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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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1.
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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
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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
40
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
50
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
