Modeling chromatographic separation of produced gas in shale wells

International Journal of Coal Geology 121 (2014) 110–122

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Modeling chromatographic separation of produced gas in shale wells Mohsen Rezaveisi a, Farzam Javadpour b,⁎, Kamy Sepehrnoori a a b

Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX, United States Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin, Austin, TX, United States

a r t i c l e

i n f o

Article history: Received 5 June 2013 Received in revised form 29 October 2013 Accepted 5 November 2013 Available online 14 November 2013 Keywords: Shale gas mixture Knudsen diffusion Slip effect Nanoporous media Nanopore Gas shale

a b s t r a c t Field evidence exists showing temporal variation in produced gas composition in shale wells. Preferential gas flow and sorption of the species in shale formations cause compositional variations in the gas produced from shale. This process is similar to gas chromatographic (GC) separation, in which the size of gas molecules and their affinity for walls cause separation. As in gas chromatography, shale gas contains molecules of different gases (methane, ethane, propane, carbon dioxide, etc.). When reservoir pressure is greater than critical sorption pressure, sorption process is negligible and the separation process is mainly due to differences in gas molecule speeds in pores. The extremely small size of the pores in shale adds different flow physics, such as Knudsen diffusion and slip flow that intensifies separation of gas components. Understanding and modeling chromatographic separation (CS) in shale can improve our knowledge and help us produce more valuable gas from gas shale. We have developed a numerical model to study temporal variations of the composition of gas produced from shale gas wells. The model is a physical transport model of single-phase multicomponent gas flow in nanoporous media. The governing equations are implemented into a one-dimensional numerical model and solved using a fully implicit solution method. A sensitivity study of the effect of different parameters such as reservoir pressure, length of the system, tortuosity, and permeability on the CS process is performed. The model results confirm strong CS process in shale. In an early stage of production, the component with the highest Knudsen diffusivity and slip coefficient is produced with a higher mole fraction than its in-situ composition. At a later time, the same component comprises a smaller mole fraction than its in-situ composition in the gas production stream. Lower Darcy permeability, a longer system, and higher reservoir pressure intensify the CS process. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Shale gas reservoirs in the United States are vast and gas from these reservoirs is already contributing to U.S. fossil energy production. New shale gas plays have been discovered in other places, including Latin America, Europe, and Asia, making the shale gas boom a truly international phenomenon. Shale reservoirs are categorized as unconventional reservoirs because they are extremely tight, with pores in the range of nanometers (Javadpour et al., 2012; Klaver et al., 2012; Loucks et al., 2012; Milliken et al., 2012) and permeabilities in the range of nanodarcys (Chalmers et al., 2012; Darabi et al., 2012; Javadpour, 2009). Other complexities of gas production in shale include gas sorption and diffusion processes (Etminan et al., in press; Javadpour et al., 2007). Thanks to advances in horizontal drilling and hydraulic fracturing, we have the ability to produce economically from these reservoirs. However, many challenges still lie ahead. These challenges include production data analysis (Clarkson, 2013) and determination of effective reservoir permeability and matrix block size after hydraulic fracturing (Azom and Javadpour, 2012).

⁎ Corresponding author. 0166-5162/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coal.2013.11.005

The objective of this paper is to model temporal composition variations of the produced gas in shale gas wells (Freeman et al., 2012; Schettler and Parmely, 1989). Compositional variations in produced gas arise from the differences in physical and chemical properties of the gas components. Important physical properties include molecular size and geometry, while important chemical properties include sorption affinity of the gas components. The differences of physical and chemical properties cause different resident time for each component; hence a form of chromatography separation (CS) occurs in the reservoir. CS can also occur as part of other reservoir engineering problems such as chemical flood, tracer injection, gas flooding, and sour acid gas injection (Gupta et al., 1988; Pooladi-Darvish et al., 2009). However, separation in shale gas reservoirs is more pronounced, because in this setting gas molecules tend to move individually rather than in bulk. The tiny size of the pores in shale makes the mean free path of the gas molecules comparable to the size of the pores, i.e., high Knudsen-number flow. In such flow conditions, Knudsen diffusion and slip flow dramatically affect gas-component flow. We will address in detail how these flow mechanisms affect CS in shale. Selective sorption of different gas components from organic material in shale affects CS (affinity CS) as well. However, affinity CS is important at the later life of a reservoir when reservoir pressure declines to below the critical

M. Rezaveisi et al. / International Journal of Coal Geology 121 (2014) 110–122

X

j-½ j-1

111

P=const. @ X=L

j+½ j

C1 C2 C3

j+1 L

no-flow b.c. Fig. 1. Schematic diagram of linear nanoporous media. Boundary conditions and numerical discretization shown.

sorption pressure. In this paper we only study the effect of fluid flow processes on CS. The issue of chromatography separation is sensible for the systems where the diameters of the pores are small and their lengths are large, i.e., pores in shale. The CS process occurs mainly in shale matrix, e.g., network of nanopores. Once gas molecules enter the fractures (open natural and induced fractures) they move as bulk towards the wellbore. However, the CS in matrix shows itself as the variation of gas composition at the wellbore. Therefore, variation of gas composition of the producing gas gives us valuable information about gas transport in the matrix. A numerical model such as the one presented in this paper can be used to analyze producing gas composition variation to determine the average length of shale matrix or apparent permeability of the matrix. In conventional systems, we usually have good information about matrix block size and transient analysis (well testing) can readily be used to determine permeability. In shale none of these two important parameters (matrix block size and permeability) are known, hence any attempt to determine these parameters improves shale reservoir characterization attempts. To model the compositional variations in the produced gas stream, a physical transport model capable of handling multi-component single-phase gas transport in porous media is necessary. Considering Knudsen diffusion and slip effects in the governing equations makes the system nonlinear and hence there is no analytical solution. We have developed a numerical model for transport of multi-component gas in porous media; the model includes the important transport mechanisms governing the flow through nanopores. First, we briefly introduce some common transport models that have been applied to describe gas flow through nanoporous media. We then introduce our model. Next, we present governing equations, discretization of the numerical model, and the input physical properties. This is followed by a sensitivity study of the effect of different parameters, such as reservoir pressure, length of the system, permeability, and tortuosity, on the CS process. 2. Gas flow models There are many models to describe gas flow in tight and ultratight porous media. In this section we evaluate three recent gas flow models. 2.1. Advective–diffusive model In the advective–diffusive model (ADM), the advective and diffusive fluxes of gas in porous media are superimposed (Webb and Pruess, 2003). For advective flow contribution, Darcy's law is used and for diffusive flow contribution, Fick's law of diffusion is employed. Slip effects are included in the flow equations by modifying permeability with the Klinkenberg parameter (Firouzi et al., in press; Klinkenberg, 1941). The flow as a result of ordinary diffusion is modified by a tortuosity factor to account for the presence of porous media (D⁎i ).

The molar flux of each gas component is given by Eq. (1) for the advective–diffusive model (Pruess et al., 1999): Fi ¼ −

keff  ρ y ∇P −Di ρg ∇yi μg g i g

ð1Þ

where ρg is molar density of gas and yi is mole fraction of component i in gas phase. The effective permeability in this equation is the Darcy permeability modified by the Klinkenberg parameter, as given by Eq. (2):
 b : keff ¼ k0 1 þ P

ð2Þ

The ADM is simple to apply, but it ignores the coupling between advective and diffusive mechanisms (Webb and Pruess, 2003). Furthermore, there is evidence that the Klinkenberg parameter is not able to predict gas flow at intermediate- and high-Knudsennumber conditions, e.g., gas flow in shales at moderate and low pressure (Bravo, 2006; Freeman et al., 2011). Using a Lattice Boltzmann simulation, Fathi et al. (2012) proposed a correction term to the Klinkenberg model to improve model predictability. 2.2. The dusty–gas model The “dusty–gas” model (DGM) applies the kinetic theory of gases to the gaseous components and the porous media (or “dust”). The porous medium is visualized as an array of dust particles held stationary in space. Knudsen diffusion and other molecule–wall interactions are inherently included in the model (Evans et al., 1961, 1962). The DGM considers advection, Knudsen diffusion, and ordinary diffusion. One of the key aspects of the DGM is the combination of diffusion (ordinary plus Knudsen) and advection. The DGM formulation in terms of total molar fluxes is given by the following equation (Webb and Pruess, 2003): T T n X xi N j −x j N i j¼1 j≠i

Dij

−

P g ∇xi NTi þ ¼ Dik RT

1þ

k0 P g DiK μ g

!

xi ∇P g : RT

ð3Þ

The first term on the left-hand side of this equation accounts for molecule–molecule interactions and is based on the Stefan–Maxwell equations. The second term accounts for molecule–particle (porous media) interactions and the right-hand side accounts for diffusion and advection. This model results in a coupled system of equations for flux of each component. For an isothermal binary system, the total mass flux of component 1 can be written as: F 1 ¼ −m1

h   i   D1K D12 P g =RT ∇x1 þ D1K ðD12 þ D2K Þx1 ∇P g =RT k0 P g ∇P g    −x1 m1 : ð4Þ μ g RT D12 þ x1 D2K þ x1 D1K

This equation shows that the flux of component 1 consists of a diffusive flux (first term) and an advective flux (second term). The

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Table 1 Reservoir parameters of the base-case model. Property

Value

Darcy permeability, kD (nD) Porosity, ϕ L (m) Initial pressure (psi) Outlet pressure (psi) Temperature (K) Tortuosity Number of grid blocks (number of refined grid blocks near the outlet)

100 0.1 4 5000 1000 373 4 150 (70)

diffusive flux consists of the contributions of ordinary and Knudsen diffusion. Compared to the ADM model, the dusty gas model captures more physics of gas transport in nanoscale. However, the main drawback of this model is the need for solving a complex system of equations in order to find the total component fluxes. This disadvantage precludes this model's direct application in a reservoir simulator. 2.3. The Knudsen–slip–advective flow model Another flow model was proposed to consider the molecule–wall interactions during flow in nanopores by Javadpour (2009). The original model was proposed for flow of a single gas component through a nanopore and later extended to a nanoporous system (Azom and Javadpour, 2012; Darabi et al., 2012; Mehmani et al., 2013). This model can be extended to multi-component flow of a gaseous mixture through porous media by the following equation for flux of component i: F i ¼ −yi ρ



 Dk;eff kD b Pyi 1 þ i ∇P− ∇ μ P RT Z

ð5Þ

where yi is the mole fraction of component i, ρ is molar density of the gas phase, kD is bulk Darcy permeability, μ is viscosity, bi is the Klinkenberg parameter of component i, and Dk,eff is the effective Knudsen diffusivity of the porous medium. The first term on the right-hand side of Eq. (5) refers to the contributions of bulk Darcy and slip flow of component i

and the second term accounts for the contribution of Knudsen diffusion. Here, bi is the extended Klinkenberg parameter, which takes different values for different components. In the form presented above, the factor bi is applied to each gas component and contributes to CS of the produced gas components. The value of bi for different components is given by (Javadpour, 2009):
bi ¼

8πRT Mi

0:5




μ Ravg

avg

 2 −1 αi

ð6Þ

where Mi is molar mass of component i and Ravg represents the average pore radius of the porous medium. The term αi is tangential momentum accommodation coefficient (TMAC). Treatment of αi in this equation is subject to questions of how to calculate the value of α for a gas component in a gas mixture. We will discuss the treatment of TMAC in Section 2.4. The value of Ravg for the shale rock must be obtained through indirect techniques such as capillary pressure tests (Javadpour et al., 2007) and/or nitrogen adsorption (Brunauer et al., 1938) and direct methods such as image analysis of the scanning electron microscope (SEM) images (Loucks et al., 2012) and topography imaging by an atomic force microscope (AFM) (Javadpour et al., 2012). Here we use a simple first-order approximation considering the bundle of capillary tubes model (Peters, 2012) for nanoporous media given by the following: Ravg ¼


 8τK D 0:5 ϕ

ð7Þ

where τ is tortuosity of the porous medium, defined as the square of the ratio of equivalent length of the capillary tubes to the length of the porous medium, and ϕ is porosity of the porous medium. In our numerical model, effective Knudsen diffusivity (Dk,eff) is obtained through a simple analogy to the effective molecular diffusion coefficient given by Dk;eff ¼ Dk

ϕ : τ

ð8Þ

This equation is similar to that suggested in Webb and Pruess (2003) for effective molecular diffusion coefficient, however, in contrast to the tortuosity definition by Webb and Pruess, we define

Fig. 2. Pressure profile across linear model for systems with different permeability values after 1 day and 5 days of production. (Base parameters listed in Table 1.)

M. Rezaveisi et al. / International Journal of Coal Geology 121 (2014) 110–122

113

Fig. 3. Methane (C1) concentration profile across linear model for systems with different permeability after 12 days and 22 days of production. (Base parameters listed in Table 1.)

tortuosity as actual length of the pore to the bed length (always N 1). In Eq. (8), Dk is Knudsen diffusivity of a gas component (Javadpour, 2009) Dk ¼


 2 8RT 0:5 Ravg : 3 πM

ð9Þ

dependencies of TMAC on Knudsen number in actual shale gas reservoirs. 3. Description of the numerical model 3.1. Governing equations

2.4. Treatment of TMAC

The general mass-conservation equation for multi-component single-phase transport in porous media is given by (Chang, 1990)

The TMAC is the fraction of gas molecules reflected diffusively from tube wall relative to specular reflections. The value of TMAC depends on the nature of the gas, pressure of the gas, material of the surface, surface cleanliness and roughness, and surface temperature. Agrawal and Prabhu (2008) reviewed the experimental techniques employed by researchers to measure the TMAC for different gases. Their review suggests that the TMAC at about 0.93 is almost constant with respect to the Knudsen number for monoatomic gases and this value remains the same for most surface materials. However, for gas mixtures, Agrawal and Prabhu proposed the following correlation for TMAC as a function of the Knudsen number:   0:7 α ¼ 1− log 1 þ K n

ð10Þ

where Kn is the Knudsen number given by (Javadpour, 2009): Kn ¼

λ k T pBffiffiffi ¼ Ravg Ravg 2πσ 2 P

! ∂wi ¼ −∇: F i þ r i ∂t

ð12Þ

where wi is the mass accumulation term and Fi is the flux of component i, and ri is the source or sink term, which may be used to accommodate sorption of gas molecules. For single-phase gas flow in porous media, wi is given by wi ¼ ϕρyi

ð13Þ

where ρ is molar density, ϕ is porosity of the reservoir, and yi is the mole fraction of component i in the gas mixture. For the flux of component i we use Eq. (5). This model considers the contributions of bulk Darcy flow, slip flow, and Knudsen diffusion, and is analogous to the DGM

ð11Þ

where T is temperature in Kelvin, kB is the Boltzmann constant in J/K, R avg is average pore size of the porous medium, σ is the collision diameter of the molecules, and P is the pressure of the system. We extend the definition of Knudsen number given by Eq. (11) to each gas component in a gas mixture (K ni ) by using collision diameter of the specific component (i) in Eq. (11). Note that the correlation given by Eq. (10) was derived based on experimental data using atmospheric gases different from those we expect in a shale gas reservoir (except for CO 2 and CH 4 ), and also for surface materials, such as metals, that are not what we expect in reservoirs. The correlation is also based on experimental data of slippage of only single-component gases not gaseous mixtures. Therefore, our numerical results inherently assume validity of Eq. (10), even though our approach remains valid for other functional

Fig. 4. Spatial pressure profile and concentration profile of different components after 12 days for the base case.

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Fig. 5. Spatial pressure profile and concentration profile of different components after 22 days for the base case.

model in the physics while offering simplicity comparable to that of the ADM model. For one-dimensional linear flow of multicomponent gas mixture in porous media, the final mass-conservation equation has the following form;



∂ðϕρyi Þ ∂ k b ∂P Dk;eff ;i ∂ Pyi þ ri ¼− −yi ρ D 1 þ i − μ P ∂x RT ∂x Z ∂t ∂x ¼ 1; 2; …; Nc

ð14Þ

    3 2 2 2 3 Z −ð1−BÞZ þ A−3B −2B Z− AB−B −B ¼ 0

ð16Þ

where A and B are dimensionless equation-of-state parameters that are calculated by applying a mixing rule over critical properties of individual components. Gas viscosity was obtained from the following correlation for natural gas viscosity, given in McCain (1989): Bρc

μ ¼ Ae

A¼

3.2. Gas physical properties For calculation of molar density of gas (gmol/m3), the equation of state for a real gas (Eq. (15)) is used: P ZRT

where P (Pa), R (8.314 J/gmol.K), T (K), and Z is the compressibility factor of the gas phase. The Peng–Robinson equation of state (Peng and Robinson, 1976) was used to obtain the gas compressibility factor. The cubic equation solved for Z based on this equation of state is given by:

for i

where bi and Dkeff,i are given by Eqs. (6) and (8), respectively. A schematic diagram of the linear system and the boundary conditions is given in Fig. 1. The left-hand side of the linear reservoir is the noflow boundary and the right-hand side is a producing boundary maintained at constant pressure. All other boundaries of the reservoir are also no-flow boundaries. Porosity of the reservoir is assumed constant during production.

ρ¼

Fig. 7. Temporal variation of normalized producing C2 concentration for systems with different permeability values. (Base parameters listed in Table 1.)

ð15Þ

ð17Þ

ð9:379 þ 0:01607  M ÞT 1:5 209:2 þ 19:26 M þ T

ð18Þ

986:4 þ 0:01009 M T

ð19Þ

B ¼ 3:448 þ

C ¼ 2:447−0:2224 B

ð20Þ

where M (g/gmol or lb/lbmol), T (Rankin), µ (cp), and is average molar mass of the gas mixture. Density in Eq. (17) is mass density (g/cm3).

Fig. 6. Temporal variation of normalized producing C1 concentration for systems with different permeability values. Note the very early data points correspond to the composition of the very first gas pockets produced from the system that shows deviation in gas composition from initial gas composition. (Base parameters listed in Table 1.)

Fig. 8. Temporal variation of normalized producing C3 concentration for systems with different permeability values. (Base parameters listed in Table 1.)

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Fig. 9. Temporal variation of normalized producing concentration of different components and volumetric average pressure for the base case.

115

Fig. 11. Temporal variation of normalized producing C1 concentration and absolute flux of C1 by different transport processes at the outlet (kD = 10 nD).

3.3. Finite difference approximation The partial differential equation describing mass conservation of each component, given by Eq. (14), is spatially and temporally discretized to obtain a nonlinear system of algebraic equations. The discretized mass conservation equation is given by −

ðϕρyi Þnþ1 −ðϕρyi Þnj j Δt
 nþ1 nþ1 P jþ1 −P nþ1 1 k b j ρyi d 1 þ i Δx j μ P jþ1=2 Δx jþ1=2
 nþ1 P nþ1 −P nþ1 k b j j−1 − ρyi d 1 þ i Þ μ P Δx j−1=2 j−1=2 " #

nþ1 nþ1 Dk;eff ;i nþ1 ðPyi =Z Þ jþ1 −ðPyi =Z Þ j 1 þ Δx j RT jþ1=2 Δx jþ1=2 #

nþ1 " nþ1 ðPyi =Z Þ j −ðPyi =Z Þnþ1 Dk;eff ;i jþ1 − Þ RT j−1=2 Δx j−1=2 þ

last one, which is obtained from the constraint that mole fractions  Nc must sum to unity ∑ j¼1 yj ¼ 1 : The value of the parameters that need to be evaluated at the spatial location of j ± 1/2 refers to the values at the boundary of the adjacent grid blocks (Fig. 1), and, as in typical reservoir engineering practice, “upwinding” is used to calculate their values. Upwinding means that the value of the properties in the upstream grid block is used instead of the value of the properties at the boundary of the neighboring grid blocks. 3.4. Initial and boundary conditions

ð21Þ

¼ 0 for i ¼ 1; 2; …; Nc and for j ¼ 1; 2; …; N Block where superscript n indicates the time level, i is the component index, and j is the grid block index. Nc and NBlock are the number of components and number of grid blocks, respectively. This equation is written for all of the grid blocks and for all the components; therefore there are Nc × NBlock independent equations. The unknowns in each grid block are pressure and mole fraction of each component except the

Initial conditions of the system are initial reservoir pressure and initial gas composition in the reservoir, P ðx;t¼0Þ ¼ P i yiðx;t¼0Þ ¼ yi0

ð22Þ for i ¼ 1; 2; …; Nc −1:

ð23Þ

At the left end of the system, (grid block #1) is the no-flow boundary condition which we specify by setting the transmissibility coefficients in the governing equation equal to zero.


k b nþ1 ¼0 ρyi d μ 1 þ P j−1=2 μ

Dk;eff ;i nþ1 ¼0 RT j−1=2

for j ¼ 1

for j ¼ 1

ð24Þ

ð25Þ

The other boundary condition (the boundary at the right hand side of Fig. 1) is the specified pressure at the outlet. However, a condition is also required for the mole fraction of the components at the outlet face. The Danckwerts boundary condition (Guiochon et al., 2006), which considers zero gradients of mole fractions (or component concentrations) at the outlet, is the appropriate boundary condition for our problem. Using a backward difference approximation for zero gradients of mole fractions at the outlet face, the Danckwerts boundary conditions can be reduced to Eq. (26): ½yi outlet ¼ ½yi NBlock Fig. 10. Temporal variation of normalized producing C1 concentration and absolute flux of C1 by different transport processes at the outlet (kD = 100 nD).

for i ¼ 1; …; Nc −1

where NBlock represents the last grid block.

ð26Þ

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Fig. 12. Temporal variation of normalized producing C1 and C2 concentration for systems with different initial reservoir pressures (Pi). (Base parameters listed in Table 1.)

Fig. 13. Temporal variation of normalized producing C1 and C2 concentration for systems with different length (L). (Base parameters listed in Table 1.)

3.5. Solution method

the linear system, and initial reservoir pressure. The reservoir parameters used in the sensitivity studies for the base case model are given in Table 1. The initial gas mixture molar composition is 60% C1, 25% C2, 10% C3, and 5% CO2. The disturbance imposed on the system from its initial resting conditions is the change in pressure at the outlet (Fig. 1). The pressure disturbance moves towards the no-flow boundary of the reservoir with time, and the average reservoir pressure decreases as the reservoir depletes. Fig. 2 shows how the pressure profile changes with time. The pressure disturbance reaches the end of the linear system quickly.

We used a fully implicit treatment because of the nonlinearity of the governing equations to solve Eq. (21). The Newton–Raphson method (Eq. (27)) was used as our nonlinear iterative solution method. x

nþ1

n

−x ¼ J

−1

ð27Þ

f

where xn + 1 is the unknown vector at time n + 1 or next time step, xn is the vector of unknowns in the previous time step, J is the matrix of derivatives of the residual equations with respect to the unknowns, and f is the residuals vector (see Appendix B for details). The Newton– Raphson method converges to the solution in about three iterations. After convergence of Newton–Raphson iterations in each time step is achieved, the values of pressure and overall composition of each component in each grid block are obtained. The calculated material balance error at each time step, which serves as a test of convergence, is very small and negligible. The number of produced moles of each component is calculated by material balance using the following equations: Niðt¼0Þ −Niðt Þ ¼ Q iðt Þ N i ðt Þ ¼

XN

Block

j¼1



VB j  ϕ j  ρ j  yi; jðt Þ

ð28Þ 

ð29Þ

where Ni(t = 0) is the number of moles of component i in the reservoir at time t = 0, Ni(t) is the total moles of component i in the reservoir at time t, and Qi(t) is the total number of produced moles of component i at time t. VBj is the bulk volume of grid block number j. The producing mole fraction of each component, which is the parameter of our interest, is calculated by dividing the incremental moles of component i produced during a time step by the total number of incremental moles produced (Eq. (30)). This method considers a constant average value of the producing mole fraction over a time step. C iðt→tþΔt Þ ¼

Q iðtþΔtÞ −Q iðt Þ Q ðtþΔtÞ −Q ðtÞ

ð30Þ

where Ci is the producing mole fraction of component i and Qi(t) is the cumulative production of component i at time t. 4. Sensitivity studies on chromatographic separation (CS) We present our model sensitivity results of the CS process in this section. We studied the effects of permeability, tortuosity, length of

4.1. Effect of reservoir permeability We varied Darcy permeability (kD) of the system over a wide range (10–1000 nD). The spatial pressure and gas component concentration variation across the model, as well as temporal gas-mixture concentration variation in the production stream, were compared in Figs. 2–9. Fig. 2 shows the pressure profile for different permeability values after 1 and 5 days of production. Average reservoir pressure drops faster for the cases with higher reservoir permeability. This rapid drop is associated with gas production by bulk Darcy flow. Fig. 3 shows concentration profile of methane (C1) at different permeability values. This figure shows that C1 produces faster in more permeable reservoirs. Figs. 4 and 5 show pressure and concentration profile of different components for the base case model after 12 and 22 days, respectively. The variation of methane (C1) concentration across the system is greater than heavier gas components (C2, C3, CO2). Fig. 6 shows temporal variation of normalized C1 mole fraction (or concentration) for the system with different permeability values. Normalized C1 is mole fraction of C1 in the produced gas stream divided by mole fraction of C1 at the initial reservoir condition (Table 1). Interestingly, for all permeability cases, normalized C1 concentration is greater than one for the early produced gas from the system and this behavior is more pronounced at systems with lower-permeability. This pattern is due to the relative contribution of Knudsen diffusion and slip flow to total flux. C1 is the lightest component in the gas mixture and has the highest Knudsen diffusion and slip flux. Therefore, C1 preferentially exits the system. The normalized C1 concentration decreases at later time because the source of C1 depletes in the system. At lower permeability values (10 nD, 50 nD, and 100 nD), the normalized C1 concentration remains greater than unity even after 120 days of gas production. The low decline in C1 concentration is a result of low production from the low-permeability system (Fig. 6). In contrast, at higher permeability values (500 nD and 1000 nD), the normalized C1 concentration reaches values smaller than one after ~ 50 and

M. Rezaveisi et al. / International Journal of Coal Geology 121 (2014) 110–122

117

Fig. 14. Pressure profile across the linear model at different times for systems with different tortuosity (τ). (Base parameters listed in Table 1.)

~30 days, respectively. The reduction trend continues with slower slope afterwards. An explanation for the aforementioned trend is that typical pore sizes are large in higher-permeability systems, and, as a result, the contribution of Knudsen diffusion and slip flow reduces and the preferential production of C1 decreases (Fig. 6). This behavior suggests that rapid variation in the composition of produced gas in wells could correlate to high-permeability reservoirs. Figs. 7 and 8 show temporal variation of normalized C2 and C3 concentrations at different values of permeability. For both components, normalized concentration is initially smaller than unity and increases as the in-situ reservoir gas becomes depleted of C1 and enriched with C2 and C3. This progression is expected because of the smaller Knudsen diffusion coefficient and slip flow coefficient of C2 and C3 relative to C1. Comparing Figs. 7 and 8, the initial value of normalized C2 is greater than that of C3. This difference exists because the magnitude of slip flow and Knudsen diffusion, which contributes to separation of components, is inversely proportional to their molar mass. Therefore, components with lower molar mass tend to move faster towards the lower-pressure zone. To better compare the relative behavior of the gas components Fig. 9 shows the temporal variation of normalized producing mole fraction of all the components along with volumetric average reservoir pressure of the base case on the same plot.

Figs. 10 and 11 superimpose the temporal variations of the flux of C1 by different flow mechanisms at the outlet on the normalized producing C1 concentration for kD = 100 nD and kD = 10 nD, respectively. Fig. 10 shows that the contribution of Knudsen diffusion to flux of C1 is significantly larger than that of slip and Darcy flow at the early time with the difference between different contributions decreasing at the late time. As the contribution of Knudsen diffusion to flow of C1 decreases, the normalized producing C1 concentration also decreases. The contribution of bulk Darcy and slip flow to C1 production is very close for the case with kD = 100 nD. Fig. 11 shows that contribution of bulk Darcy flow to C1 production is small compared to Knudsen and slip flow for kD = 10 nD. The contribution of Knudsen flow and Slip flow is very close and decreases at the late simulation time.

4.2. Effect of initial reservoir pressure Three simulations were performed at different initial reservoir pressures: 4200, 4600, and 5000 psi. Other reservoir parameters are base-case parameters (Table 1). The pressure of the producing end was adjusted so that the difference between initial reservoir pressure and producing-end pressure remains constant at 4000 psia. Fig. 12 shows temporal variation of normalized C1 and C2 concentration at different initial reservoir pressure values. The larger the initial pressure, the higher the initial normalized C1 concentration, because contribution of both the slip effect and Knudsen diffusion to flow increase at higher producing pressures. The normalized C1 concentration drops faster when initial reservoir pressure is lower, because the total amount of gas in place is higher when the reservoir pressure is larger and, therefore, depletion of C1 takes a longer time. The normalized C2 concentration follows a trend opposite of that of C1. Normalized producing C2 concentration starts with initial values less than one and reaches to

Table 2 Correlations to determine the shift in critical properties as a function of molar mass (M) for different sizes of nanopores (Devegowda et al., 2012). Pc shift

Tc shift Fig. 15. Temporal variation of normalized producing C1 concentration for systems with different tortuosity (τ). (Base parameters listed in Table 1.)

2 4 5 2 4 5

nm nm nm nm nm nm

ΔPc ΔPc ΔPc ΔTc ΔTc ΔTc

= 0.085 ln(M) − 0.0693 = −0.085 ln(M) + 0.1193 = −0.077 ln(M) − 0.041 = 0.0636 M0.2129 = 0.0229 M0.2319 = 0.0153 M0.241

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Fig. 16. Effect of confinement correction (Cr.) on temporal variation of normalized producing C1 concentration for systems with different permeability. (Base parameters listed in Table 1.)

higher than unity values at late time. The normalized producing C3 and CO2 concentrations follow the same trend as that of C2.

4.3. Effect of length Four simulations were performed for reservoirs with different lengths: 2, 4, 6, and 8 m. The rest of the parameters were the base-case parameters (Table 1). Fig. 13 shows temporal variation of normalized producing C1 and C2 concentrations. Initial producing C1 concentration is the same for all values of length. For longer systems, the normalized producing C1 concentration remains greater than unity for a longer period of time. For the 2 m system, the normalized producing C1 concentration reaches to less than 1 after only 35 days; however, for the 8 m system, C1 concentration will take much more than 120 days to reach a value less than 1. The longer system holds more gas in place and thus depletion takes longer. This behavior could have practical application in shale gas reservoir development when the length (or size) of the hydraulically fractured blocks is estimated. Smaller fractured blocks lead to rapid decrease in the producing mole fraction of the lightest component. The normalized producing C2 and C3 concentrations follow a trend opposite to that of C1.

Fig. 18. Effect of confinement correction (Cr.) on normalized producing C1 concentration for systems with different tortuosity (τ). (Base parameters listed in Table 1.)

4.4. Effect of tortuosity We performed four simulation cases with tortuosity values of 2, 4, 10, and 40. The rest of the simulation parameters were the base case parameters (Table 1). Larger tortuosity for a constant permeability implies larger average pore size (Eq. (7)). Hence, for a given permeability, the Knudsen number decreases (Eq. (11)), TMAC increases (Eq. (10)), and slip coefficient decreases (Eq. (6)). Therefore, both slip flow and Knudsen diffusion effect decrease significantly when tortuosity increases. Fig. 14 shows spatial pressure profile at different times for different tortuosity values. Pressure depletion is higher when tortuosity is lower. Note that the contribution of bulk Darcy flow is the same, because we assume that permeability is constant and the factors determining degree of reservoir depletion are the contributions of Knudsen diffusion and slip flow. Fig. 15 shows temporal variation of normalized producing C1 concentration at different tortuosity values. Fig. 15 shows that initial producing C1 concentration is larger when tortuosity is smaller. For the systems with smaller tortuosity values, the producing C1 concentration decreases rapidly due to depletion. Normalized producing C2 and C3 concentrations follow a trend opposite to that of C1. 5. Effect of pore confinement on fluid properties and CS Evidence indicates that the phase behavior of fluids changes under confinement, e.g., nanopores in shale. Thermodynamic properties such as critical temperature and pressure, density, and surface tension inside the nanoscale systems are known to deviate significantly from their values in an unconfined state, as a result of increasing influence of pore walls on the molecules (Devegowda et al., 2012; Singh et al.,

Table 3 Reservoir and well system parameters used in approximating the field data (from Freeman et al., 2012).

Fig. 17. Effect of confinement correction (Cr.) on temporal variation of normalized producing C1 concentration for systems with different initial reservoir pressure (Pi). (Base parameters listed in Table 1.)

Property

Value

Darcy permeability, kD (nD) Porosity, ϕ L (m)

100 0.0276 15.24

Initial pressure (psi) Outlet pressure (psi) Temperature (K) Tortuosity Fluid composition

4568 3626 359.55 3.3 80% C1, 7% C2, 5% C3, 5% C4, 3% C5

Notes

Length was taken to be equal to the fracture spacing in the field. Reservoir pressure Well pressure

M. Rezaveisi et al. / International Journal of Coal Geology 121 (2014) 110–122

Fig. 19. The simulated temporal variation of normalized producing mole fraction of different components for the field parameters in Freeman et al. (2012).

2009). This issue is of particular importance in gas condensate fluids in shale formations where pore sizes are small. Condensate banking in the near-wellbore region, a phenomenon that adversely affects well productivity in conventional reservoirs, is significantly alleviated in lowpermeability shale gas reservoirs because of favorable modification of fluid properties under confinement (Devegowda et al., 2012; Sapmanee, 2011). Devegowda et al. (2012) presented a critical review of the previously published research on modeling the changes in critical properties under confinement in nanopores. Singh et al. (2009) conducted numerical simulations to study thermophysical properties (including critical properties) of methane, ethane, propane, n-butane, and n-octane in bulk and silt pores. Devegowda et al. (2012) extended the numerical experimental data of Singh et al. (2009) in 2-, 4-, and 5-nm pores to other hydrocarbons on the basis of their molar mass. Singh et al. (2009) modeled the effect of confinement on phase behavior of the fluid through a shift in critical properties: ΔT c ¼

T cb −T cp T cb

ð31Þ

ΔP c ¼

P cb −P cp P cb

ð32Þ

where P cb and T cb are critical pressure and critical temperature in bulk fluid and P cp and T cp are critical pressure and temperature in confined pores. Devegowda et al. (2012) used the shifts in critical properties given in Table 2 to simulate gas condensate production with confinement effects. The effect of confinement in nanopores on CS process and production behavior of shale gas reservoirs can be incorporated in our numerical model by shifting the critical properties as specified in Table 2. In this process we only consider the effect of nanopores on shifting the critical properties of the fluid for pores smaller than 10 nm in size. For CO2 this effect was assumed to be negligible. We used linear interpolation between the pore sizes reported in Table 2 to obtain the required shift in critical properties for pores sizes that are not tabulated. The temporal variation of normalized producing C1 concentration at different permeability values with and without the confinement effect is presented in Fig. 16. The curves for equivalent permeability are displayed in the same color. The dashed-line curves show the confinement effect. The temporal variations of normalized producing C1 and other component concentrations including the confinement effect follow the same trend as we observed when the confinement effect was not present. Under the assumptions that we have made and with the base-case parameters in Table 1, Fig. 16 shows that only for

119

permeabilities of 50 nD and 100 nD is there a significant effect of confinement on producing behavior of C1. For permeabilities of 500 nD and 1000 nD, the average pore size of the reservoir is greater than 10 nm and therefore no effect of confinement is expected. Upon consideration of the effect of confinement, no change in producing behavior for the 10 nD reservoir was observed; however, this lack of observed change stemmed from a different pattern of influences. For 10 nD permeability, average pore radius of the reservoir is 1.8 nm. Table 2 suggests that the corresponding relative shifts in critical temperature and pressure are 10.63% and 11.19%, respectively. It seems that the relative changes in critical pressure and temperature have compensatory effects on producing mole fraction of the components. A similar sensitivity study was performed to investigate the effect of confinement on production behavior of the reservoir at different initial reservoir-pressure values. Fig. 17 shows the temporal variation of normalized producing C1 concentration at different values of initial pressure. At larger reservoir pressure, the difference in production profiles between the simulations with and without confinement effects becomes more pronounced. The general behavior of the production curves in this plot suggests that considering the effect of confinement decreases the producing mole fraction of the lightest component and therefore alleviates the CS process. Fig. 18 shows temporal variation of normalized producing C1 concentration with and without confinement effects at different tortuosity values. The effect of considering the shift in critical properties becomes more significant for larger tortuosity values. The trends of the normalized producing mole fraction curves are similar to the cases without confinement effect. 6. Comparison of simulation trends with field data Even though compositional variation of the produced gas in shale wells is well known, there is a lack of reported compositional measurements of the produced gas in shale wells in the literature. Freeman et al. (2012) examined production data and compositional analysis of the produced gas in five wells in a North American shale gas play. We qualitatively compare the trends in the composition of the produced gas in these wells with our simulation results. An exact quantitative comparison is not possible because of (i) the difference between defined boundary conditions in our model and the studied field data, (ii) infinite size of the studied field compared to our bounded model, (iii) differences in flow geometry of the field and our model, and most importantly (iv) presence of sorption process in the field that is ignored in our model. The field data reported by Freeman et al. (2012) indicates that the normalized producing C1 concentration is greater than one and it is either at a stabilized value or has a slightly increasing trend. The normalized producing C1 varies approximately from 1.02 to 1.06 for different wells in their plots. Also, the normalized producing concentration of the rest of the components is generally less than one (except where there is an operational change) and generally the heavier the component, the lower is their normalized producing concentration. The heavier components show an obvious decreasing trend of normalized producing concentration. We used the reservoir parameters reported by Freeman et al. (2012), given in Table 3 to qualitatively compare our simulation results with trends observed in the field. Our simulation results show similar trends in normalized producing concentration of different components. We observe a stabilized trend of normalized producing C1 early in the production (Fig. 19) at a value of approximately 1.025 with a very small decreasing trend at later time (the normalized producing concentration of C1 decreases only 0.0001 over the 120 days of production). The slight decreasing trend in our simulations is because of the finite size of our model porous medium and depletion of the source of C1. The normalized producing concentration of all other heavier components is less than one and similar to the field data the heavier the component the lower is the normalized producing

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concentration. We do not observe a clear increasing trend in our normalized producing C1, most probably because we did not include sorption process in our model. Sorption can also affect the compositional variations of the produced gas stream in shale reservoirs. Based on isotopic signature of the produced gas, Freeman et al. (2012) observed that there is a potential change from strong desorption influence early in the production to more diffusion influence as production pressure becomes more stable. Furthermore, initial pressure of a shale reservoir may be well above critical sorption pressure and sorption may not happen at the early life of the reservoir; therefore slip and Knudsen diffusion may be the only reasons of chromatography separation. Our proposed model here, although incomplete in terms of sorption, is applicable to many shale systems at least as long as the pressure is higher than critical sorption pressure or under stabilized pressure conditions in the reservoir.

manuscript. Publication is authorized by the Director, Bureau of Economic Geology. Appendix A The finite difference scheme for discretization of Eq. (14) is given below for different terms in the equation. For temporal derivative backward difference is used.

ðϕ ρ yi Þnþ1 −ðϕ ρ yi Þnj ∂ðϕρyi Þ j ¼− Δt ∂t

ðA1Þ

The spatial derivatives are discretized with central difference approximation with a step size of Δx 2 as follows:

7. Conclusions Modeling the variations of the producing gas composition helps in characterizing the reservoir through history-matching analysis of the produced gas composition. Important reservoir parameters such as permeability, matrix block size, porosity, and contribution of the sorbed gas to gas production may be subject to history-matching analysis and therefore valuable information may be obtained by applying the results of this research. We developed a numerical transport model for multi-component gas flow through ultralow-permeability linear porous media. The model captures the flow behavior in nanopores encountered in shale gas reservoirs. Knudsen diffusion, slip flow, and bulk Darcy flow are taken into account. Some highlights of the model results are: 1. The simulation results confirm variations in the mole fraction of the produced gas components with time. Light components are preferentially produced until certain depletion in the system. 2. The normalized producing mole fraction of the lightest component is initially greater than one; this value decreases monotonically with time as the reservoir depletes. For other components with lower Knudsen diffusivity and slip coefficient, the trend of variations in the producing mole fractions is opposite to that of the lightest component. 3. The normalized producing mole fraction of methane (the lightest component) is initially larger and will decrease at a smaller rate for larger initial pressure and for lower Darcy permeability. For high Darcy permeability and lower initial reservoir pressure, the normalized producing mole fraction of methane becomes smaller than one in a shorter time. 4. The initial normalized producing mole fraction of methane is the same for systems with different lengths. However, as the system length increases, the time required for the normalized producing mole fraction of methane to become smaller than one increases greatly. 5. For a given permeability, the larger the tortuosity, the smaller the normalized producing mole fraction of methane at early time. 6. Consideration of the effect of confinement on critical properties of the fluid alleviates CS process in the system.





 nþ1 ∂ k b ∂P 1 k b ∂P nþ1 ρ yi d 1 þ i ¼ ρyi d 1 þ i j μ P ∂x μ P jþ1=2 ∂x jþ1=2 Δx j ∂x 
 nþ1 k b ∂P nþ1 − ρyi d 1 þ i j μ P j−1=2 ∂x j−1=2

¼

1 Δx j

!
 nþ1 nþ1
 nþ1 P jþ1 −P nþ1 P nþ1 −P nþ1 k b k b j j j−1 ρ yi d 1 þ i − ρyi d 1 þ i μ P jþ1=2 Δx jþ1=2 μ P Δx j−1=2 j−1=2

ðA2Þ

ðA3Þ

where Eq. (A3) is obtained by substituting Eq. (A4) for pressure derivatives into Eq. (A2).   nþ1 P nþ1 P nþ1 −P nþ1 ∂P nþ1 ∂P nþ1 jþ1 −P j j j−1 ¼ and  ¼ :  Δx jþ1=2 Δx j−1=2 ∂x jþ1=2 ∂x j−1=2

ðA4Þ

In a similar manner,


∂ Dk;eff ;i ∂ Pyi ∂x RT ∂x Z nþ1  nþ1 3 2 i i

Dk;eff ;i nþ1 6 Pyi = Z jþ1 − Pyi = Z j 7 1 ¼ 4 5 Δx j RT jþ1=2 Δx jþ1=2 nþ1  nþ1 3 2 ! i i

Dk;eff ;i nþ1 6 Pyi = Z j − Pyi = Z j−1 7 − 4 5 : RT j−1=2 Δx j−1=2

ðA5Þ

Upwind method was used to evaluate the value of the parameters at the cell interfaces according to the following equations:


 nþ1
 nþ1 k b k b ρyi d 1 þ i ¼ ρyi d 1 þ i μ P jþ1=2 μ P j

ðA6Þ

ðA7Þ

Acknowledgments


 nþ1
 nþ1 k b k b ρyi d 1 þ i ¼ ρyi d 1 þ i : μ P j−1=2 μ P j−1

This work was supported partly by the NanoGeosciences Lab at the Bureau of Economic Geology (UT-Austin). Chris Parker edited the

Similar equations were used for the other nonlinear coefficients at the cell interfaces.

M. Rezaveisi et al. / International Journal of Coal Geology 121 (2014) 110–122

121

Appendix B For a system with NC components and NB gridblocks, different terms in Eq. (27) are given below where f and x are vectors of residuals and unknowns, respectively and J is the Jacobin matrix. Ri,j is the residual of conservation equation (Eq. (27)) of component i in block j. yi,j is mole fraction of component i in block j and Pj is pressure in block j. 2 6 6 6 6 6 6 6 6 6 6 6 ! 6 f ¼6 6 6 6 6 6 6 6 6 6 6 6 4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ! 6 x ¼6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

R1;1 R2;1 : RNc;1 R1;2 R2;2 : RNC;2 : : : R1;NB R2;NB : RNC;NB

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

y1;1 y2;1 :

yNc−1;1 P1 y1;2 y2;2 : yNC−1;2 P2 : : : y1;NB y2;NB : yNC−1;NB P NB

2

∂R1;1 6 6 ∂y1;1 6 6 ∂R2;1 6 6 ∂y 6 1;1 6 6 ∂R : 6 NC;1 6 6 ∂y1;1 6 : 6 6 : J¼6 6 : 6 6 ∂R 1;NB 6 6 6 ∂y1;1 6 : 6 6 ∂RNC−1;NB 6 6 ∂y 1;1 6 6 ∂R 4 NC;NB ∂y1;1

ðB1Þ

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

∂R1;1 ∂y2;1 ∂R2;1 ∂y2;1 : ∂RNC;1 ∂y2;1 : : : ∂R1;NB ∂y2;1 : ∂RNC−1;NB ∂y2;1 ∂RNC;NB ∂y2;1

ðB2Þ

:: :: : : : : : :: : :: ::

∂R1;1 ∂yNC−1;1 ∂R2;1 ∂yNC−1;1 : ∂RNC;1 ∂yNC−1;1 : : : ∂R1;NB ∂yNC−1;1 : ∂RNC−1;NB ∂yNC−1;1 ∂RNC;NB ∂yNC−1;1

∂R1;1 ∂P 1 ∂R2;1 ∂P 1 : ∂RNC;1 ∂P 1 : : : ∂R1;NB ∂P 1 : ∂RNC−1;NB ∂P 1 ∂RNC;NB ∂P 1

:

: :

:

: :

:

: :

:

: :

: : :

: : : : : :

:

: :

:

: :

:

: :

:

: :

∂R1;1 ∂y1;NB ∂R2;1 ∂y1;NB : ∂RNC;1 ∂y1;NB : : : ∂R1;NB ∂y1;NB : ∂RNC−1;NB ∂y1;NB ∂RNC;NB ∂y1;NB

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:: :: : :: : : : :: : :: ::

∂R1;1 ∂yNC−1;NB ∂R2;1 ∂yNC−1;NB : ∂RNC;1 ∂yNC−1;NB : : : ∂R1;NB ∂yNC−1;NB : ∂RNC−1;NB ∂yNC−1;NB ∂RNC;NB ∂yNC−1;NB

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