Estimation of shale apparent permeability for multi-mechanistic multi

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Estimation of shale apparent permeability for multi-mechanistic multi-component gas production using rate transient analysis Erfan Mohagheghian, Hassan Hassanzadeh, and Zhangxin Chen Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b04159 • Publication Date (Web): 01 Feb 2019 Downloaded from http://pubs.acs.org on February 12, 2019

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Energy & Fuels

1

Estimation of shale apparent permeability for multi-mechanistic

2

multi-component gas production using rate transient analysis

3

Erfan Mohagheghian, Hassan Hassanzadeh*, Zhangxin Chen

4 5 6 7 8

Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada

9

Gas-producing shale and ultra-tight reservoirs are playing a key role in the energy industry and the

10

global gas market. Compositional simulation of gas production from shale media in the presence

11

of different mechanisms such as viscous flow, slip flow (Klinkenberg effect), Knudsen diffusion,

12

sorption, pore radius variation and real gas effect is a computational challenge. In this work, we

13

present a model that takes into account all the noted mechanisms of gas transport in shale media.

14

It is shown that the compositional effect of gas in shale media can be lumped into a single

15

component by introducing an apparent gas permeability, which can be estimated from the

16

conventional rate transient analysis. The main contribution of this study is a workflow

17

incorporating the relevant physics into a single term (apparent permeability) which will substitute

18

Darcy permeability. This procedure reduces the simulation runtime substantially and will find

19

applications in reservoir characterization and simulation of production from shale gas reservoirs.

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Keywords: Shale gas; Apparent gas permeability; Sorption; Knudsen diffusion; Slip flow

Abstract

21 22

*

Corresponding author: [email protected]

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1

Introduction

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As production from conventional gas resources declines, unconventional shale and ultra-tight gas

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reservoirs have attracted a great deal of attention, especially in the United States 1, 2. The promising

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future of these reservoirs is caused by recent advances in horizontal drilling and hydraulic

27

fracturing technology 3. Different transport mechanisms involved in shale gas production make the

28

gas system unique and complex. A range of 1 to 200 nm for the pore size and permeability in the

29

order of nanoDarcys 4-6 invalidate the applications of Darcy’s law and the Navier-Stokes equations

30

7, 8.

31

Viscous flow of the compressed free gas provides a small portion of the production, whereas

32

adsorbed and dissolved gas in the kerogen are the two main contributors to the flow

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Desorption, which is pronounced after pressure is declined below a critical sorption pressure, could

34

be the source of up to half of the total gas production 12. The mean free path of gas molecules is

35

almost in the same order of magnitude as the size of the pores, resulting in a high Knudsen number

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flow 13. Thus, Knudsen diffusion is not negligible, especially at lower pressures and smaller pore

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radii 4. The no-slip boundary condition is no longer valid 8; and pore enlargement due to pressure

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depletion would also play a role in gas production. The above mechanisms in shale media lead to

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gas production rates higher than the values predicted by continuum models.

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Many attempts have been made to model the apparent gas permeability of shale reservoirs and

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lump the effects of different transport mechanisms in one expression, which can substitute for

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intrinsic (Darcy) permeability. Most of these models are either correlations, which are derived

43

empirically based on matrix permeability, or use ideal gas law in a set of capillary tubes

44

Civan used the single pipe model presented by Beskok and Karniadakis 17 and developed a simple

9-11.

5, 14-16.


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Hagen-Poiseuille type correlation including a second-order approximation for slip flow; however,

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it involves using parameters such as rarefaction coefficient and specific grain surface which have

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to be determined experimentally 15. Recently, simple analytical models have been presented, the

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majority of which do not include all the contributing transport mechanisms. Apparent permeability

49

function (APF) and nonempirical apparent permeability (NAP) are well-known examples of this

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category. APF includes the effects of slip flow and Knudsen diffusion, however, ignoring

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desorption 2. NAP, which is derived based on the Navier-Stokes equations and contains no

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empirical parameters, takes into account the effects of Knudsen diffusion and sorption, while

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overlooking slip flow. Surface diffusion, pore surface roughness and mineralogy are deemed to

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have negligible effects on shale gas production 11.

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Molecular dynamics (MD) and lattice Boltzmann methods (LBM) are the most reliable approaches

56

to describe flow in shale reservoirs

57

restrictions on their use

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been fairly successful in shale gas flow modeling although most of them are complex and/or ignore

59

one or more of the transport mechanisms

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dimensional linear numerical model to study the effect of fluid flow processes (Knudsen diffusion,

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viscous and slip flow) on the change in the composition of the produced gas. Different components

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are produced at different rates depending on their physicochemical properties leading to

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chromatographic separation in the shale porous medium. The effects of sorption and pore radius

64

change were, however, ignored 26. A brief tabulated survey of most well-known models can be

65

found elsewhere 21.

66

Conventional well test analysis methods are derived based on Darcy’s law, hence are not

67

appropriate means to accurately estimate shale gas reservoir parameters

21.

18-20;

however, high computational costs impose major

On the other hand, computationally cheaper numerical models have

22-25.

For example, Rezaveisi et al. developed a one-

27.

The mechanisms 3

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contributing to shale gas production cause the apparent permeability of shale reservoirs to be

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higher than Darcy (liquid equivalent) permeability. Decline curve analysis and empirical

70

correlations are the classical tools of production data analysis 28. The decline equations have been

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extended and type curves have been generated using dimensionless variables 29. Rate transient data

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analysis is an efficient tool to reduce the uncertainty of gas production evaluation and is widely

73

used to estimate long-term production from shale gas reservoirs producing under constant wellbore

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pressure

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shale gas reservoirs, has been introduced to evaluate parameters such as permeability 30. Later on,

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several authors tried to improve the accuracy of predictions by including and/or modifying pseudo

77

variables 31, 32.

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As noted earlier, several mechanisms play role in gas transport through shale media. Developing

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a comprehensive model, which can capture the effects of all the contributing mechanisms,

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especially at field scale is a complicated task and the computational demand is high. The purpose

81

of this study is to develop a model which accounts for the role of all significant mechanisms

82

contributing to multi-component shale gas production. The numerical reservoir model presented

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in the later sections includes the effect of viscous flow, slip flow (Klinkenberg effect), Knudsen

84

diffusion, sorption, pore radius change and real gas effect on shale gas production.

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In the following, the model including the governing equations and boundary conditions as well as

86

reservoir input properties are described. The developed numerical model is then used to find an

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apparent gas permeability based on the rate transient data analysis. The apparent permeability,

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which theoretically lumps the effects of different transport mechanisms, reduces the computational

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time substantially, while keeps the accuracy of predictions within an acceptable range compared

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to the simulation results where all mechanisms are accounted for individually. The model actually

30.

The square-root-of-time plot for linear flow, which is the dominant flow regime in


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represents a shale matrix block producing multi-component gas from the adjacent fracture. In the

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presence of experimental gas production data from shale core plugs, the model can be applied to

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obtain gas apparent permeability, which can replace Darcy permeability and lump the effects of

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all the involved mechanisms so as to expedite the simulation process. To the best of our knowledge,

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such a model incorporating the simultaneous effects of all the above mechanisms on a multi-

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component gas system has not been presented in literature.

97 98 99

2

Mathematical formulation 2.1 Model description

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A few models have been proposed in literature to model multi-component gas flow in shale

101

reservoirs

102

account the effects of advection and diffusion, respectively. Permeability is corrected using one

103

single Klinkenberg parameter for the whole system and this fact reduces the flexibility of the

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available models, especially at low pressures 35-37.

105

The dusty-gas model employs kinetic theory of gases to include ordinary and Knudsen diffusion

106

as well as advection for the flux of each component of the gas 38. Although the dusty-gas model

107

yields more robust results, the interaction between molecules leads to a coupled system of partial

108

differential equations, hence the problem becomes complex and the computational cost increases.

109

A comparison between the above two models can be found elsewhere 35.

110

The model used in this study is an extension of the model proposed by Rezaveisi et al.

111

modified to include the effect of sorption and pore radius change. To this aim, the non-revised

33, 34.

The advective-diffusive model employs Darcy’s law and Fick’s law to take into

26

and


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Langmuir adsorption model is employed to account for the adsorption capacity of different

113

components 39. The flux of each component Fi is modeled as follows:





Fi   yi 

Dkn ,i   Pyi k D  bi   1 P       P  RT  Z

(1)

 , 

114

where yi is the mole fraction of component i,  is the molar density of the gas mixture, k D is

115

Darcy permeability,  is viscosity, bi is the Klinkenberg parameter for component i , Dkn ,i is

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effective Knudsen diffusion coefficient of component i , P is pressure, R is the universal gas

117

constant, T is temperature and Z is gas compressibility factor.

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As pressure declines during production, free gas is produced and the pressure difference created

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between shale bulk matrix (kerogen) and porous space causes desorption of the gas 40. Langmuir

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isotherm adsorption model has proved successful in fitting the experimental shale sorption data 12.

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The non-revised Langmuir model is the most commonly used means to predict adsorption

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capacities of different components of a gas mixture. The following equation models the adsorbed

123

moles of component i per unit volume of shale  qa ,i 

qa ,i 

s P

sc

RT sc

VL ,i

39, 40:

(2)

Pyi PL ,i , Nc y j

1  P j 1

PL , j

124

where  s is the shale density, P sc and T sc are standard pressure and temperature, respectively, VL ,i

125

is the Langmuir volume of single component i, which is the maximal volume of the gas component

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adsorbed to the unit mass of shale, PL ,i is the Langmuir pressure or the pressure at which half of VL ,i

127

is reached and N c is the number of components. 6 ACS Paragon Plus Environment

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128

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The Klinkenberg parameter for component i is given by

bi 

(3)

 8 RT   2   1 , M i reff   i 

129

where M i is the molar mass of component i and reff is the effective pore radius, which is updated

130

versus changes of pressure and composition in the course of simulation. Parameter  i is the

131

tangential momentum accommodation coefficient (TMAC) of component i. TMAC is the ratio of

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diffusive to specular (mirror like) reflection of molecules from a surface and it depends on

133

physicochemical properties of the surface, the type of the gas, pressure and temperature. The

134

following correlation has been proposed by Agrawal and Prabhu

135

and is extended for a multi-component mixture to be used in our model.

 i  1  log 1  Kni0.7  ,

41

based on experimental data

(4)

136

Kni is the Knudsen number of component i, which is defined as the ratio of the mean free path of

137

gas molecules to the characteristic length of the system (pore radius), and is modeled as given

138

below 5:

Kni 

(5)

k BT ,  2reff  i2 P

139

where  i is the molecular diameter of component i.

140

The following equation is utilized to account for the effect of pore enlargement caused by pressure

141

depletion 40:


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i

Nc

reff  rav   i 1

142

(6)

Pyi PL ,i Nc

yj

j 1

PL , j

1  P

Page 8 of 30

,

where rav is the average pore radius as given by Equation 7 42:

rav 

8 k D



(7)

,

143

where  and  are the porosity and tortuosity of shale 43, respectively.

144

Dkn ,i (effective Knudsen diffusivity) is the modified molecular diffusion coefficient 5 to account

145

for the effects of porosity and tortuosity 35 as follows:

Dkn ,i 

146

16 3

(8)

RTk D .  M i

The properties of the base case linear reservoir model are presented in Table 1.

147 148

Table 1. Parameters used in the base case reservoir model Property

Value

Darcy permeability, k D  nD 

100

Porosity,   

0.1

Length, L  m Initial pressure  psia  Bottom hole pressure  psia  Temperature, T C  Tortuosity,    Mole fractions of C1, C2, C3, CO2 ( yi )

4 5000 1000 100 4 0.6, 0.25, 0.1, 0.05

149 150

2.2 Governing equations 8 ACS Paragon Plus Environment

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The temporal partial differential form of the mass accumulation term considering free and

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adsorbed gas in a porous medium with constant porosity containing multi-component single-phase

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gas is as follows:



    yi   1    qa ,i  t t



1     s P VL,i    yi   t RT sc PL ,i sc

(9)   Pyi  Nc y t  j  1 P   j 1 PL , j 

  .   

154

According to mass conservation, the above term must be equal to the negative divergence of the

155

flux of component i disregarding sink/source term. The flow vector term of the system considering

156

viscous flow, slip flow and Knudsen diffusion in a 1D linear model is given by Equation 10.  

 . F i 

k  b  P Dkn ,i   Pyi   yi  D 1  i      P  x RT x  Z x 

  . 

(10)

157

Using the real gas law for molar density    P / ZRT  and isothermal flow assumption, the final

158

form of the mass conservation equation in the system of interest is obtained as given by Equation

159

11.

  sc Pyi   Pyi  1     sTP VL ,i      sc Nc y t  Z  t  T PL ,i j 1  P   j 1 PL , j    yi k D P  Py   P  bi   Dkn,i  i   .  x  Z  x x  Z  

     

(11)

160


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161

The above equation is actually a system of coupled partial differential equations which is solved

162

for pressure and the mole fractions of ( N c  1) components, considering that mole fractions must

163

add up to unity ( yi  1) . i

44

164

The gas compressibility factor is obtained using the Peng-Robinson equation of state

and the

165

Lee correlation is utilized to calculate gas viscosity 45.

166

Appropriate initial and boundary conditions are prescribed to the system. The pressure and gas

167

composition are initially set at the values presented in Table 1. Transmissibilities are set to zero at

168

the left end of the model (no-flow boundary condition). Constant pressure of 1000 psia and zero

169

concentration gradient of the components are the conditions at the right boundary.

170 171 172

2.3 Rate transient analysis The gas diffusivity equation can be expressed as

 c  k

t

(12)

  2 ,

173

where  is the real gas pseudopotential presented as the following integral from a reference

174

pressure ( pref ) to an arbitrary pressure ( p )

 ( p)  2

p



pr ef

46:

(13)

P dP. Z

175

The solution to the gas linear flow equation in an infinite reservoir producing under a constant

176

pressure condition is given by 47


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(14)

1  2  t DA , qD

177

where the dimensionless gas flow rate (qD ) and dimensionless time based on cross-sectional area

178

to flow ( Ac ) are defined below.

1 k Ac  ( pi )  ( pw )   , qD 1424qg T

t DA 

(15)

(16)

0.006328kt ,  ct i Ac

179

where k is the reservoir permeability in mD, pi and pw represent the initial reservoir and constant

180

flowing pressure in psia, respectively, qg denotes the gas production rate in Mscf / day,  is the

181

gas viscosity in cp, c is the gas isothermal compressibility in psia 1 and subscript i refers to the

182

initial state.

183

The following procedure is employed in this study to obtain the apparent gas permeability for the

184

numerically simulated cases. The reciprocal of the gas production rate versus time on a log-log

185

plot results in a half-slope straight line. The half-slope line determines the beginning and end of

186

the linear flow period. The point at which the solution deviates from half slope is the start of the

187

boundary-dominated flow. After substitutions and rearrangement of Equations 14 to 16 and

188

recognizing the fact that the plot of the reciprocal of the gas production rate versus the square root

189

of time generates a straight line, the following equation is obtained to find the apparent gas

190

permeability for different case studies 30: k Ac 

1262T

 ( pi )  ( pw ) mcp  c i

,

(17)


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191 192

Page 12 of 30

where mcp denotes the slope of the constant-pressure straight line.

2.4 Methodology

193

The transport mechanisms considered here make the system of equations highly nonlinear, hence

194

a numerical solution approach is inevitable. The equations are discretized using backward

195

differences in time and centered differences in space, and upwinding is used to estimate the values

196

of the parameters at block interfaces. There are N c unknowns/independent equations per grid

197

block, so a system of algebraic equations with a dimension of N c  N b should be solved at each

198

time step, where N b is the number of grid blocks. The Newton-Raphson iterative approach is used

199

to obtain the unknown values (pressure and composition) at the next time step fully implicitly and

200

march in time. The convergence criterion is the norm of the residual vector (the difference between

201

the temporal and spatial sides of Equation 11) to be lower than a threshold    104  .

202

The procedure to obtain the apparent gas permeability is described as follows. For a fixed set of

203

reservoir shale properties, two scenarios are simulated using the same Darcy permeability (k D ).

204

One scenario is the simulation of a gas production from a shale matrix block filled with methane

205

under the effect of pure Darcy flow, and the second scenario is the simulation of methane

206

production where all the involved transport mechanisms are considered. Gas production rates

207

versus time are obtained and the linear flow period is determined for each scenario from the log-

208

log plot of the reciprocal of gas production rate versus time. Next, the square-root-of-time plots

209

are generated and the slope of the constant-pressure straight line over the linear flow period is

210

measured on each plot (mcp1 and mcp 2 ). The slope is inversely proportional to the square root of

211

permeability as shown in Equation 17. Assuming a permeability equal to the input Darcy 12 ACS Paragon Plus Environment

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212

permeability (k D ) for the first scenario, an apparent permeability (kapp ) is obtained for the second

213

scenario from the ratio of the constant-pressure line slopes. This apparent permeability once

214

incorporated into a Darcy model with single-component gas with the average properties of the

215

reservoir gas is expected to recover the production behavior of a multi-component and multi-

216

mechanistic gas production leading to a significant saving in the computational time. This

217

procedure is validated by comparing the simulation results of a multi-component and multi-

218

mechanistic case with Darcy permeability (k D ) against those of a single-component Darcy flow

219

with the apparent permeability (kapp ). A flowchart summarizing the above procedure is presented

220

in Figure 1.

221

In the following, the applicability of the proposed apparent gas permeability is validated and then

222

tested on two case studies with two sets of rock properties and different gas compositions. In case

223

that gas production data from a shale core plug are available, this procedure can be applied on our

224

model with the shale properties to obtain the apparent permeability and substitute it for Darcy

225

permeability in the grid blocks of the reservoir simulation model. The advantage of the apparent

226

gas permeability is a large reduction in computational time without imposing a significant error.


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Fluid data (composition) and Rock data (Darcy permeability, porosity, tortuosity, medium length, sorption properties)

Multi-mechanistic flow simulation of methane production with kD (1)

Darcy flow simulation of methane production with kD (2)

Apply rate transient analysis to the simulation results (mcp1, mcp2)

Multi-mechanistic and multicomponent flow simulation with kD

Apparent gas permeability kapp=kD (mcp2/mcp1)2

Darcy flow simulation of gas production with kapp

227 228

Figure 1. The flowchart of the procedure of obtaining apparent gas permeability and validation.

229

3

230

Results and discussion 3.1 Apparent gas permeability (Case study I)

231

In this section, two simulation scenarios are performed in the first step to obtain the apparent

232

permeability. The reservoir rock properties, pressures and temperature are those from Table 1. A

233

grid spacing of 0.02 m (200 grid blocks for the length of 4 m) was selected as the optimal size

234

after conducting sensitivity analysis. One scenario is methane production under the effect of pure

235

Darcy flow. The second is methane production including the effects of all the transport 14 ACS Paragon Plus Environment

Page 15 of 30

236

mechanisms including viscous flow, slip flow (Klinkenberg effect), Knudsen diffusion, sorption,

237

pore radius change and real gas effect on shale gas production. The log-log plots of the reciprocal

238

of the gas production rate versus time are generated and the beginning and end of the half slope

239

line is determined for each case. The above plot for the second scenario is shown in Figure 2. The

240

half-slope line is tangent to the curve after very early time to the point where the boundary

241

dominated flow starts. This period is known as the linear flow period.

242 243

1e-2

1e-3 1/q (day/Mscf)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1e-4

2

1e-5

1e-6 0.001 244

0.01

0.1

1

10

100

1000

t (day)

245

Figure 2. The log-log plot of the reciprocal of gas efflux versus time for the multi-component multi-

246

mechanistic gas flow in shale medium including viscous flow, slip flow (Klinkenberg effect), Knudsen

247

diffusion, sorption, pore radius variation and real gas effect with Darcy permeability of 100 nD.


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Then, the reciprocal of rates are plotted versus the square root of time, the slope of the constant-

249

pressure lines (mcp ) are measured and the apparent permeability is obtained from Equation 17

250

(from the ratio of the slopes of the constant-pressure lines following the procedure explained in

251

the “Methodology” section) for the second scenario where all the mechanisms are involved. The

252

square-root-of-time plot for the second scenario is presented in Figure 3 as an example. This

253

process yields a permeability of about 190 nD for the second scenario, which is an indication that

254

the apparent permeability is higher than Darcy permeability.

0.0025

0.0020 1/q (day/Mscf)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 30

0.0015

0.0010

0.0005 mcp=8.4x10-5 (day)0.5/Mscf

0.0000 0

2

4

6

8

10

12

t1/2 (day1/2)

255 256

Figure 3. The log-log plot of the reciprocal of gas efflux versus square-root-of-time for multi-mechanistic

257

case with Darcy permeability of 100 nD.

258 259

The above apparent permeability is used in the later simulations of this case study. In the first

260

experiment, one simulation is conducted with the multi-component gas mixture and reservoir

261

properties of the base case (refer to Table 1) where all the mechanisms are included. Another 16 ACS Paragon Plus Environment

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simulation is performed with a Darcy permeability of 190 nD including one single lumped

263

component with the average gas properties of the same mixture. The only mechanism included in

264

this scenario is Darcy flow. The computational time of the latter is by an order of magnitude lower

265

than that of the former (three hours compared to 45 hours on a desktop quad core computer). It is

266

expected that the results of the single-component case (only Darcy flow, permeability of 190 nD)

267

compares well with the multi-component multi-mechanistic case (permeability of 100 nD).

268

The temporal variation of pressure at the no-flux boundary of the medium and gas production rate

269

versus time are shown in Figures 4 and 5, respectively. As can be viewed, the error caused by

270

lumping the components into a single one and replacing Darcy permeability with the obtained

271

apparent permeability is negligible compared to the case with a permeability of 100 nD where a

272

multi-component gas is produced by the coupled effects of viscous flow, slip flow, Knudsen

273

diffusion and sorption. The pressure deviation is insignificant and the temporal average pressures

274

are almost the same. The gas production rates also match perfectly except a small difference

275

observed towards the end of the simulation time.


Energy & Fuels

5000 4500 multi-component, 100 nD

Pressure (psia)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4000 3500 3000

single component, 190 nD

2500 2000 1500 0

276

20

40

60

80

100

120

t (day)

277

Figure 4. Temporal variation of pressure at the no-flux boundary for the multi-component multi-

278

mechanistic case (k  100 nD ) and single component Darcy flow (k  190 nD ). The multi-mechanistic

279

case simulates flow of multi-component gas in shale medium including viscous flow, slip flow

280

(Klinkenberg effect), Knudsen diffusion, sorption, pore radius variation and real gas effect with the

281

properties of the base case. The single component case simulates Darcy flow of a single-component gas

282

with an apparent permeability obtained from the multi-mechanistic case.

283


Page 19 of 30

1e+6

multi-component, 100 nD single component, 190 nD

1e+5 q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

1e+4

1e+3

1e+2 0.001

0.01

0.1

1

10

100

1000

t (day)

284 285

Figure 5. Efflux of gas versus time for the multi-component multi-mechanistic base case (solid line) and

286

single component Darcy flow (dashed line). The apparent permeability used in Darcy flow simulations is

287

190 nD, which is obtained from the multi-mechanistic case.

288

The above upscaling task was carried out to validate the procedure of simplifying the simulations

289

by using an apparent permeability. The apparent permeability depends on pressure, average pore

290

size and gas type

291

generalization of the proposed upscaling notion. A confident answer to this question and the

292

sensitivity of the apparent permeability to different parameters require a comprehensive

293

investigation; however, to check the applicability of the concept, gas composition is selected as a

294

test measure.

295

In the second experiment of this case study, simulations were conducted using the data provided

296

in Table 1 with different methane mole fractions of 0.7 and 0.8. The apparent permeability

11.

An important question that needs to be addressed is related to the


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obtained from the previous step (kapp  190 nD) was used on a single component with average gas

298

properties and including only the effect of Darcy flow. Figures 6 and 7 show the results of the two

299

scenarios. The gas production rate from the simplified scenario with an apparent permeability of

300

kapp  190 nD is reasonably accurate when compared with the complex multi-mechanistic case. In

301

a second case study, the applicability of the procedure will be tested on a different set of properties

302

for the reservoir model.

1e+6 multi-component, 100 nD single component, 190 nD

1e+5 q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1e+4

1e+3

1e+2 0.001

303

0.01

0.1

1

10

100

1000

t (day)

304

Figure 6. Efflux of gas versus time for the multi-component multi-mechanistic case (solid line) with

305

k  100 nD, yC1  0.7 and single component Darcy flow (dashed line) with k  190 nD .


Page 21 of 30


1e+5 q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

1e+4

1e+3

1e+2 0.001

306

0.01

0.1

1

10

100

1000

t (day)

307


308

k  100 nD, yC1  0.8 and single-component Darcy flow (dashed line) with k  190 nD .

309 310

3.2 Apparent gas permeability (Case study II)

311

The sole effect of composition was investigated in the previous case study. To confirm the validity

312

of the model, a system with different properties than those of the base case is selected. A

313

permeability of 200 nD, porosity of 0.15, medium length of 6 m, Initial pressure of 4500 psia,

314

temperature of 70C and tortuosity of 10 are assigned to the second case study. A grid spacing

315

of 0.024 m (250 grid blocks for the length of 6 m) was selected as the optimal size after

316

conducting sensitivity analysis. The same procedure explained under the section of

317

“Methodology” was repeated; i.e., one simulation of methane production under the effect of pure

318

Darcy flow and another including all the contributing mechanisms were performed and the slopes


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319

of the constant-pressure lines on the square-root-of-time plot were employed to obtain the apparent

320

gas permeability. The acquired apparent permeability is 280 nD.

321

Then three experiments are carried out with mole fractions of 0.7, 0.8 and 0.9 for methane with

322

Darcy permeability of 200 nD and including all the mechanisms involved, namely viscous flow,

323

slip flow, Knudsen diffusion, adsorption/desorption, pore enlargement and real gas effect. For each

324

experiment, a simulation on a single-component gas with the average properties of the original

325

mixture and apparent permeability of 280 nD is conducted. The results of the temporal change of

326

gas efflux for the above experiments are depicted in Figures 8 to 10.


q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

1e+5

1e+4

1e+3 0.001

0.01

0.1

1

10

100

1000

t (day)

327 328


329


330 331 332 333


Page 23 of 30


q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

1e+5

1e+4

1e+3 0.001

0.01

0.1

1

10

100

1000

t (day)

334 335


336


337 338 339


Energy & Fuels


q (Mscf/day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 30

1e+5

1e+4

1e+3 0.001

0.01

0.1

1

10

100

1000

t (day)

340 341


342


343 344 345

As can be viewed in the above figures, the proposed proxy acts perfectly well and the model is

346

capable of yielding accurate results by approximately an order of magnitude faster. The

347

experiments with lower methane mole fractions were also conducted for our own test and

348

acceptable results were achieved. The validity of the simplifying assumptions and the applicability

349

of representing all the contributing mechanisms in a single term, namely apparent permeability,

350

are realized. This can provide a basis and guideline to reduce computational load in large-scale

351

numerical reservoir simulations of gas production from shale media. Although the model is one-

352

dimensional, if compositional gas production data are available from experimental measurements

353

on shale core plugs, the model and procedure developed in this study can be applied to assign

354

different apparent permeabilities, including the effects of multiple transport mechanisms, to

355

heterogeneous shale gas reservoirs. 24 ACS Paragon Plus Environment

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356 357

4

358

A numerical model was presented to simulate efflux of gas from shale matrix blocks. Various

359

contributing transport mechanisms including viscous and slip flow, Knudsen diffusion, gas

360

adsorption/desorption, pore enlargement and real gas effect are included in the simulations. A

361

combination of all of these mechanisms responsible for flow of multi-component gas in shale

362

media have not been reported in literature. In particular, we have included the effect of sorption

363

and pore enlargement. These are significant phenomena, ignoring whose effect could cause a

364

significant error in the prediction of gas efflux from matrix blocks of shale media and

365

underestimation of the shale apparent permeability.

366

Compositional simulation of multi-mechanistic gas flow in shale media is computationally very

367

challenging. An important question that was addressed in this work is the possibility of lumping

368

all the components into a single one while maintaining accuracy of the prediction of gas efflux

369

from shale media. The results show that the analysis of multi-component gas efflux versus time

370

(the reciprocal of gas efflux versus the square root of time) from shale media can be used to obtain

371

an apparent permeability. The obtained apparent permeability when used in a single-component

372

Darcy flow model enables us to predict the gas efflux from a shale matrix block with high accuracy.

373

The model is one-dimensional and homogeneous which does not capture the effects of three-

374

dimensional phenomena, heterogeneity and water film in shale matrix; however, it can be suitably

375

utilized to simulate the experimental gas production data and obtain apparent gas permeability for

376

grid blocks of a shale gas reservoir model. The proposed lumping scheme can significantly reduce

377

the computational burden of multi-component multi-mechanistic gas flow simulation and will find

378

applications in large-scale numerical simulation of gas production from shale media.

Summary and Conclusions


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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379

Acknowledgments

380 381 382

The authors would like to thank two reviewers for their constructive comments. The financial support of NSERC/Energi Simulation and Alberta Innovates (iCORE) Chairs in the Department of Chemical and Petroleum Engineering at the University of Calgary is greatly appreciated.

383


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References

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Graphic for TOC Fluid data (composition) and Rock data (Darcy permeability, porosity, tortuosity, medium length, sorption properties)

Multi-mechanistic flow simulation of methane production with kD (1)

Darcy flow simulation of methane production with kD (2)

Apply rate transient analysis to the simulation results (mcp1, mcp2)

Multi-mechanistic and multicomponent flow simulation with kD

Apparent gas permeability kapp=kD (mcp2/mcp1)2

Darcy flow simulation of gas production with kapp

507


Estimation of shale apparent permeability for multi-mechanistic multi

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