# Flow Mechanism and Simulation Approaches for Shale Gas Reservoirs: A Review

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## Abstract

The past two decades have borne remarkable progress in our understanding of flow mechanisms and numerical simulation approaches of shale gas reservoir, with much larger number of publications in recent 5 years compared to that before year 2012. In this paper, a review is constructed with three parts: flow mechanism, reservoir models and numerical approaches. In mechanism, it is found that gas adsorption process can be concluded into different isotherm models for various reservoir basins. Multi-component adsorption mechanisms are taken into account in recent years. Flow mechanism and equations vary with different Knudsen numbers, which could be figured out in two ways: molecular dynamics (MD) and lattice Boltzmann method (LBM). MD has been successfully applied in the study of adsorption, diffusion, displacement and other mechanisms. LBM has been introduced in the study of slippage, Knudsen diffusion and apparent permeability correction. The apparent permeability corrections are introduced to improve classic Darcy’s model in matrix with low velocities and fractures with high velocities. At reservoir-scale simulation, gas flow models are presented with multiple porosity classified into organic matrix with nanopores, organic matrix with micropores, inorganic matrix and natural fractures. A popular trend is to incorporate geomechanism with flow model in order to better understand the shale gas production. Finally, to solve the new models based on enhanced flow mechanisms, improved macroscopic numerical approaches, including the finite difference method and finite element method, are commonly used in this area. Other approaches like finite volume method and fast matching method are also developed in recent years.

## Keywords

Shale gas Flow mechanism Flow model coupled with geomechanics Adsorption Apparent permeability correction## 1 Introduction

Shale gas reservoir is playing an important role in the world energy market, due to its significant advantages of less pollution in combustion compared with conventional fuel resources like oil and coal. Starting from the beginning of the twenty-first century (Le 2018; Wang et al. 2014; Hefley and Wang 2016; Zhang et al. 2015b), shale gas exploitation has become an essential component to bridge the growing gap between domestic production and consumption and thus secure the energy supply in North America (Hefley and Wang 2016). The USA successfully became the largest natural gas producer in 2009, thanks to the high progress in shale gas production (Bros 2012; Wang and Hefley 2016. In another large energy exporter, shale gas resources in Canada are estimated with an amount larger than 1000 tcf (tera cubic feet). A paradigm shift has been made toward the exploration of shale gas in one of the main reservoir blocks, the Western Canada sedimentary basin (WCSB) (Cipolla et al. 2010; Rubin 2010). With the development and popularity of shale gas exploration all over the world, there have also been other countries and areas reported with great potential of exploitation. For example, shale gas resources in China are estimated about \(31\times 10^{12}\,\mathrm{m}^{3}\) (Wang and Wang 2011)

This paper is designed to conclude and comment on the flow mechanism and simulation approaches. First, the adsorption and deadsorption process is introduced, as well as the flow regime description. Besides, numerical simulation on micro- and mesoscales, e.g., molecular dynamics and lattice Boltzmann method, are reviewed. Meanwhile, apparent permeability correction, which is the macroscopic focus, is concluded. Afterward, we focus on the gas flow simulations at reservoir scales, including numerical models and the effect of geomechanics. Finally, the common macroscopic numerical simulation approaches, including finite element method, finite difference method and other schemes, will be presented.

## 2 Flow Mechanism of Shale Gas

Permeability is always relatively low in shale gas reservoirs, generally less than 1 md, and stratigraphic composition can be divided into different types of intervals (e.g., Devonian, Jurassic and Cretaceous strata) (Reis et al. 2001; Barree and Conway 2004). The stress-sensitive parameters, including organic richness, porosity, thickness and lateral extent, can vary significantly with in situ stress changes. Consequently, the fluid flow and geomechanics’ impacts are always effected by the change (Babadagli et al. 2015). Such extremely tight rock formations in shale gas reservoirs with different parameters result in the gas transportation which occurs through them by different mechanisms. With more efforts been devoted to the researches of such flow mechanisms, the inherent limitations of the conventional macroscopic methods used in petroleum industry have been overcome and new microscopic and mesoscopic approaches including molecular dynamics (MD) and lattice Boltzmann method (LBM) are introduced.

### 2.1 Adsorption/Desorption Mechanism

There are three states of gas reserved in shale reservoir: free gas, adsorbed gas and dissolved gas (Pollastro 2007). In previous study, it is found that adsorbed gas is the main state among the above three states, with statistical results, indicating that \(20-80\%\) of the total gas is adsorbed in reservoirs (Lu et al. 1995; Huan-zhi and Yan-qing 2010; Ross and Marc Bustin 2007; Curtis 2002). Adsorption properties can provide critical information to help characterize shale structures and optimize hydraulic fracturing. With the decrease in environment pressure, adsorbed gas will become free gas in the early period of exploitation (Grieser et al. 2009). As a result, gas adsorption/desorption description is of great importance to investigate the well production.

Comparison of different isotherm adsorption models

Classification | Isotherm model | Basin | References |
---|---|---|---|

Langmuir | \(V=\frac{V_\mathrm{L}P}{P_\mathrm{L}+P}\) | Barnett, the USA | |

Freundlich | \(V=Kp^{x}\) | Mansouri, Iran | |

Langmuir–Freundlich | \(V=\frac{V_\mathrm{L}\left( bp\right) ^{m}}{1+\left( bp\right) ^{m}}\) | Longmaxi, China | |

D-R | \(V=V_{0}\mathrm{exp}\left[ -D\mathrm{ln}^{2}\left( P_\mathrm{s}/P\right) \right] \) | Qaidam, China | Luo et al. (2015) |

BET | \(n_\mathrm{a}=\frac{1}{\frac{1}{n_{o}c}+\frac{c-1}{n_{o}c}\frac{P}{P_{0}}}\frac{1}{\frac{P}{P_{o}}-1}\) | Marcellus, the USA | |

Toth | \(V=\frac{V_\mathrm{L}bp}{\left[ 1+\left( bp\right) ^{k}\right] 1/k}\) | Bornholm, Denmark |

Shale permeability will be changed, due to the desorption of gas in the production process (Guo et al. 2017a). For example, gas desorption process is found in organic grids, known as kerogen, where pressure drop occurs (Guo et al. 2014). Meanwhile, pressure difference will be generated between the bulk matrix and the pores, with the pore pressure decreasing in the free gas production process; thus, the desorption on the surface of bulk matrix is reinforced.

A large number of gas adsorption isotherm models have been proposed in previous studies, such as Langmuir’s type model, Freundlich-type model, Langmuir–Freundlich-type model, D-R-type model, BET-type model and Toth-type models (Herzog 2010; Langmuir 1918; Brunauer et al. 1938; Dubinin 1960, 1965; Yang et al. 2014; Su et al. 2008; Bae and Bhatia 2006; Dada et al. 2012). Most available adsorption models, including their basic equation and the basins where they are applied, are listed in Table 1. In this table, *V* denotes the adsorbate volume, *P* denotes the pressure, *K* denotes an associated equilibrium constant, *k* denotes the Henry’s constant, *b* denotes the adsorption affinity, *D* denotes the empirical binary interaction parameter, *x* and *m* denote a constant for a given absorbate and absorbent at a particular temperature, and *c* is a constant related to the adsorption net heat. The subscript L in \(P_\mathrm{L}\) and \(V_\mathrm{L}\) denotes the Langmuir pressure and Langmuir volume.

*p*(Pa) is the gas pressure, \(p_\mathrm{L}\) (Pa) is the Langmuir gas pressure, and \(M_\mathrm{g}\) (kg/mol)is the molecular weight of gas.

In reservoir scale, the effect of gas adsorption capacity is highly extrapolated in regions. As a result, gas-in-place evaluation and production prediction are quite easy to be overestimated or underestimated and then severely impact the energy industry and social economy (Saulsberry et al. 1996; Ge et al. 2018). However, the existing models are still in developing and continuous optimization. For example, the original BET model is seldom used at present due to the weak theoretical foundations. It has been found that some assumptions in these models, like multilayer formation, small pore capillary condensation, adsorbed liquid phase and saturation pressure, are no longer suitable for special flow mechanisms of shale gas fluid (Zhou and Zhou 2009; Dubinin 1960). Another shortcoming of the classical model is that extrapolated data beyond the test range cannot be fully relied due to different empirical correlations in different temperature regimes. The original physical meanings inside these models, coming from the well-designed experiments, are weakened due to the introduction of some empirical constants. These constants are manually corrected to improve the fitting performance but make the models less reliable (Tang et al. 2016). There remains a lot to do to meet the realistic industry conditions better and to help the industry with more accuracy on the production forecast and control.

It has been pointed out that gas-in-place volumes in reservoirs are often incorrectly determined for cases with multi-component sorbed gas phase (Ambrose et al. 2011; Hartman et al. 2011; Fathi and Akkutlu 2014). Especially for shale gas fluid flow with high composition of varieties of hydrocarbons (C2+) and subsequently high total organic content (TOC), the adjustment of taking multi-component effect into account has been more necessary in the gas-in-place predictions. Compared to conventional approach, the new multi-component model will show a 20 percent decrease in total gas storage capacity calculations (Ambrose et al. 2011). Besides, multi-component sorption phenomena, in particular in the primary (micro-) pore structure of the shale matrix, e.g., co- and counter diffusion and competitive adsorption process, are the fundamental interests in the study of CO\(_2\) sequestration and enhanced shale gas recovery (Fathi and Akkutlu 2014). However, the current multi-component adsorption model is still limited on just modifications based on classical single-component Langmuir sorption model (Jiang et al. 2014; Hartman et al. 2011). A more uniform and widely applicable model is still in urgent requirement to meet the complex physical and chemical environment of shale gas reservoirs. With the rapid development of fully coupled multi-component multi-continuum compositional simulator which incorporates several transport/storage mechanisms of shale gas reservoirs, a more comprehensive adsorption/desorption model is needed to capture and predict the transport process in shale gas reservoirs.

### 2.2 Flow Mechanisms of Gas Transport in Shale Gas Reservoir

It is important to study the flow mechanism of gas transport in shale gas reservoir. A particular interest has been focused on the multiscale flow simulation on the subsurface porous media with pore size ranging from macroscale (\(>1\) mm) to nanoscale (\(<100\) nm) (Gerke et al. 2015; Di and Jensen 2015). Different pore-scale characteristics are presented with different flow regimes identified by Knudsen number (McCain 1990; Kärger 1996; Zhao et al. 2014; He et al. 2013). Slippage and diffusion processes are often viewed as the main flow mechanisms (Berkowitz and Ewing 1998). New approaches, including molecular dynamics (MD) and lattice Boltzmann method (LBM), are rapidly developed in these years to study the flow mechanisms.

#### 2.2.1 Flow Regime

*Kn*) is a parameter introduced in gas flow description to identify flow regimes with different rarefaction degrees of gas encountered. Generally, four regimes are characterized based on

*Kn*: continuous flow (\(Kn<10^{-3}\)), slip flow (\(10^{-3}<Kn<10^{0.1}\)), transition flow (\(0.1<Kn<10\)) and Knudsen flow (\(Kn>10\)) (Beskok et al. 1996). Different interfacial effects are found effective in different flow regimes in small porous structure. For large tube diameter, the gas flow is mainly viewed as continuous flow with only slip regime near the wall (Ozkan et al. 2010). Strong interfacial effects are found in shale nanotubes, which are believed to be caused by two important flow regimes including Knudsen flow and transitional flow. It should be noted that the flow pattern of single-phase gas flow and gas–water two-phase flow is of big difference (Zhao et al. 2012). In this paper, we focus on the single-phase flow (Fig. 2).

*Kn*varies from 0.001 to 0.1, gas transportation is in the regime of slip flow and slip boundary condition should be incorporated into Navier–Stokes equation or lattice Boltzmann method (LBM) to take into account the slippage on the gas solid interface. When

*Kn*is in a higher range of 0.1 and 10, gas flow enters the transitional regime, where neither Navier–Stokes equation nor lattice Boltzmann model is applicable any more. Then, Burnett equation based on higher-order moments of Boltzmann equation should be solved or numerical method of direct simulation Monte Carlo (DSMC) should be used to represent the fluid flow behavior. As Kn goes beyond 10, gas stream is considered as free molecules, and molecular dynamics (MD) must be adopted to capture the physics controlling the gas flow.

Knudsen number and flow regimes with applicable mathematical models

Particle model | Boltzmann equation (BE) | Collisionless BE | ||
---|---|---|---|---|

Continuum model | LBM/NS equation | DSMC | Molecular dynamics | |

| \(\left( 0,0.001\right) \) | \(\left( 0.001,0.1\right) \) | \(\left( 0.1,10\right) \) | \(>10\) |

Flow regimes | No slip | Slip | Transitional | Free molecular |

For gas flow in nanotubes, it has been demonstrated that slippage effects will change the flow regime identification (Tian et al. 2012). The concept of slip has a long history, starting from the famous scientist Navier (Neto et al. 2005; Matthews and Hill 2008) , and has been used in a large range of practices. For fluid flow passing rough surface, slip boundary condition is often applied with the slip length relevant to roughness height. When \(Kn<10^{-2}\), flow is in continuous regime and Darcy’s law is enough to describe the flow. As *Kn* increases from 0.01, diffusive flux is no longer ignorable and additional term should be considered in the flow equations, which makes it nonlinear.

Geometrical properties of fractured porous media are vital to predict and evaluate the hydraulic transport properties of fracture networks (Wei and Xia 2017; Tan et al. 2017). Although a variety of subjects have been studied related to geometrical, fractal and hydraulic properties of fractured porous media such as rock masses and reservoirs, a gap still exists between theoretical knowledge and field practice (Yinghao et al. 2018). It is of great importance to seek new theoretical and numerical studies and advances in various subjects addressing flow and transport mechanism as well as hydrocarbon recovery improvement, such as innovative stimulation techniques, reservoir characterization and other approaches. Specifically, not all the length distribution of fractures and fracture networks follows the fractal law. They may be multi-fractal and even non-fractal. Thus, more elaborate explorations are needed for adequately characterizing the complex fractured networks. As we discussed in the above section, fractal dimension is one of the most important parameters to quantitatively characterize the complexity of fractures. However, fractal dimension is sensitive to prediction methods; even some irrational values may be obtained (Cai et al. 2017). Future works also should be focused on the influence of fracture surface roughness, hydraulic gradient, and coupled thermo-hydro-mechanical-chemical processes.

#### 2.2.2 Molecular Dynamics for Shale Gas Transportation

Molecular dynamics (MD) simulation approaches recognize the fluid flow as a swarm of discrete particles and is suitable for flow simulation with high *Kn* number. It is often seen as an accurate approach due to the deterministic (Hadjiconstantinou 2006) or probabilistic (Heer 2012) calculation of the particle properties at every time step (Bird 1994; Koplik and Banavar 1995; Hadjiconstantinou 2006). These properties include particle inertia, position and state. Boltzmann distribution is often used to describe individual particle dynamics at different temperatures. Newton’s equation of motion is integrated numerically to determine the two-body potential energy and transient evaluation of two particles and then to find the particle positions.

Five high-citing papers of MD simulation of shale gas reservoirs

Authors | Year | Interest | References | Cited by |
---|---|---|---|---|

Sharma et al. | 2015 | Adsorption/diffusion | Sharma et al. (2015) | 35 |

Zhehui et al. | 2015 | Molecular velocity in nanopores | Jin and Firoozabadi (2015) | 22 |

HengAn et al. | 2015 | Adsorption/displacement | Wu et al. (2015) | 30 |

Mahnaz et al. | 2014 | Pore size distribution | Firouzi et al. (2014) | 49 |

Quanzi et al. | 2015 | Enhanced recovery | Yuan et al. (2015) | 26 |

It should be noted that modern computation capability, represented by supercomputers, is still not enough to handle a reasonable, practical and very detailed flow simulation through nanotubes network in time and space scale of the real production process in shale gas reservoir. Although MD models are designed to capture microscopic interactions, which are the foundation of macroscopic phenomena, time steps are generally strictly limited to femtoseconds (\(10^{-15}\) s), which results in the limitation of simulation timescale generally ranging from picoseconds (\(10^{-12}\) s) to nanoseconds (\(10^{-9}\) s) (Hadjiconstantinou 2006)

#### 2.2.3 Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has been proved to be a useful and efficient approach to study the shale gas reservoirs (Zhang et al. 2014; Chen et al. 2015). Knudsen diffusion has already been incorporated in the general LBM flow models to describe transport properties of shale gas fluid flows (Zhang et al. 2014) For multiphase flow, the famous Shan–Chen model of single-component multiphase flow is commonly used (Chen et al. 2015)

The first attempt to take Knudsen diffusion into account of the fluid flow using LBM simulation approach is said to be in Chen et al. (2015). In their study, compared to commonly used shale tortuosity, which is an important component of Bruggeman equation, the improved model will lead to a much higher tortuosity result and consequently the intrinsic permeability is said to be extremely lower (Chen et al. 2015). For relative permeability, it is found that the countercurrent relative permeabilities, as a function of wetting saturation, usually seem smaller than the concurrent ones with a Lattice Boltzmann scheme derived for two-phase steady-state flow (Huang and Lu 2009).

*Kn*value. Permeability is increased as the result of velocity enhancement caused by slippage effect on pore walls. Adsorptive and cohesive forces among particles in gas fluid flow are used to simulate molecular-level interactions accounting with LBM scheme in Fathi and Akkutlu (2012). With slip boundary condition of Langmuir type at organic pore walls, mass transport along the tube walls is partitioned into two components: hopping of adsorbed gas molecules and slippage of free gas molecules. Hopping is the process of surface transport. In Table 4, we listed five recent papers with high citing rates relevant to lattice Boltzmann simulation of shale gas reservoirs. The citations of each paper are searched from Web of Science Core Database.

Five high-citing papers of LBM simulation of shale gas reservoirs

Authors | Year | Interest | References | Cited by |
---|---|---|---|---|

Chen et al. | 2015 | Knudsen diffusion | Chen et al. (2015) | 70 |

Fathi et al. | 2012 | Slippage and hopping | Fathi and Akkutlu (2012) | 24 |

XIaoling et al. | 2014 | Apparent permeability | Zhang et al. (2014) | 31 |

Ebrahim et al. | 2012 | Klinkenberg effect | Fathi et al. (2012) | 58 |

Song et al. | 2015 | Gas flow rate | Song et al. (2015) | 13 |

Previous researches have shown that approaches belonging to Lattice Boltzmann scheme are still limited in the application of rapidly recovering the imaging of pore structure and furthermore in the simulation and visualization of fluid flow in porous media, especially less effective in three dimensions. Pore network models, which is also a mesoscopic approach, are capable of simplifying detailed large-scale pore structures into a readable network constituting of pore bodies connected by pore throats (Ma et al. 2014; Cao et al. 2015; Huang et al. 2016; Wua et al. 2015). Each pore body is associated with different numbers of attributes, which are called coordinate numbers, and the spatial location is then specified explicitly. In this way, the highly irregular porous space is reduced to a network with topology and geometry easily captured (Zhang et al. 2015b).

### 2.3 Apparent Permeability Correction

*v*is assumed to be determined by global permeability

*k*and the pressure gradient \(\nabla p\) across the media

*k*is a macroscopic parameter defined to describe the relation between gas flow and pore structure. Same as many other classical macroscopic theories, Darcy’s law was first concluded from experiments conducted by Darcy (2007). It is proved that Darcy’s law can also be derived from Navier–Stokes equation as a simplification and extension in porous media (Li et al. 2015).

However, the long history research of shale gas reservoir has brought insights of special percolation characteristics and flow mechanisms in the tight rock structures. The original Darcy’s equation is no longer capable of explaining these phenomena. A strict limitation of flow velocity is found in the application of classical Darcy’s law. For highly fractured reservoir structures, gas flow is at relatively high velocity and the original Darcy’s law will lead to misleading results, sometimes with an overprediction of productivity as much as \(100\%\) (Nguyen 1986). To facilitate the inclusion of this phenomenon into reservoir simulators, many multipliers are generated to correlate the apparent permeability to the absolute permeability in different flow regimes.

*ka*to the absolute permeability \(k_{\infty }\) via

*b*is the Klinkenberg factor and

*p*represents average pressure across the core. The Klinkenberg factor is usually obtained by matching experimental data. Klinkenberg’s correction can be applied in the low Knudsen number range (\(<0.1\)); therefore, it is widely adopted for simulating low permeability gas reservoirs. The Klinkenberg factor is often calculated by a function of the absolute permeability and the rock porosity. Different expressions of Klinkenberg factor

*b*can be found in Jones and Owens (1980), Sampath and Keighin (1982), Civan (2010).

*Kn*, and many other correlations have been developed based on it:

Comparison of different apparent permeability correction models

Model | Equation | Regime | Ref |
---|---|---|---|

Klinkenberg | \(k_\mathrm{a}=\left( 1+\frac{b}{p}\right) k_{\infty }\) | Low Knudsen number | Klinkenberg (1941) |

Beskok | \(f(Kn)=(1+\alpha Kn)\left( 1+\frac{4Kn}{1-bKn.}\right) \) | Transitional flow | Beskok and Karniadakis (1999) |

Pour | \(f\left( K{n}\right) ={\left\{ \begin{array}{ll} 1+5K{n} &{} \\ 0.8453+5.4576K_{n}+0.1633K_{n}^{2} &{} \end{array}\right. }\) | Transitional flow | Sakhaee-Pour and Bryant (2012) |

Sun | \(k_\mathrm{a}=\left( 1+\frac{b_{\alpha }}{p}\right) k_{\infty }\) | free molecular flow | Sun et al. (2015) |

He | \(k_\mathrm{app}=\frac{Fr^{2}}{8}+\frac{\mu D_\mathrm{T}}{p}+\frac{\mu D_{s}\varsigma _\mathrm{ms}RTC_\mathrm{s}}{p^{2}}\) | free molecular flow | He et al. (2017) |

*f*(

*Kn*) is a flow condition function given as a function of the Knudsen number

*Kn*, the dimensionless rarefaction coefficient \(\alpha \) and the slip coefficient

*b*, which is an empirical parameter by:

*u*is gas molecules thermal velocity,

*R*is the gas constant, and \(M_\mathrm{A}\) is the gas molecular weight. The derivation of Knudsen diffusion coefficient can be found in Pollard and Present (1948). By rearranging the above equation, we can get an apparent gas permeability formulation similar to the Klinkenberg’s correlation:

### 2.4 Improved Darcy Model in Fractures

*k*to introduce the nonlinearity. The improved model with the two coefficients can be written as

The Barree and Conway model (BCM) has been widely applied in modern petroleum industry as a basic mathematical model of shale gas reservoir simulator. A 3D single-phase fluid flow scheme is derived according to Forchheimer and BCM equations to simulate pressure transient analysis in fractured reservoirs (Al-Otaibi and Wu 2011). On combining both the equations, an equivalent non-Darcy flow coefficient can be calculated to describe all non-Darcy flow phenomena coupling with near-wellbore effects. Besides, the BCM has already been extended to model the multiphase flow in porous media, which is widely used in practical shale gas reservoir simulator (Wu et al. 2011).

## 3 Gas Flow Simulation at Reservoir Scale

### 3.1 Flow Models

Shale is generally viewed as sediments with very fine grains and obvious fissility (Javadpour 2009). The porous media, constituting of pores with diameters ranging from nanometer to micrometer, are classified into inter-particle and intra-particle pores. The intra-particle pores are associated with organic matter pores within kerogen and mineral particles (Loucks et al. 2012).

Different approaches have been developed to capture the shale properties in reservoir simulations. Multiple interacting continua (MINC) and explicit fracture modeling are proposed to generate an efficient scheme with single porosity for shale gas simulation (Wu et al. 2009). A coarse-grid model incorporating numerical dynamic skin factor is presented for shale reservoirs with hydraulic fractures (Ding et al. 2013). To handle practical well performance in a long term as well as common transient behavior, the coarse-grid model is improved to better describe the fractures and wells (Ding et al. 2014).

*K*is the media permeability, \(q_\mathrm{a}\) is the mass of gas adsorbed on unit volume of media, and \(\varphi \) is the porosity of the porous media. The first term on the left-hand side of above equation represents the Fickian diffusion flux, and the second term represents the Darcy flow flux. On the right-hand side, the first term refers to the compressed gas in all the grids and the second term refers to the accumulation of desorbed gas in organic grid blocks.

### 3.2 Flow Model Coupled with Geomechanics

Geomechanics is of critical importance to be incorporated in the reservoir simulation to better describe the underground pressure and velocity distribution. An efficient and reliable prediction on hydrocarbon production is closely relevant to an accurate description of rock physics which might be changed by field operations.

*R*is the gas constant,

*T*is the absolute temperature, \(\alpha \) is the Biot’s coefficient, \(\sigma _{0}\)is the initial total stress tensor, \(\lambda \) is the first Lame’s constant, \(\mu \) is the second Lame’s constant, and \(tr\left( \varepsilon \right) \) is the trace of strain tensor.

Necessary properties needed for production workflow, like pressure and deformation process, can be better provided from the coupling models (Gupta et al. 2012). It is found in previous research (Yu et al. 2014) that gas production will be overestimated if geomechanics is not incorporated in the flow models. The production rate in naturally fractured reservoirs is proved to be highly sensitive to fracture aperture changes (Moradi et al. 2017). With the introduction of stress sensitivity, well production will be reduced. The effect of total organic carbon (TOC) on gas production is studied with a model coupling geomechanics and flow, and the cumulative production is said to be increased if TOC is larger (An et al. 2017).

Generally, only linear elasticity is considered in geomechanics numerical model, which leads to the disability of recovering nonlinear elastic behaviors caused by hydrocarbon depletion and stress changes in shale gas reservoirs (An et al. 2017; Shovkun and Espinoza 2017). To handle this problem, an enhanced coupling model is proposed recently to consider nonlinear elasticity (Wei et al. 2018). It is found that as rocks are being compacted and consolidated during the production process, permeability values are quite different and meet experiment data better than linear elasticity models on samples obtained from the Longmaxi formation in China. It is indicated that permeability will be overestimated by 1.6 to 53 times if nonlinear elasticity is not considered.

## 4 Macroscopic Numerical Simulation Approaches

Analytical methods are not capable of solving the mathematical formulas constituting flow models of shale gas reservoirs. As a result, numerical methods are strongly needed to solve the model. In petroleum industry, numerical simulations can go back to the 1950s, and now they have been applied in a wide range of complex fluid flow processes. Except for microscopic and mesoscopic approaches discussed in Sect. 2, macroscopic approaches are also common methods to provide numerical solutions of fluid flow in shale gas reservoirs. Due to the long history of the application, some macroscopic approaches, like finite difference method (FDM) and finite element method (FEM), are more commonly used and well developed. Recently, efforts have also been paid on other methods including finite volume method (FVM) and fast matching method(FMM).

### 4.1 Finite Difference Method

In reservoir simulation, and even larger scale of flow simulation, FDM is always viewed as the most commonly used and best developed method. Discretization of ordinary and partial differential equations modeling flow in reservoirs is the first procedure in the technique. Afterward, a finite difference grid should be constructed on the simulated reservoir area and the method implementation is conducted on the grids. For boundary conditions, pressure information is commonly used at each boundary point at the block (Shabro et al. 2012). It is found that the accuracy of numerical results using finite difference methods is deeply relevant to the grid division and boundary conditions (Nacul et al. 1990). Truncations on Taylor series expansion are used to solve unknown velocity and pressure distribution with spatial derivatives (Ertekin et al. 2001).

The main advantage of FDM over FEM is the efficiency and simplicity. Rectangular and triangular grids, uniform and non-uniform meshes, Cartesian and curvilinear coordinates have all been proved to be easy to implement in reservoir simulations extended from 1D to 3D. Especially for 3D complex flow problems, FDM is said to be far superior, although problems like numerical dispersion and grid dependence may occur (Firoozabadi 1999).

### 4.2 Finite Element Method

Compared to FDM, FEM is said to be more accurate in reservoir simulations. Opposed to piecewise constant approximation, FDM results in a linear approximation solution (Jayakumar et al. 2011). Besides, the flexibility of accommodations to unstructured meshes is demonstrated in studies using FEM. As a result, FEM is more capable of describing flow properties in complex porous structures in reservoir geometry from fracture to matrix, and excellent efficiency could still be preserved (Logan et al. 1985; Jayakumar et al. 2011).

Upscaling or homogenization techniques are widely used in traditional researches to develop effective parameters that represent subscale behavior in an averaged sense on a coarser scale as flow modeling needs to be concerned on a wide range of spatial and temporal scales in practical reservoir simulation (Yang et al. 2018; Chung et al. 2015; Alotaibi et al. 2015; Chen et al. 2017). In an attempt to overcome some of the limitations of upscaling methods, the so-called multiscale discretization methods have been proposed over the past two decades to solve second-order elliptic equations with strongly heterogeneous coefficients (Efendiev and Hou 2009). This includes methods such as the generalized finite element methods (Babuška et al. 1994), numerical subgrid upscaling (Arbogast 2002), multiscale mixed finite element methods (Chen and Hou 2003) and mortar mixed finite element methods (Arbogast et al. 2007). The key idea of all these methods is to construct a set of prolongation operators (or basis functions) that map between unknowns associated with cells of the fine geo-cellular grid and unknowns on a coarser grid used for dynamic simulation. Over the past decade, there have primarily been main developments in this direction focusing on the multiscale mixed finite element (MsMFE) method. The main process is to make this method as geometrically flexible as possible and developing coarsening strategies that semiautomatically adapt to barriers, channels, faults and wells in a way that ensures good accuracy for a chosen level of coarsening. In order to produce high-quality approximate solutions for complex industry standard grids with high aspect ratios and unstructured connections, a new multiscale formulation has been presented recently (Møyner and Lie 2016), which could guarantee the robustness, accuracy, flexibility as well as simplification on the implementation. Besides, many works have been done on the weighted Jacobi smoothing on interpolation operators with a large degree of success in the algebraic multigrid (AMG) community where fast coarsening is combined with simple operators constructed via one or two smoothing steps (Vanek et al. 1994; Vaněk et al. 1996; Brezina et al. 2005) as an inexpensive alternative to the interpolation operators used in standard AMG (Stüben 2001). Many high-performance multigrid solvers have been proposed to support smoothed aggregation as a strategy for large, complex problems (Gee et al. 2006) due to the inexpensive coarsening and interpolation strategies.

### 4.3 Other Methods

For reservoir simulations incorporating complex rock geometries, finite volume method (FVM) is said to be more easily implemented with unstructured grids. It is a fairly new developed technique and mainly focusing on discretization methodologies (Reis et al. 2001). It is proved that to get numerical approximations at the same level of accuracy, FVM is easier and faster compared to FEM. Compared to FDM, FVM is believed to have better versatility.

## 5 Conclusion

This paper reviews the flow mechanism and numerical simulation approaches of shale gas reservoirs. Investigation of gas adsorption/desorption is important to predict well production, and gas adsorption isotherm can be concluded into different models. With the classification of flow regimes based on Knudsen number, different governing equations and numerical approaches are suitable for different gas transport mechanisms. Microscopic and mesoscopic approaches, represented by molecular dynamics (MD) and lattice Boltzmann method (LBM), have successfully been applied in the study of shale gas mechanisms, in particular interests of studying Klinkenberg effect, Knudsen diffusion, molecular velocity and many other details of special mechanisms of shale gas flow in reservoirs. Due to the special mechanisms and percolation characteristics of shale gas transport, classical Darcy’s law should be corrected and the concept of apparent permeability is introduced. For flow at high rate, e.g., in fractures, improved Darcy’s model with high-order terms of velocity is presented to better describe the gas flow.

In reservoir-scale flow models, shale is usually classified into four types: inorganic matter, organic matrix (kerogen), natural fractures and hydraulic fractures. Various models of gas transport in shale with detailed description on fracture characterizations have been developed, including single porosity, dual porosity and many others. Besides, it is critical to incorporate general reservoir flow model with geomechanics in order to better understand the pressure and reservoir performance in hydrocarbon development. Due to the long history and better visibility, the petroleum industry is more familiar with macroscopic numerical approaches, including the finite difference method (FDM) and finite element method (FEM). Recent progress has also been made on other methods, like finite volume method (FVM) and fast matching method(FMM).

## Notes

### Acknowledgements

The research reported in this publication was supported in part by funding from King Abdullah University of Science and Technology (KAUST) through the grant BAS/1/1351-01-01. The authors are also grateful for financial support from the Beijing Nova Program under Grant No. Z171100001117081 and the Fundamental Research Funds for the Central Universities under Grant No. FRF-TP-17-001C1.

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