Simulation of a Multistage Fractured Horizontal Well with Finite

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Simulation of A Multistage Fractured Horizontal well with Finite Conductivity in Composite Shale Gas Reservoir through Finite-element Method Ruihan Zhang, Lie-Hui Zhang, Ruihe Wang, Yu-Long Zhao, and Rui Huang Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.6b01565 • Publication Date (Web): 10 Oct 2016 Downloaded from http://pubs.acs.org on October 12, 2016

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Energy & Fuels

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Simulation of A Multistage Fractured Horizontal well with Finite Conductivity in Composite Shale Gas Reservoir through Finite-element Method

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Rui-han Zhang1∗, Lie-hui Zhang1, Rui-he Wang2, Yu-long Zhao1*, Rui Huang2

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The State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500

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2. China National Oil and Gas Exploration and Development Corporation, Beijing, 00724

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Abstract: Different from oil properties, gas properties (gas formation factor, viscosity,

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z-factor etc.) have nonlinear behaviors with pressure changes. However, many scholars use

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the average pressure or pseudo pressure concept to simplify the phenomenon for easier

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solutions. Gas flow in shales is believed to be a complex process with multiple flow

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mechanisms including continuum flow, slip flow, diffusion, ad-desorption, and the

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stress-sensitivity of fractures (natural or induced) permeability in multi-scaled systems of

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nano- to macro- porosity. Multistage hydraulic fracturing not only creates the stimulated rock

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volume (SRV) to improve production, but also makes the flow in shales more complex. In this

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paper, a rectangular composite model for a multistage fractured horizontal well (MFHW) with

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finite conductivity in shale gas considering the multiple flow mechanisms and

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multi-nonlinearities is developed. Comparing with the existing models for MFHW in shale,

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the model presented here takes strong nonlinearity of gas properties, hydraulic fracture

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asymmetry, fracturing efficiency and SRV region into account, which is more in line with

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field practice. Numerical simulation of fully implicit control volume finite element (CVFE)

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based on unstructured 3D tetrahedral mesh is proposed to obtain the production performance

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of MFHW. Sensitivity analysis focuses on the effects of nonlinearity, Langmuir volume,

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stress-sensitivity, finite conductivity and SRV type on the production performance. The

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research and the numerical results obtained in this paper can provide theoretical guidance to

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efficient and scale development for shale gas reservoir.

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Key words: Composite shale gas reservoir; Multi-nonlinearity; Multiple flow mechanisms;

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CVFE method; Stimulated reservoir volume (SRV)

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1. Introduction

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At present, countries around the world are actively responding to global climate change.

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As gas being a kind of clean energy, gas reservoir especially unconventional gas reservoir



Corresponding Author: Rui-han Zhang Email: [email protected] Yu-long Zhao Email: [email protected]. 1

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such as shale gas is gradually becoming a key component in the world's energy supply.1, 2 In

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the exploration and development of gas reservoir, gas properties (gas formation factor,

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viscosity, z-factor etc.) behave nonlinearly with pressure changes. 3 These characteristics have

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great impacts on gas production performance, but also bring challenges for researchers to

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obtain analytical solutions.4, 5

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Laboratory results of the shale matrix show that the average pore radius range of

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nanopores is between 1 and 100nm, conventional Darcy flow cannot adequately describe the

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various non-viscous gas flow regimes that may be present.6, 7 Although many researchers used

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microscopic experiments and molecular dynamics simulations to describe the flow in

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nanopores,8, 9 the much difference of space scale and flow mechanisms makes it unable to

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combine the molecular simulations on nano scale flow and commercial simulators on field

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scale. In order to quantify the effects of these non-Darcy flow mechanisms, Ertekin et al.10

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and Beskok and Karniadakis11 proposed the concept of equivalent apparent permeability

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based on micro-continuum theory, which made it feasible for using the equation of motion to

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describe the multi-scale flow mechanisms in shale. After that,Javadpour et al.12 and

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Javadpour13 used a linear-integrated way to combine the gas slippage in nanopores and

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Knudsen diffusion. Roy et al.14 and Li et al.15 extended the model of Javadpour and classified

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the diffusion into three forms: Knudsen diffusion, Fick diffusion and transition diffusion

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according to the Knudsen number (Kn). However, Javadpour model is a simple linear sum

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without considering a weighted sum on the basis of corresponding contribution. 16 Another

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type of correlation model was proposed by Civan17 and Civan et al.18 The model of Civan

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took the molecular kinetic theory into account, and used the Hagen-Poiseuille equation11 to

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deduce an equivalent apparent permeability formula which could incorporate the multiple

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transport mechanisms. However, Civan model is valid only for Kn≤1.19 Moreover, dues to the

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high reservoir pressure of shale gas especially in North American and China, gas transport

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mechanisms through nanopores are obviously difference from idea gas under a low pressure

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condition.20 Accounting of real gas effect, a bulk gas transport model was proposed on the

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basis of the weighted superposition of slip flow and Knudsen diffusion.16 Unlike the

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conventional reservoirs, as reported, most of the gas is adsorbed on the surface of organic-rich

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rock while less portion is stored in pores of these rocks as free gas. Many researchers21, 22

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counted in the desorption quantity as part of total compressibility and used the Langmuir

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isothermal adsorption theory for the study.23 However, the existing models above neglected

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the Langmuir desorption expression had a strong nonlinearity with pressure changes and

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acquired analytical solution under initial formation conditions, which would result in some 2

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errors.15

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Shale gas reservoir is a kind of self-sourced and self-preserved unconventional reservoir

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with complex natural fracture network and extremely tight matrix. The permeability of shale

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rock is extremely low to 10-150 nanodarcies, field practice indicated that the only way to

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obtain economic gas production from shale gas reservoir is the fractured horizontal well

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technology.1, 13 Some existing models of MFHW in conventional reservoir were applied to

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describe the flow dynamics and mechanisms of fractured horizontal wells in shale gas

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reservoir. Al-Ahmadi et al.24 described the type curves and production decline of fractured

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horizontal wells in shale gas reservoir based on the linear flow model without considering the

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effect of desorption and diffusion. Tivayanonda and Wattenbarger25 treated the shale gas

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reservoir as tri-porosity media containing three interconnected systems: nano-micro matrix

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pore system, nature fracture system and hydraulic fractures, and obtained the pressure

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performance of MFHW in shale. Wang5 proposed a well testing model for MFH well in shale

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gas reservoir based on the tri-porosity media, which took multiple flow mechanisms, stress

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stress-sensitivity of natural fractures and hydraulic fracture orientation into account. However,

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unlike creating a few highly conductive dominant fractures in conventional reservoir,

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stimulation treatments in shale reservoir should use low viscosity fluid to activate and connect

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existing natural fractures so as to develop large fracture network system, which is defined as

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stimulated reservoir volume (SRV).26, 27

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As micro seismogram of multiple fractured horizontal well shows (see Figure 1), the

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fracture network (natural or induced) around the horizontal well is very complex. In order to

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concisely describe the fracture network, Zhang et al.22 and Zhao et al.28 assumed the MFH

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well in a circular composite reservoir where the fracture network is simplified by

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dual-porosity model29 to describe the performance of well in shale gas reservoir. However,

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these models assumed that the hydraulic fractures were infinite conductivity without

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considering the effect of conductivity on production performance. Some scholars presented

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the effective trilinear flow or extended five region flow model where the SRV is inner region

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to describe the performance of finite conductivity fractured horizontal well in rectangular

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composite shale reservoir.4, 30 These models took into account the multiple flow mechanisms

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of shale gas and were easy to solve. However, these models assumed the hydraulic fractures

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were uniform; ignored the fracturing efficiency and interference between fractures. In order to

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reflect the complete flow regime response of finite conductivity fractured horizontal well in

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rectangular composite shale reservoir, Fan et al.31 proposed a coupled dual continuum for

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composite reservoir and discrete fracture for hydraulic fracture model and used the finite 3

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element model to obtain the numerical solution. However, the inter-porosity transfer from the

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matrix to fracture system was calculated explicitly in this model; and the average pressure and

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production of grids containing well were used to substitute the actual BHP and production of

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the well without considering the well model of finite conductivity fractured horizontal well in

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numerical stimulation. Li et al.15 and Zhao et al.32 proposed a modified Peaceman’s equation33

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for finite conductivity fractured horizontal well in shale gas reservoir without considering the

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SRV.

a

b

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Figure 1. MFH well with SRV in composite shale gas reservoir. Micro seismogram and MFHW model

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with whole inner SRV region around the horizontal well (a).34 Micro seismogram and MFHW model with

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asymmetric SRV surrounding the fractures (b).32

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In view of this, this paper proposes a coupled model of dual-porosity dual-permeability

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continuum media and discrete fractures to simulate the production performance of finite

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conductivity MFH well in rectangular composite shale gas reservoir. Nonlinearity of gas

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properties under high pressure condition, multiple flow mechanisms including slip flow,

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diffusion, ad-desorption, stress sensitivity of fracture network (natural or induced), and

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stimulated reservoir volume are all taken into account. The control volume finite element

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(CVFE) method with fully implicit and sequential iterative algorithm, based on the

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unstructured 3D tetrahedral meshes, is adopted. Furthermore, the two widely applied SRV

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model shown in Figure 1: the whole inner SRV region and asymmetric dominant hydraulic

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fractures around with multiple branch fractures are compared to obtain the effective

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stimulation treatment.35 The main purpose of this paper is using our simulator to investigate 4

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the impact of key parameters on production performance and to give suggestions on shale gas

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efficient development.

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2. Mathematical model and numerical solution

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2.1 Model assumption z

y

Matrix

x Hydraulic fracture

Wellbore

Micro fracture Flow in matrix Inter-porosity flow Flow in micro fracture Flow in hydraulic fracture b

a

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Figure 2. Sketch of a coupled continuum and discrete fracture model of MFH well. A MFHW in actual

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reservoir (a). Diagram of dual-porosity dual-permeability continuum media and discrete fractures model

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(b).

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In this paper, shale gas reservoir is treated as three regions: outer region, stimulated region

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and hydraulic fractures. Each of them has distinct reservoir properties. The stimulated region

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is simplified as dual-porosity dual-permeability media including matrix, natural fractures and

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induced fractures. The outer region is a dual-porosity dual-permeability media which is not

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influenced by hydraulic fracturing. The rectangular matrix geometry chosen here is not a

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requirement; the model can also be extended to other matrix shapes such as sphere.36

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To make this mathematical model more tractable and easy to understand, the following

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assumptions and descriptions are applied:

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(1) Gas flow is isothermal and in single-phase;

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(2) Shale gas reservoir is simplified as dual-porosity dual-permeability continuum media, in

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which there are two set of reservoir properties for matrix and fracture (natural or induced)

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system respectively (see Figure 2);

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(3) Shale gas flows from matrix into fracture system at a quasisteady state and then into hydraulic fractures;37 (4) Dominant hydraulic fractures with induced fracture system are asymmetry, could have 5

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different lengths, SRVs and fracturing efficiency (see Figure 1).

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2.2 Mathematical model

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2.2.1 Nonlinearity of gas properties

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In this section, we take into account the strongly nonlinearity of gas properties with

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pressure changes, which is always simplified to be under original reservoir conditions or

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average pressure for analytical solution.5, 22 We first use the numerical approximate method

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proposed by Lee et al.38 and Dranchuk and AbouKassem39 to calculate the changes of

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viscosity µg and Z-factor under different pressures. And then the gas formation factor Bg and

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compressibility Cg are respectively calculated by eq 1 and eq 2. Moreover, for convenience,

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the cubic spline interpolation is selected to obtain the first derivatives of variation functions.

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As Figure 3 shows, gas properties are quite sensitive to pressure changes and different

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properties behave different. For instance, Bg and Cg decrease a lot with pressure increases;

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however, gas viscosity µg increases along and Z-factor decreases and then increases with

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pressure increases. Such behavior of gas properties would affect production performance

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significantly and should not be ignored.3

0.1

0.03

0.08 0.02

T=373.15K rg=0.65

0.06 0.04

0.01

0.02 0

0 0

159 160

10

20 30 Pressure, MPa

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1.4

compressibility-cg cubic spline-cg Z-factor cubic-Z

0.8

1.15

0.6 0.9 T=373.15K rg=0.65

0.4

0.65

0.2 0

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Z-factor

0.12

1 Compressibility -cg,MPa-1

0.04

gas formation factor-Bg cubic spline-Bg gas viscosity-µg cubic spline-µg

Gas viscosity-µg,cp

0.14

Volume cofficient-Bg

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0.4 0

10

a

20 30 40 Pressure, MPa

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b

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Figure 3. Parameters varitions with pressure changes. Varitions of gas formation factor and gas viscosity

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with pressure changes (a). Varitions of compressibility and Z-factor with pressure changes (b).

Bg =

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Cg =

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Z ( p ) Tpsc pTsc

1 1 dZ ( p ) − p Z ( p ) dp

(1)

(2)

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where Bg is the gas formation factor, cg is the gas compressibility, Z is the Z-factor, T is the

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temperature of reservoir. 6

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2.2.2 Multiple flow mechanisms for shale gas reservoir Flow in matrix system

b

Adsorbed gas Desorbed gas Free gas

a

c Darcy flow

Diffusion d

Slippage

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Figure 4. Multiple flow mechanisms of MFHW in shale gas reservoir. Gas flows from small scale fractures

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into hydraulic fractures (a). Gas flows from matrix to fracture system (dual continuum media model) (b).

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Gas desorbs from organic matter surface into matrix pores (c). Flow mechanisms in micro-nano pores

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include Darcy flow, slippage and diffusion (d).

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As a special type of gas reservoir, multiple types of pores such as nanopores in organic

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and inorganic matrix, small-scale natural fractures and large-scale hydraulic fractures exist in

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shale gas reservoir, which results in multiple flow mechanisms of gas flow in shale (see

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Figure 4).5 To describe the gas flow in such a multiscale porosity system, the Knudsen

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number (calculated by eqs 3-4) is used to classify the flow types.19

λ =31.6 ×

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µg

π ZRT

p

2rg M air

Kn =

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λ 2rn

(3)

(4)

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where Kn is the Knudsen number, λ is the average free path of gas, µg is the gas viscosity, R is

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the gas constant, Mair is the molecular weight of air, rg is the gas specific gravity, rn is the

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nanopore radius.

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The Knudsen number is the ratio of average free path of gas and matrix nanopore diameter, 7

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which is a physical parameter characterizing the proportion relationship between gas

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intermolecular collision and gas molecules-walls collision in nanopores.40 As shown in Figure

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5, when Kn is less than 0.001, average free path of gas being much bigger than nanopore

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diameter, the gas intermolecular collisions dominate the gas flow, and can be modeled by

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Darcy’s law; as pressure or pore size decreases, the gas intermolecular collisions decrease,

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while the molecule/wall collisions increase, and gas molecules slip on the wall. When

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0.001