ds 1 × 10-7 m2 s-1 Gas molecular mass, M 0.016 kg mol-1 Molar volume in standard conditions, Vstd 0.0224 m3 mol-1 Shale density, ρr 2580 kg m-3 Gas molecular diameter, dm 3.8 × 10-10 m p1 179 MPa Confining pressure, pc 103 MPa m 0.5 — α 0.5 — Parameter Value Units Model dimensions 916 × 290 × 90 m × m × m Grid block number 16158 — Initial reservoir pressure, pi 21 MPa Bottom hole pressure, pwf 3.5 MPa Well radius, rw 0.1 m Reservoir temperature, T 340 K Reservoir porosity, φ0 0.06 — Initial gas saturation, Sg 0.7 — Initial permeability, k0 1.4 × 10-19 m2 Tortuosity, τ 1.5 — Number of hydraulic fractures 28 — Hydraulic fracture half-length 40 m Hydraulic fracture spacing 30 m Hydraulic fracture height 90 m Hydraulic fracture width 0.003 m Hydraulic fracture conductivity 2 × 10-16 m3 Langmuir volume, VL 2.72 × 10-3 m3 kg-1 Langmuir pressure, pL 4.48 MPa Surface diffusion coefficient, Ds 1 × 10-7 m2 s-1 Gas molecular mass, M 0.016 kg mol-1 Molar volume in standard conditions, Vstd 0.0224 m3 mol-1 Shale density, ρr 2580 kg m-3 Gas molecular diameter, dm 3.8 × 10-10 m p1 179 MPa Confining pressure, pc 103 MPa m 0.5 — α 0.5 — View Large Figure 4. View largeDownload slide Comparison between simulation data and field data for the Barnett Shale. Figure 4. View largeDownload slide Comparison between simulation data and field data for the Barnett Shale. As shown in figures 5–7, it is clearly seen that when a well is open for production for about 100 d, gas flows from the shale matrix into hydraulic fractures linearly (figure 5). This is referred to as the formation linear flow period. With the continuation of the depressurization process, the compound linear flow period may be observed, as shown in figure 6. Finally, a long period of production decline is required before pseudo-elliptical flow appears, owing to the ultralow permeability of shale reservoirs (figure 7). Figure 5. View largeDownload slide Gas pressure distribution in Barnett Shale at 100 d (in MPa). Figure 5. View largeDownload slide Gas pressure distribution in Barnett Shale at 100 d (in MPa). Figure 6. View largeDownload slide Gas pressure distribution in Barnett Shale at 1000 d (in MPa). Figure 6. View largeDownload slide Gas pressure distribution in Barnett Shale at 1000 d (in MPa). Figure 7. View largeDownload slide Gas pressure distribution in Barnett Shale at 10 000 d (in MPa). Figure 7. View largeDownload slide Gas pressure distribution in Barnett Shale at 10 000 d (in MPa). From these figures, we can see that with the progress of production the pressure wave spreads to more areas. The pressure at different locations varies dynamically, which can further affect the paths available for gas transport according to equation (11). Therefore, the porosity and permeability change during the process of gas production. 5. Results and discussion On the basis of the results of matching Barnett Shale data, a base case model was established to perform sensitive analyses. The parameters of the base case are identical to the data listed in table 1. In this study, to better understand the behavior of gas flow in shale gas reservoirs during gas production, analyses of the impacts of gas transport mechanisms (including SF, KD, and SD), the effects of the adsorbed layer, geomechanical effects, ad-/desorption phenomena, SD coefficients, and real gas effect are conducted in this section sequentially. Figure 8 shows the gas production rate and cumulative gas production performance considering SF, KD, and SD. Geomechanical effects (G) are involved in all cases in figure 8. As shown in figure 8, cumulative gas production without considering KD increases by 1.46% in comparison to the base case. Without considering SD, cumulative gas production decreases by 0.93%. From figure 9, we can see that KD and SD contribute smaller on gas transport share for the base case. And SF dominates in the whole period of pressurization. Thus, KD and SD may have small effects on the gas production rate and cumulative gas production in the present model. When the relationship between SF and KD uses linear superposition, cumulative gas production increases by 17.28%, which means that linear superposition of SF and KD may result in a forecast of larger cumulative gas production. Figure 8. View largeDownload slide Gas production rate and cumulative gas production performance considering multiple transport mechanisms. Figure 8. View largeDownload slide Gas production rate and cumulative gas production performance considering multiple transport mechanisms. Figure 9. View largeDownload slide Contribution share with pressure for different transport mechanism. Figure 9. View largeDownload slide Contribution share with pressure for different transport mechanism. Figures 10 and 11 illustrate the impacts of KD and SD, respectively, on cumulative gas production for different pore radii. As shown in figure 10, cumulative gas production undergoes a notable improvement with an increase in the pore radius of the shale matrix, and KD has a small influence on cumulative gas production for different pore radii in comparison to the base case. This is because that under a high reservoir pressure when pore radius is smaller, the SD and SF restricts each other. When pore radius is larger, the contribution shares of SD and SF do not dominate, as shown in figure 9. From figure 11, we can see that the impact of SD on cumulative gas production increases as the pore size decreases, and there is a remarkable impact on cumulative gas production when the pore radius of the shale matrix is less than 5 nm, without considering SD of adsorbed gas. Figure 10. View largeDownload slide Impact of Knudsen diffusion on cumulative gas production for different pore radii. Figure 10. View largeDownload slide Impact of Knudsen diffusion on cumulative gas production for different pore radii. Figure 11. View largeDownload slide Impact of surface diffusion on cumulative gas production for different pore radii. Figure 11. View largeDownload slide Impact of surface diffusion on cumulative gas production for different pore radii. Figure 12 shows the impacts of adsorbed gas molecules and geomechanical effects on gas production performance. As shown in figure 12, in comparison to the base case, cumulative gas production increases by 11.31% when the adsorbed layer is ignored, which means that the adsorbed layer has a notable impact on gas production forecasts. In addition, it may cause a significant increase in production forecasts that do not consider the influence of the adsorbed layer. Moreover, geomechanical effects also have a small influence on cumulative gas production, as the increase in cumulative gas production is about 2.98% when this effect is not considered. This is because that when reservoir pressure decreases, pore spaces increase. Thus, when geomechanical effects is neglected, cumulative gas production increases. Furthermore, this may lead to forecasts of higher production when geomechanical effects and impacts of the adsorbed layer are ignored simultaneously, owing to an increase in cumulative gas production of about 14.47% in comparison to the base case. Figure 12. View largeDownload slide Gas production rate and cumulative gas production performance considering the impacts of adsorbed gas molecules and geomechanical effects. Figure 12. View largeDownload slide Gas production rate and cumulative gas production performance considering the impacts of adsorbed gas molecules and geomechanical effects. Figures 13 and 14 show the impacts of the adsorbed layer and geomechanical effects, respectively, on cumulative gas production for different pore radii. As shown in figure 13, the impact of the adsorbed layer on cumulative gas production increases as the pore size decreases. Furthermore, when the pore radius of the shale matrix is less than 15 nm, adsorbed gas molecules may exert a notable negative effect on cumulative gas production. With a decrease in pore size, this phenomenon will be more prominent because there is a greater similarity in size between the pore radius and the diameter of gas molecules. According to figure 14, it can be observed that geomechanical effects have a slight impact on cumulative gas production for different pore radii. Figure 13. View largeDownload slide Impact of the adsorbed layer on gas production for different pore radii. Figure 13. View largeDownload slide Impact of the adsorbed layer on gas production for different pore radii. Figure 14. View largeDownload slide Impact of geomechanical effects on gas production for different pore radii. Figure 14. View largeDownload slide Impact of geomechanical effects on gas production for different pore radii. In the base case, the Langmuir volume VL and Langmuir pressure pL are 2.72 × 10-3 kg m-3 and 4.48 MPa, respectively. The Langmuir pressure is regarded as constant, whereas the Langmuir volume is changeable. Three scenarios, namely, 0 kg m-3 (no adsorption), 2.72 × 10-3 kg m-3 (base case), and 5 × 10-3 kg m-3 are analyzed. As shown in figure 15, it can easily be found that cumulative gas production increases with an increase in the Langmuir volume. The reason for this is that an increase in the Langmuir volume means an increase in adsorbed gas reserves. When adsorbed gas is ignored, gas production is underestimated by 18.62% in comparison to the base case after 10 000 d. Thus, the Langmuir volume is an important parameter in forecasts of gas production. Figure 15. View largeDownload slide Impact of Langmuir volume on gas production rate and cumulative gas production performance. Figure 15. View largeDownload slide Impact of Langmuir volume on gas production rate and cumulative gas production performance. Then, the Langmuir volume is treated as a constant, whereas the Langmuir pressure is assigned different values, namely, 2.48 MPa, 4.48 MPa (base case), and 6.48 MPa. From figure 16, we can see that cumulative gas production increases by 5.78% and 8.94% for increases in the Langmuir pressure from 2.48 MPa–4.48 MPa and 6.48 MPa, respectively, after 10 000 d. Thus, from figures 15 and 16, we can see that gas desorption has a significant effect on the later flow period of gas production owing to the relatively low reservoir pressure. Figure 16. View largeDownload slide Impact of Langmuir pressure on gas production rate and cumulative gas production performance. Figure 16. View largeDownload slide Impact of Langmuir pressure on gas production rate and cumulative gas production performance. Figure 17 shows the influence of the SD coefficient on well production performance. Four scenarios are presented to study its impact after 10 000 d. As shown in figure 17, cumulative gas production increases as Ds increases. When Ds is assigned a value of 1 × 10-5 m2 s-1, cumulative gas production increases by about 48.9% in comparison to the base case. The effect of Ds on cumulative gas production can be negligible when the SD coefficient is less than 1 × 10-7 m2 s-1. Thus, a reasonable SD coefficient for real gas reservoirs is very important for predicting gas production performance. Figure 17. View largeDownload slide Impact of surface diffusion coefficient on gas production rate and cumulative gas production performance. Figure 17. View largeDownload slide Impact of surface diffusion coefficient on gas production rate and cumulative gas production performance. Figure 18 illustrates the impact of real gas effects on the gas production rate and cumulative gas production performance. As shown in figure 18, when the gas deviation factor Z and gas viscosity μg are assigned values of 1 (for an ideal gas) and 2 × 10-5 Pa s, respectively, cumulative gas production is underestimated by 11% in comparison to the base case after 10 000 d. If we define Z as unity, cumulative gas production will decrease by 6.98%. In summary, the gas deviation factor and gas viscosity play crucial roles in predictions of gas production. Ignorance of real gas effects may result in dramatic underestimates of well production. Figure 18. View largeDownload slide Impact of real gas effects on gas production rate and evolution of cumulative gas production. Figure 18. View largeDownload slide Impact of real gas effects on gas production rate and evolution of cumulative gas production. 6. Conclusion In this paper, a unified model of apparent permeability was established. A comprehensive mathematical model, which included the model of apparent permeability, was derived to simulate the shale gas production process. Validation of the mathematical model and numerical discretization was conducted by matching production data from the Barnett Shale. Finally, a sensitive analysis was performed to quantitatively estimate the impacts of different influencing factors on gas production. According to this study, the following conclusions can be deduced: The gas transport mechanisms play a key role in the gas transport process. A linear superposition relationship of SF and KD may lead to a notable increase in gas production forecasts. When the pore radius is greater than 5 nm, SD can be ignored. KD may result in a negligible impact on gas production in the proposed model. The impact of the adsorbed layer increases as the pore size decreases. The impact of the adsorbed layer can be neglected when the pore radius is greater than 15 nm. Geomechanical effects in shale gas reservoirs have slight impacts on cumulative gas production. The Langmuir parameters significantly affect the evolution of cumulative gas production. Cumulative gas production increases as the Langmuir volume VL and Langmuir pressure pL increase. The desorption process contributes more to gas production in the late period of gas production. An artificial increase in the SD coefficient Ds, which may be inconsistent with real gas reservoir conditions, can result in significant overestimates in production forecasts, and the impact of SD can be neglected when the SD coefficient is less than the order of 10-7 m2 s-1. Acknowledgments This work was supported by the National Natural Science Foundation of China (Key Program) (Grant No. 51534006), National Natural Science Foundation of China (Grant No. 51704247 and 51674211) and the Scientific Research Starting Project of SWPU (No. 2015QHZ003). 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TI - Numerical solution of fractured horizontal wells in shale gas reservoirs considering multiple transport mechanisms
JO - Journal of Geophysics and Engineering
DO - 10.1088/1742-2140/aa93b2
DA - 2018-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/numerical-solution-of-fractured-horizontal-wells-in-shale-gas-u5wlcmlhLP
SP - 739
VL - 15
IS - 3
DP - DeepDyve
ER -