Shale gas simulation considering natural fractures, gas desorption, and slippage flow effects using conventional modified model
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Abstract
This paper presents the theory and application to modify the conventional simulator to describe the effects of gas adsorption and gas slippage flow in shale gas. Because of the local desorption of gas and the assumptions of gas desorption instantaneously with the decrease in pore pressure, we define one fictitious immobile “pseudo” oil with dissolved gas. The dissolved gas–oil ratio is calculated from the Langmuir adsorption isotherm constants and shale gas properties. Additional modifications required in the input data are the porosity and relative permeability curves to account for the existence of “pseudo” oil. The input rock table considers the changes of rock permeability versus pressure to describe the gas slippage flow effects. In addition, dualporosity dualpermeability models coupled with local grid refinement method are used to distinguish the impacts of natural fractures and hydraulic fractures on shale gas production with the comparison of vertical well, fractured vertical well, horizontal well, and multistage fractured horizontal well production. This proposed simulation approach shows enough accuracy and outstanding time efficiency. Results show that ignoring gas desorption and slippage flow effects would bring significant error in shale gas simulation The existence of natural fractures also imposes great effects on the productivity of shale gas.
Keywords
Shale gas simulation Conventional modified model “Pseudo” oilList of symbols
 B_{g}
Gas volume factor
 B_{o}
Oil formation volume factor
 b
Langmuir pressure constant
 c_{m}
Matrix compressibility, psia^{−1}
 c_{g}
Gas compressibility, psia^{−1}
 c_{o}
Oil compressibility, psia^{−1}
 c_{w}
Water compressibility, psia^{−1}
 c_{f}
Rock compressibility, psia^{−1}
 h
Reservoir thickness, m
 h_{f}
Hydraulic fracture height, m
 k_{m}
Matrix permeability, md
 k_{nf}
Natural fracture permeability, md
 k_{hf}
Hydraulic fracture permeability, md
 L_{x}L_{y}L_{z}
The average size of cleaved matrix blocks, m
 p
Shale gas reservoir pressure, psia
 P_{L}
Langmuir pressure, psia
 P_{i}
Reservoir pressure, psia
 R_{s}
Immobile oil solution gas–oil ratio
 S_{d}
Dualporosity model shape factor, reflecting the geometry of matrix grids and controlling the flow between matrix and fracture, ft^{−2}
 S_{g}
Gas saturation in real shale gas reservoir
 S_{gm}
Gas saturation in shale gas simulation model
 S_{o}
Oil saturation in real shale gas reservoir
 S_{om}
Oil saturation in shale gas simulation model
 V
Amount of gas adsorbed in a unite volume, ft^{3}
 V_{L}
Langmuir volume, scf/ton
 x_{f}
Hydraulic fracture half length, ft
 \(\rho_{B}\)
Gas density, g/cm^{3}
 \(\phi_{\text{hf}}\)
Hydraulic fracture porosity
 \(\phi_{\text{m}}\)
Porosity in shale gas simulation model
 \(\phi_{{}}\)
Porosity in real shale gas reservoir
 \(\phi_{\text{nf}}\)
Natural fracture porosity
Introduction
Recently, tight gas and shale gas have achieved great attention because of advancements of horizontal well drilling and largescale fracturing technology. Recoverable reserves of shale gas in USA are estimated to be 862 TCF (Kuuskraa et al. 2011; Yuan and Wood 2015a, b). One of the issues yet not solved is the lack of predicative tools for shale gas production. The complex shale gas flow mechanisms in nanopores cannot be simply described as Darcy flow equation. In addition, due to largescale fracturing in shale gas reservoirs with natural fractures, the conventional single porosity model is not enough to simulate the characteristics of shale gas reservoirs. Furthermore, as advanced simulation methods summarized by (Wood et al. 2015), rigorous simulation techniques, such as direct simulation, Monte Carlo method, and Molecular dynamic simulation, are very computationally expensive (Moghanloo et al. 2015; Yuan et al. 2015). Hence, there are two alternative approaches to model shale gas: (1) application of analytical methods to characterize the primary characteristics of shale gas (Ghanbarnezhad Moghanloo and Javadpour 2014), and (2) extending the conventional simulator to effectively model flow from shale gas reservoirs. In this paper, we present the theory and application to modify the conventional simulator to describe the effects of Langmuir gas desorption and gas slippage flow effects (Shabro et al. 2011; Waqas 2012). We also present the dualporosity dualpermeability model coupled with local grid refinement method to distinguish the impacts of natural fractures and hydraulic fractures on shale gas production.

When K _{ n } < 0.001, the Navier–Stokes equation with noslip boundary conditions are presented to describe the fluid flow. This regime is called continuum flow regime.

When 0.001 < K _{ n } < 10, the Navier–Stokes equation will not fully perform well and the flow regimes of rarefied gas, which is neither continuum flow nor freemolecular flow.

When 0.001 < K _{ n } < 0.1, gas molecules tend to slip at the surface of rock (slippage condition), and as a result, gas flux will increase. However, the slippage is just limited to the vicinity adjacent to pore surfaces. Knudsen layer are defined between the fluid and pore surface with the order of one molecular mean free path. Therefore, this Knudsen layer can be neglected compared with the size of pore throat, flow velocity is linearly extrapolated toward to the pore surface. The finite nonzero flow velocity exists on the wall of pore throats. The shale gas flow will deviate from Darcy flow equation with the modification of Klinkenberg’s modification.

When 0.1 < K _{ n } < 10, the Knudsen layer will increase. The Navier–Stokes equation will not applicable. Furthermore, Knudsen diffusion increases because the molecular mean free path becomes comparable with the size of pore throat; in other words, the molecule/wall collisions will dominate particle/particle collisions. (Fathi et al. 2012) introduced a correction of double slip to Klinkenberg’s slippage model.

For K _{ n } > 10, due to the occurrence of free molecule flow, kinetic effects become dominate. The collisions among gas molecules are smaller that of molecule/wall collision. Continuum equations becomes fully ineffective, and the ballistic flow happens (Hadjiconstantinou 2006).
In this paper, the gas slippage (Klinkenberg effect) will be described with the commonly used Klinkenberg gas slippage factor within the Darcy flow equation. In addition, in tight gas reservoirs, there are significant incidences of natural complex mineralized or plugged fractures. Fracturing of horizontal wells with hybrid fluid or slick water (SWF) (Britt and Smith 2009) not only increases fracture formation contact areas by creating SRVs, but also reactivates natural fractures to some extent, which is vital to tight gas production (Ozkan et al. 2009; Clarkson 2013).
In this paper, we present conventional modified simulation approach to simulate the performance naturally fractured shale gas reservoirs. The main advantage of our work is to provide an equivalent approach to simulate shale gas in conventional black oil model by considering gas desorption, gas flow slippage, all of which are the particular characteristics of shale gas different from conventional gas reservoirs. The comparison with previous conventional methods indicates the accuracy of our proposed equivalent methods. This approach can also cover the primary geological characteristics of natural fractured shale gas, and the difference of the natural fractures around the well nearby and those hydraulic fractures.
Based on the theory of fluid flow in naturally fractured reservoirs developed by Barenblatt et al. (1960), Warren and Root (1963) presented the concept of dualporosity models to describe the matrix as the source of interconnected fractures. Kazemi et al. (1976) introduced the dualporosity model into numerical model to describe fluid flow on large scale. During the largescale hydraulic fracturing process, there is reactivation of preexisting (cemented) natural fractures and activation of planes of weakness, and even connected with hydraulic primary and secondary fractures, which is socalled fracture networks or stimulated reservoir volume.
Mathematical model and simulation approximation
Gas desorption
 1.Matrix System:$$\begin{aligned} & {\text{Oil}}\,{\text{Phase:}} \\ & \frac{\partial }{{\partial t}}\left( {\phi _{{\text{m}}} \rho _{{\text{o}}} \frac{{S_{{{\text{om}}}} }}{{B_{{\text{o}}} }}} \right)  \nabla \cdot \left( {\frac{{kk_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{\rho _{{\text{o}}} }}{{B_{{\text{o}}} }}\nabla p_{{{\text{matrix}}}} } \right) + \frac{{k_{{{\text{matrix}}}} k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{\rho _{{\text{o}}} }}{{B_{{\text{o}}} }}S_{{\text{d}}} \left( {p_{{{\text{matrix}}}}  p_{{{\text{frac}}}} } \right) = 0 \\ & {\text{Gas}}\,{\text{Phase:}} \\ & \frac{\partial }{{\partial t}}\left( {\phi _{{\text{m}}} \left( {\frac{{\rho _{{\text{g}}} S_{{{\text{gm}}}} }}{{B_{{\text{g}}} }} + \frac{{R_{{\text{s}}} \rho _{{\text{g}}} S_{{{\text{om}}}} }}{{B_{{\text{o}}} }}} \right)} \right)  \nabla \cdot \left( {k_{{{\text{matrix}}}} \left( {\frac{{k_{{{\text{rg}}}} }}{{\mu _{{\text{g}}} }}\frac{{\rho _{{\text{g}}} }}{{B_{{\text{g}}} }} + \frac{{k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{R_{{{\text{sm}}}} \rho _{{\text{g}}} }}{{B_{{\text{o}}} }}} \right)\nabla p} \right) + k_{{{\text{matrix}}}} \left( {\frac{{k_{{{\text{rg}}}} }}{{\mu _{{\text{g}}} }}\frac{{\rho _{{\text{g}}} }}{{B_{{\text{g}}} }} + \frac{{k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{R_{{{\text{sm}}}} \rho _{{\text{g}}} }}{{B_{{\text{o}}} }}} \right)S_{{\text{d}}} \left( {p_{{{\text{matrix}}}}  p_{{{\text{frac}}}} } \right) = 0 \\ \end{aligned}$$(2a)
 2.Fracture System:$$\begin{aligned} & {\text{Oil}}\,{\text{Phase:}} \\ & \frac{\partial }{{\partial t}}\left( {\phi _{{\text{f}}} \rho _{{\text{o}}} \frac{{S_{{{\text{om}}}} }}{{B_{{\text{o}}} }}} \right)  \nabla \cdot \left( {\frac{{k_{{\text{f}}} k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{\rho _{{\text{o}}} }}{{B_{{\text{o}}} }}\nabla p_{{{\text{frac}}}} } \right)  \frac{{k_{{{\text{frac}}}} k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{\rho _{{\text{o}}} }}{{B_{{\text{o}}} }}S_{{\text{d}}} \left( {p_{{{\text{matrix}}}}  p_{{{\text{frac}}}} } \right) = 0 \\ & {\text{Gas}}\,{\text{Phase:}} \\ & \frac{\partial }{{\partial t}}\left( {\phi _{f} \left( {\frac{{\rho _{{\text{g}}} S_{{{\text{gm}}}} }}{{B_{{\text{g}}} }} + \frac{{R_{{\text{s}}} \rho _{{\text{g}}} S_{{{\text{om}}}} }}{{B_{{\text{o}}} }}} \right)} \right)  \nabla \cdot \left( {\left( {\frac{{k_{{\text{f}}} k_{{{\text{rg}}}} }}{{\mu _{{\text{g}}} }}\frac{{\rho _{{\text{g}}} }}{{B_{{\text{g}}} }} + \frac{{k_{{\text{f}}} k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{R_{{{\text{sm}}}} \rho _{{\text{g}}} }}{{B_{{\text{o}}} }}} \right)\nabla p} \right)  \left( {\frac{{k_{{\text{f}}} k_{{{\text{rg}}}} }}{{\mu _{{\text{g}}} }}\frac{{\rho _{{\text{g}}} }}{{B_{{\text{g}}} }} + \frac{{k_{{\text{f}}} k_{{{\text{ro}}}} }}{{\mu _{{\text{o}}} }}\frac{{R_{{{\text{sm}}}} \rho _{{\text{g}}} }}{{B_{{\text{o}}} }}} \right)S_{{\text{d}}} \left( {p_{{{\text{matrix}}}}  p_{{{\text{frac}}}} } \right) = 0 \\ \end{aligned}$$(2b)
The amounts of gas in the simulation model and actual shale gas reservoirs must be equal to meet with the material balance (Seidle and Arri 1990), and therefore, the saturation is related to each other by:
The sum of phase saturation must be unity, and therefore,
In addition, in the real shale gas reservoirs, assuming \(\phi\) is hydrocarbon filled porosity:
Equations 2a and 2b are simplified to be
This equation relates the shale gas reservoirs porosity with porosity value in the simulation model. The immobile oil saturation can be arbitrary, but once chosen, it must be constant throughout the simulation process. The gas saturation in the model becomes:
Because we have defined oil phase, we have to define the relative permeability curve to describe the flow of gas in the shale gas model. The relative permeability corresponding to the actual gas saturation must be altered to the new gas saturation in model. Actually, in shale gas reservoir, there does not exists mobile oil, and therefore, the immobile oil defined in our model is characterized by very small relative permeability (close to zero) and very large viscosity, in the paper, 10^{7} cp.
In the physical shale gas reservoirs, the amount of gas adsorbed in a unite volume is defined as:
In the simulation model, \(V_{\text{m}}\), the amount of gas dissolved in the same unit volume is:
Values of parameters in shale gas simulation
Matrix porosity, ϕ _{m}  0.06  Reservoir Pressure, Pi, psia  5000 
Matrix permeability, k _{m}, md  0.00005  Gas volume factor, B _{g}, RB/MCF  1.38 
Natural fracture porosity, ϕ _{nf}  0.50  Gas compressibility, c _{g}, psia^{−1}  0.0006 
Natural fracture permeability, k _{nf}, md  100  Oil compressibility, c _{o}, psia^{−1}  0.00001 
Hydraulic fracture porosity, ϕ _{hf}  0.60  Model porosity, ϕ _{m}  0.066666667 
Hydraulic fracture permeability, k _{hf}, md  5100.75  Model oil saturation, S _{om}  0.1 
Hydraulic fracture half length, x _{f}, ft  148  Gas density, g/cm^{3}  0.0007354 
Reservoir thickness, h, ft  100  Langmuir factor, psia^{−1}  0.0025 
Hydraulic fracture height, hf, ft  100  Langmuir pressure, p _{m}, psia  400 
Matrix compressibility, c _{m}, psia^{−1}  1E−6  Langmuir volume, V _{m}, scf/ton  100 
Reservoir depth, D, ft  5000  Reservoir temperature, T, °F  212 
Equivalent dissolved gas–oil ratio with amounts of gas adsorption (PVTO input values)
R _{s} (Mscf/stb)  P (psia)  B _{o} (rb/stb)  Uo (cp) 

0.001073353  500  1.0452896  1.00E + 08 
0.001380025  1000  1.0456264  1.00E + 08 
0.001525291  1500  1.0458285  1.00E + 08 
0.001610029  2000  1.0459632  1.00E + 08 
0.001665547  2500  1.0460594  1.00E + 08 
0.001704736  3000  1.0461316  1.00E + 08 
0.001733877  3500  1.0463808  1.00E + 08 
0.001756395  4000  1.0465421  1.00E + 08 
0.001774318  4500  1.0467033  1.00E + 08 
0.001788921  5000  1.0468646  1.00E + 08 
0.001788921  6000  1.0467233  1.00E + 08 
The other shale gas parameters are chosen as the data provided in the literature review (Kale et al. 2010).
Gas slippage flow
The gas slippage effect is commonly defined by Klinkenberg slippage factor (Klinkenberg 1941; Britt and Smith 2009).
Here, the p is the mean pressure, psia; a is the Klinkenberg factor. Using the transmissibility multiplier with respect to the actual shale gas permeability, Tr:
Traditionally, the slippage factor, a, is assumed as constant. (Jones and Owens 1980) studied more than 100 samples from various tight gas formation from the USA and propose the following empirical relation between slippage factor and reference permeability.
In this paper, a is calculated to be 331 psia^{−1}.
Transmissibility multiplier as ROCKTABLE input
P, psia  T _{r} 

5000  1.06638694 
4500  1.073763267 
4000  1.082983675 
3500  1.094838486 
3000  1.1106449 
2500  1.13277388 
2000  1.16596735 
1500  1.2212898 
1000  1.3319347 
500  1.6638694 
Natural fractured shale gas reservoirs
Well type and fracturing design
Conclusion
 1.
Gas dissolved in an immobile oil is modeled as adsorption of gas, to mimic the gas desorbs instantaneously from matrix, which is based on conventional modified model. The solution gas–oil ratio of immobile “pseudo” oil is calculated from the Langmuir adsorption isotherm constants and shale gas properties.
 2.
Shale gas adsorption and gas slippage effects has been considered in the model, the proposed approach can offset the computationconsuming drawback of conventional modified simulator. The effects of shale gas slippage and shale gas adsorption on shale gas simulation are very significant. The neglect of those two effects can also bring great error in shale gas simulation.
 3.
In dualporosity shale gas model, the magnitude of natural fracture density and permeability has great effects on shale gas productivity. The natural fractures properties are the great decisive factors on the productivity of shale gas. Both natural fractures and matrix permeability have great effects on dualpermeability shale gas model.
 4.
The choice of different well type is the decisive factors on the production and recovery of shale gas. The long horizontal well drilling and largescale fracturing technology can bring great benefit for shale gas reservoirs.
Notes
Acknowledgements
This work was supported by National Basic Research Program of China (2014CB239103), China Postdoctoral Science Foundation funded project (2016M602227), National Natural Science Foundation of China (51674279). The authors would like to acknowledge the technical support of ECLISPE in this paper.
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