## Abstract

In order to investigate the influence on shale gas well productivity caused by gas transport in nanometer-size pores, a mathematical model of multi-stage fractured horizontal wells in shale gas reservoirs is built, which considers the influence of viscous flow, Knudsen diffusion, surface diffusion, and adsorption layer thickness. A discrete-fracture model is used to simplify the fracture modeling, and a finite element method is applied to solve the model. The numerical simulation results indicate that with a decrease in the intrinsic matrix permeability, Knudsen diffusion and surface diffusion contributions to production become large and cannot be ignored.

#### Authors:

##### Wei Wang^{1}, Jun Yao^{1}, Hai Sun^{1}, Wen-Hui Song^{1}

^{1}School of Petroleum Engineering, China University of Petroleum, Qingdao 266580, Shandong, China.

Received: 11 May 2015, Published online: 19 September 2015

©The Author(s) 2015.

The existence of an adsorption layer on the nanopore surfaces reduces the effective pore radius and the effective porosity, resulting in low production from fractured horizontal wells. With a decrease in the pore radius, considering the adsorption layer, the production reduction rate increases. When the pore radius is less than 10 nm, because of the combined impacts of Knudsen diffusion, surface diffusion, and adsorption layers, the production of multi-stage fractured horizontal wells increases with a decrease in the pore pressure. When the pore pressure is lower than 30 MPa, the rate of production increase becomes larger with a decrease in pore pressure.

## 1 Introduction

With the increasing demand for world energy and the improvement of oil and gas exploitation technology, unconventional oil, and gas resources such as coal-bed methane, tight sandstone gas, shale oil, and gas have been paid more and more attention, especially for shale gas which has achieved commercial production (Wu et al. 2013).

Compared with conventional reservoirs, the pore size in shale gas reservoirs is nano-scale (Javadpour et al. 2007; Loucks et al. 2009; Clarkson et al. 2013), which results in low porosity and ultra-low permeability. The matrix porosity generally is lower than 0.1, and the matrix permeability ranges from 1 nanoDarcy to 1 microDarcy (Wang et al. 2009). The shale formation acts as both the source rock and the reservoir rock, so adsorbed gas and free gas coexist in shale reservoirs (Yao et al. 2013a), where free gas is stored in the matrix pore space, and adsorbed gas can make up to 20 %–85 % of the total gas reserve (Hill and Nelson 2000). It is different pore sizes and different gas storage patterns in shales that makes the mechanisms of shale gas transport in nanopores extremely complicated. These include viscous flow, Knudsen diffusion, and molecular diffusion (which only happens when the gas is multi-component) (Bird et al. 2007). If adsorbed gas exists in porous media, surface diffusion, adsorption, and desorption should also be considered amongst the shale gas transport mechanisms (Ho and Webb 2006; Akkutlu and Fathi 2012).

Because of nano-scale porous media in shale gas reservoirs, the traditional Darcy’s law cannot accurately describe the mechanism of gas transport. The existence of nanopores in shale reservoirs makes the gas slippage effect more apparent (Swami et al. 2012). However, the Klinkenberg model only applies to low-pressure gas (Klinkenberg 1941). Based on the model of micro-nano pipes, Beskok and Karniadakis put forward a volumetric flow rate formula with different flow regimes (Beskok and Karniadakis 1999). The Beskok model that describes gas flow in a single tube has been applied to research into tight gas and shale gas flow by Civan et al. in which the Knudsen number is used to describe the transport mechanism that incorporates viscous flow and Knudsen diffusion (Civan 2010; Civan et al. 2010, 2011).

Javadpour (2009) considered that Darcy’s law and Fick’s law can describe gas transport mechanisms in micropores, and Knudsen diffusion is the main gas transport mechanism in nanopores. Thus, the Javadpour model which takes viscous flow and Knudsen diffusion into account in nanopores was established (Javadpour 2009), and used the pore radius to characterize gas transport in nanopores. Yao et al. studied matrix viscous flow, Knudsen diffusion, molecular diffusion, and adsorption and desorption based on the double porosity model and used the finite element method (FEM) for numerical simulation of vertical shale gas well production (Yao et al. 2013a, b).

Because of the nano-scale effect, the thickness of the adsorption layer greatly affects the matrix porosity and permeability (Xiong et al. 2012). Meanwhile, the thickness of the adsorption layer is a function of pressure (Sakhaee-Pour et al. 2011). Due to the existence of concentration gradients, adsorbed gas itself undergoes surface diffusion (Sheng et al. 2014), especially in nano-scale porous media, where surface diffusion is an important transport mechanism. In ultra-tight nano-scale porous media, surface diffusion even dominates the gas transport mechanisms (Sheng et al. 2014; Etminan et al. 2014; Mi et al. 2014; Ren et al. 2015).

Based on previous research, in this paper, we overall consider the shale matrix micro-nano-scale effect, which includes the influence of viscous flow, Knudsen diffusion, surface diffusion, and adsorption layer thickness and build a mathematical model of multi-stage fractured horizontal wells in shale gas reservoirs. Furthermore, the discrete-fracture model (DFM) is used to simplify the fracture description, and the FEM is applied to solve the model. Finally, the influences of different transport mechanisms on shale gas reservoir production are studied, and a parameter sensitivity analysis is used to investigate the change on production and the transport mechanism contribution to well production.

## 2 Shale gas transport mechanisms in the reservoir matrix

Gas transport in shale nanopores consists of several transport mechanisms, as shown in Fig. 1 (Sun et al. 2015). Molecular diffusion is caused by collision between different component gas molecules. For a single gas species, collision between molecules results in viscous flow, and Knudsen diffusion is generated from collision between molecules and the pore walls, while surface diffusion happens when adsorbed gas molecules creep along the pore surface.

**Fig. 1** Single-component gas transport in porous media.

Considering that there only exists a single-component, methane gas, in the shale gas reservoir and free gas coexists with adsorbed gas in the shale matrix, gas transport mechanisms in the shale matrix is determined by mutual effect of viscous flow, Knudsen diffusion and adsorption layer surface diffusion.

### 2.1 Viscous flow

When the mean-free path of gas molecules is very small compared to the pore diameters, the probability of collisions between molecules is much higher than collisions between molecules and pore walls; thus, single-component gas transport is mainly governed by viscous flow caused by the pressure gradient. Viscous flow can be modeled by Darcy’s law (Kast and Hohenthanner 2000):

where *N*_{v} is the mass flux of viscous flow, kg/(m^{2} s); *k*_{∞} is the intrinsic permeability of the porous media, m^{2}; *p*_{m} is the matrix gas pressure, Pa; *ρ*_{m} is the matrix gas density, kg/m^{3}; and *μ*_{m} is the matrix gas viscosity, Pa s.

### 2.2 Knudsen diffusion

When the pore space is so narrow that the mean-free path of gas molecules is very close to the pore diameter, collisions between molecules and pore walls dominate. Knudsen diffusion can be expressed as (Florence et al. 2007)

*C*_{m} can be given by *C*_{m} and *ρ*_{m} is obtained by *ρ*_{m}=*ρ*_{m}/*M*_{g}=(*p*_{m}*M*_{g}*)/(**ZRT)*

where *N*_{k} is the mass flux of Knudsen diffusion, kg/(m^{2} s); *M*_{g} is the molecular weight of gas, kg/mol; *D*_{k} is the Knudsen diffusivity, m^{2}/s; *C*_{m} is the concentration of free gas in the porous media, mol/m^{3}; *Z* is the gas compressibility factor; *R* is the ideal gas constant, 8.314 J/(mol K); *T* is the gas reservoir temperature, K; *ϕ*_{m} is the shale matrix porosity; and *c* is a constant close to 1. In this paper, we set *c* = 1.

### 2.3 Adsorption and desorption

Shale gas adsorbed on the surfaces of nanopores follows the mono-layer Langmuir isotherm adsorption equation (Civan et al. 2011):

where *q*_{ads} is the mass of gas adsorbed per solid volume, kg/m^{3}; *ρ*_{s} denotes the shale matrix density, kg/m^{3}; *V*_{std} is the molar volume of gas at standard temperature (273.15 K) and pressure (101,325 Pa), std m^{3}/mol; *V*_{L} is the Langmuir gas volume, std m^{3}/kg; *p*_{L} is the Langmuir gas pressure, Pa.

The Langmuir isotherm adsorption is based on a mono-layer model, and the adsorbed layer makes the effective diameter of the nanopore decrease. The modified effective pore radius caused by the mono-layer adsorption on the pore surface can be written as (Sun et al. 2015)

where *r*_{eff} is the effective pore radius, m; *r* is the pore radius, m; *d*_{m} is the diameter of a methane molecule, m.

The decrease in the pore diameter leads to a reduction in porosity. Thus, the effective porosity can be expressed as

By combining Eqs. (6) and (7), the effective intrinsic permeability that takes the adsorption layer thickness into account can be given by

where *k*_{∞eff} is the effective intrinsic permeability of the shale matrix, m^{2}; *τ* is the tortuosity.

### 2.4 Surface diffusion

Surface diffusion only occurs in porous media where the gas is adsorbed onto the pore wall, and can be expressed as

where *N*_{s} is the mass flux of surface diffusion, kg/(m^{2} s); *D*_{s} is the surface diffusivity, m^{2}/s; *C*_{s} is the concentration of adsorbed gas, mol/m^{3}. Gas adsorbed on pore surfaces follows the Langmuir isotherm adsorption and can be expressed as (Xiong et al. 2012)

*C*

_{s max}is the maximum adsorbent concentration, mol/m

^{3}.

## 3 Establishment and solution of the fractured horizontal well mathematical model in shale gas reservoirs

### 3.1 Transport equation for the matrix system

Mass transport in nano-scale porous media is the concurrent result of viscous flow, Knudsen diffusion, surface diffusion, and gas desorption. The advective–diffusive model (ADM) and dusty gas model (DGM) are generally used to incorporate the coupling mechanisms. Although DGM considers coupling effect between viscous flow and diffusion more comprehensively, the transport equation built by ADM and DGM is the same for a single-component gas (Yao et al. 2013c). Thus, ADM is applied to build the transport equation for the shale matrix:

k_{m,app} is the apparent permeability of the shale matrix; bm is the Klinkenberg coefficient that considers Knudsen diffusion b_{m}=(D_{k}μ_{k}*)*/m_{∞}.

### 3.2 Continuity equation for the matrix system

We assume that there are no natural fractures in the gas reservoir and the gas is only single-component methane. Adsorbed gas and free gas coexist in the shale matrix, and the reservoir temperature is assumed to stay constant during production. Gas adsorbed on the matrix surface conforms to the Langmuir isotherm adsorption equation. According to the mass conservation law, the continuity equation for a single-porosity matrix system can be obtained:

where *N*_{t,m} is the mass flux, kg/(m^{2} s); *q*_{ads} is the mass of gas adsorbed per solid volume, kg/m^{3}; *q*_{m} is the source sink term, kg/s; *δ*(*M* − *M*′) is the delta function which is equal to zero at all points except point *M*′, *δ*(*M* − *M*′) = 1.

### 3.3 Mathematical model for the matrix system

Assuming that the pressure is equal on the boundary of hydraulic fractures and the matrix. Г_{1} is the outer boundary of the gas reservoir, Г_{2} is the inner boundary of the production well, and Г_{3} is the boundary between hydraulic fractures and the matrix. In this study, we assume that the outer boundary Г_{1} is sealed, the inner boundary Г_{2} is under a constant pressure, and the pressure is equal across boundary Г_{3}. Substituting Eqs. (5) and (11) into Eq. (13), the mathematical model for the single-porosity matrix can be derived:

where γ=M_{g}/(ZRT); pi is the initial pressure in the gas reservoir, P_{a}; p_{w} is the wellbore pressure, Pa; and p_{F} is the pressure in fractures, Pa.

### 3.4 Mathematical model for the hydraulic fracture system

We suppose that only free gas exists in artificial fractures and that the gas obeys Darcy’s law within the fracture. The pressure is equal across the boundary of hydraulic fractures and the matrix. Thus, the mathematical model for the hydraulic fracture can be represented as

where *ρ*_{F} is the gas density within the fracture, kg/m^{3}; *ϕ*F is the porosity of the hydraulic fracture, which equals zero when there are no proppants within the fracture; *μ*_{F} is the gas viscosity, Pa s; *k*_{F} is the fracture permeability, m^{2}, which can be calculated by the following equations on condition of regular fractures:

where *h*_{F} is the fracture aperture, m.