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    UNIVERSITY OF OKLAHOMA

    GRADUATE COLLEGE

    UNCERTAINTIES IN SHALE GAS IN-PLACE CALCULATIONS: MOLECULAR SIMULATION APPROACH

    A THESIS

    SUMMITTED TO THE GRADUATE FACULTY

    in partial fulfillment of the requirement for the

    Degree of

    MASTER OF SCIENCE

    By

    MERY DIAZ CAMPOSNorman, Oklahoma

    2010

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    UNCERTAINTIES IN SHALE GAS IN-PLACE CALCULATIONS: MOLECULAR SIMULATION APPROACH

    A THESIS APPROVED FOR THEMEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING

    BY

    ___________________________________I. Yucel Akkutlu, Chair

    ___________________________________Carl Sondergeld

    ___________________________________Chandra Rai

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    Copyright by MERY DIAZ CAMPOS 2010All Rights Reserved.

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    iv

    ACKNOWLEDGEMENTS

    First of all, I thank Dr. Yucel Akkutlu for supervising my research during these two years.

    Without his new ideas it would not have been possible to develop the investigation to its

    present stage. Thanks for the constant encouragement and the academic freedom he gave to

    me during my research. Thanks for all his advice, both technical and personal. It was very

    encouraging for me every time he showed such enthusiasm looking at the results of the

    simulations as myself.

    I would like to thank my committee members Drs. Chandra Rai and Carl Sondergeld for

    their encouragements to start molecular-level investigations related to reservoir phenomena at

    the University of Oklahoma.

    The technical support I received during the development of this work has been

    indispensable to improve the quality of the molecular dynamics simulations developed in this

    work. In particular I would like to mention Dr. Alberto Striolo and Dr. Richard Sigal they deserve

    much credit in generating an appropriate ensemble for the study developed in the present

    thesis. Thanks to all the faculty of PE for sharing with me the enthusiasm of learning about this

    new molecular level approach in order to understand some physical chemistry properties of

    hydrocarbon fluids when confined in micro- and nano-porous media.

    It was a pleasure for me to discover that Dr. Akkutlu and I were not the only ones

    interested in the results of the molecular simulations involving nanopores, but also my family

    and friends. I would like to thank my uncle and my mother, in particular, who always follow up

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    v

    my performance with greatest interest. Thanks to my entire family, although being far away,

    they have been my inspiration to continue learning. Thanks to my good friends Chau Diep,

    Yoana Walschap, Pinar Akkutlu, Serita and all my Sooner teammates, for making me feel like

    being at home.

    The simulation studies were performed using parallel computing at the OU

    Supercomputing Center for Education & Research (OSCER). I thank the director Dr. Henry

    Neeman for his support of my studies.

    My education and research has been supported by Devon Energy. I am grateful for this

    support.

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    iv

    TABLE OF CONTENTS

    Page

    Acknowledgements iv

    Table of contents . iv

    List of Tables vi

    List of Figures vii

    ABSTRACT ix

    CHAPTERS

    1. INTRODUCTION1.1Overview .. 11.2Purpose of study .......... 62. MOLECULAR MODELING AND SIMULATION2.1Molecular Dynamics Simulation .. 92.2Monte Carlo Simulation 112.3Numerical Procedures for Gas Adsorption 132.4Numerical Procedure for Methane Solubility in Nanopores 193. NUMERICAL RESULTS AND DISCUSSION

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    v

    3.1Methane Adsorption in Organic Pores using MolecularSimulations.. 22

    3.2Methane Solubility Enhancement in Water confined to nano-scale pores using MolecularDynamics Simulations 32

    4. CONCLUSIONS AND RECOMMENDATIONS4.1Conclusions ... 364.2Recommendations ... 37

    REFERENCES 38

    NOMNECLATURE . 42

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    vi

    LIST OF TABLES

    Page

    CHAPTER 1

    Table 1.1- Typical TOC contents of North American Shale Gas Plays . 1

    Table 2.1- Lennard-Jones Potential Parameters for Methane-Graphite System ... 17

    Table 2.2- Lennard-Jones potential parameters for Methane-Water-Graphite System 22

    Table 3.1- Methane density at different pressures. Comparison with the bulk density

    values obtained using SUPERTRAPP under the same pressure and

    temperature conditions. .. 28

    Table 3.2- Calculated values of the Solubility Parameters l, L and Henrys Law Constant

    kH at 75.5oC .. 37

    Table 3.3- Effect on the solubility parameters l, L at 75.5oC by changing the wall-water

    interaction. The pore width = 1.62nm . 40

    Table 3.4- Shale properties, low and high sorption capacity, taken from (Ambrose et al.,

    2010), including three different density phases for comparison . 41

    Table 3.5- Free and total storage capacity for Shale A and B, at different adsorbed

    density phase values . 43

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    vii

    LIST OF FIGURES

    Page

    CHAPTER 1

    Figure 2.1 Molecular simulation cell consisting of graphite walls and OPLS-UA methane

    .. 14

    Figure 2.2 Methane insertion at a vacant spot . 25

    Figure 2.3 Total energy (left) and temperature (right) fluctuation over the time for a pure water

    system in thermodynamic equilibrium in a nanopore at 75.5oC . 25

    Figure 3.1- Number density (left) and discrete density (right) profiles for methane in 3.9 nm

    graphite slit-pore at 176

    o

    F . 29

    Figure 3.2 Number density (top) and discrete density (bottom) profiles for methane in 2.31 nm

    graphite slit-pore at 176oF .. 27

    Figure 3.3 Number density (left) and discrete density (right) profiles for methane in 1.14 nm

    graphite slit-pore at 176oF .. 32

    Figure 3.4 Number density (top) and discrete density (bottom) profiles for methane in 3.93nm

    slit-pore as a function of temperature ... 33

    Figure 3.5 Density profiles for methane in three different-size slit-pore 1.14, 2.3 and 3.8nm

    using MC simulation ... 34

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    viii

    Figure 3.6 Number density (top) and discrete density (bottom) profile for carbon dioxide in

    3.8nm slit-pore as function of distance from the wall to the center of the pore, at p =

    1980 psi .. 35

    Figure 3.7 Estimated Ostwald coefficient, L (x103), of methane in water confined to slit-pores

    with varying sizes compared with the solubility in bulk water (top). Corresponding

    SCF Methane/BBl water (down) at 200psi and 168F 38

    Figure 3.8 Estimated Ostwald coefficient, L (x103), of methane in water confined to 1.62 nm

    pore with varying pore-wall wettability 40

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    ix

    ABSTRACT

    Shale gas reservoirs are a major source for the global energy market in the US. In 2008, dry

    natural gas proved reserves attributable to shale reservoirs grew to a total of 32.8 tcf from 21.7

    tcf in 2007 (Annual Energy Outlook 2010). On the other hand regarding natural gas production,

    shale gas is currently and will be the largest contributor to its growth. EIA predicts growth from

    2.7tcf to 6tcf during 2010 and 2035 (Long, G. 2010).

    Shale gas-in-place calculation consists of a volumetric component representing

    hydrocarbons stored in the pore space of the reservoir system as free gas. In addition, a surface

    component is necessary to account for the gas physically adsorbed on large internal surface area

    of the organic microporous matrix. A second volumetric component, which is not considered

    within the calculation is gas dissolved in the formation water or liquid hydrocarbons. For such

    reservoirs, free gas is typically quantified using volumetric method. Adsorbed gas has generally

    been estimated from laboratory studies using equilibrium adsorption isotherms, under

    laboratory conditions and requires a correction for the adsorbed gas fraction present under

    reservoir temperature and pressure conditions. The amount of dissolved gas in formation water

    has not been considered important, based on bulk solubility calculations.

    The thesis is based on a series of theoretical studies using molecular modeling and

    simulations to investigate physical chemical properties of fluids confined to organic micropores.

    The amounts of adsorbed, dissolved and free natural gas are predicted using fluids in carbon slit

    pores with varying size and pore-wall wettability under typical reservoir pressure and

    temperature conditions. It is found that gas adsorption and solubility in liquid are enhanced due

    to dominant pore wall effects. These effects are controlled to a varying degree by the micropore

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    x

    size, temperature, pressure and fluid composition. The implication of these numerical

    observations is that the sorbed gas in the formation may represent a significant portion of the

    total gas reserves in unconventional reservoirs.

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    CHAPTER 1

    INTRODUCTION

    1.1OverviewProduction of natural gas from organic-rich shale makes up an increasingly greater portion of

    the total production in North America. According to the US Department of Energy's (DOE),

    Energy Information Association (EIA), the gross production from shale plays in the US nearly

    doubled from 2007 to 2008 from 1,180 bscf to over 2,000 bscf, and has continued to grow (EIA,

    2010). The main portion of the shale gas production is maintained by the large US shale plays

    like the Barnett, while growth is due to new production coming on line in the Woodford,

    Haynesville, Marcellus, Eagle Ford, and Horn River, Avalon to name a few.

    Natural gas in these reservoirs is stored by two mechanisms, free gas and adsorbed gas.

    The adsorbed gas is stored mainly on the internal surfaces of the organic material. Table 1.1

    shows average total organic content (TOC) by weight percent of the shales (Pollastro et al, 2007;

    Jenden et al., 1993; Mancini et al., 2006; Comer and Hinch, 1987; Reynolds and Munn, 2010).

    Shale or Play TOC (wt %)

    Barnett 4%Marcellus 1-10%Haynesville 0-8%Horn River 3%Woodford 5%

    Table 1.1- Typical TOC contents of North American Shale Gas Plays

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    When conducting a reservoir study on a natural gas field, one of the primary concerns is

    the estimation of initial gas-in-place. The estimate is the basis for disclosure of gas reserves and

    it is important for reservoir engineering analysis such as gas production forecast. Tank-type and

    multi-dimensional (simulation-based) material balance calculations are common industry

    approaches in predicting the gas-in-place when sufficient field performance data are available.

    For disclosure purposes, a deterministic volumetric method, in which a single average

    value is selected for each parameter in the reserves calculations, is most commonly used in

    North America. Probabilistic methods, on the other hand, are increasingly used worldwide and

    give the ability to describe the full range of values for each parameter in order to somewhat

    reflect spatial variability in the parameters and structural intricacies in reservoir architecture. In

    the present research it is shown that any volumetric approach for shale gas reserve estimation

    has added complexity, because in shale the natural gas (mostly methane) exists in different

    thermodynamic states namely: adsorbed, absorbed (or dissolved) and free gas, and that an

    accurate estimation of the gas pore volume in shale reservoirs should not be considered

    independent of these states.

    A simple petrophysical model of shale matrix is illustrated in (Ambrose et al., 2010). The

    model is used to quantify gas-in-place and it is typically done by methods developed specifically

    for tight rocks and other low-permeability formations such as Shales (Luffel et al., 1992; Luffel et

    al., 1993; and GRI, 1997). However, the effective pore volume is not directly determined in

    these studies; rather total porosity, total water volume, and total oil volume (by weight

    difference and an assumed oil density of 0.8 g/cc) are determined. Nonetheless, as shown here

    in equations (1) and (2), the total and effective gas porosity values are equivalent.

    ..... (1)( ) ) ==

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    ...... (2)

    The bulk volume, Vb, in equations (1) and (2) is determined by mercury displacement of

    competent cores. Grain volume, on the other hand, is determined on crushed cores via helium

    porosimetry. The difference between these two volumes yields the total void volume, Vv,

    (associated with the total porosity ) available for all in-situ fluids, i.e., mobile hydrocarbons,

    free and bound water, adsorbed gas, solution gas, and free gas.

    For total gas storage, the shale gas-in-place volumes are generally considered in terms

    of the following:

    1. A volumetric component, Gf, involving hydrocarbons stored in the pore space asfree gas. The free gas volume is quantified by modifications of standard

    reservoir evaluation methods.

    2. A surface component, Ga, with the gas physically adsorbed on large surface areaof the micro- and mesopores (see definition of pore size classification in Chapter

    2). The adsorbed gas amount has generally been quantified from the sorption

    isotherm measurements by establishing an equilibrium adsorption isotherm

    (Mavor et al., 2004). Note that there is no direct measurement of adsorbed gas

    phase. It is estimated indirectly from the measured temperature and pressure

    conditions.

    3. A volumetric component, Gso, involving gas dissolved into the liquidhydrocarbon. This volume is usually combined with adsorbed gas capacity in

    reservoirs that contain a large fraction of liquid hydrocarbon in the pore space.

    4. A volumetric component, Gsw, involving gas dissolved into the formation water.The amount of dissolved gas is estimated from the bulk solubility calculations.

    = ( ) ( )=

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    Although it has traditionally not been considered important, a recent study is

    available discussing significant enhancement in gas solubility in formation

    liquids when confined to small pores (Diaz-Campos et al., 2009).

    Hence, we have Gst as the total gas-in-place, as the sum of all of these volumetric

    contributions:

    ..... (3)

    where the components of storage on the right-hand-side are defined as:

    ........ (4)

    In the current industry standard calculations, the Gso and Gsw terms are not applied.

    The solution gas in mobile hydrocarbons and water, and the adsorbed gas within organic matter

    are combined in the adsorption isotherm analysis; therefore equation (3) is reduced to:

    ...... (5)

    The current volumetric approaches for shale gas are built on the premise that the two

    volumes on the right hand side of equation (5), being associated with the inorganic pores and

    organic solid, respectively, can be estimated independently of one-another. Accordingly, the

    adsorbed gas is associated with the organics, the pore volume of which is considered to be

    = +

    ( )

    =

    =

    +=

    =

    +++=

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    negligible and, therefore, the volume does not need to be accounted for during the calculations

    of free gas; whereas, all of the free gas is associated with the inorganic voids such as

    macropores, fissures, fractures etc. Recently Ambrose has proposed a new petrophysical model

    for Shale Gas in Place calculations (Ambrose et al., 2010), specifically includes a change in Gf

    where the density of adsorbed gas in the organic pore, s, must be known, equation (6).

    ...... (6)

    Direct measurements of the adsorbed phase density s at pressure and temperature

    conditions corresponding to a shale gas reservoir was not carried out yet. Previous research

    done by Mavor et al point out that the density is either assumed equal to the liquid density at

    1atm and -161.6oC, this is 0.4223 g/cc (Mavor et al., 2004); or 0.375g/cc which corresponds to

    the van der Waals equation of state co-volume constant. There is also a recommendation of

    using Kobayashi values. Haydel and Kobayashi measured the adsorbed methane density,

    0.374g/cc, however, their values were encountered using an inorganic adsorbent (silica gel) and

    at low pressures and temperature conditions, 1000psi and 104oF. In this research a density

    values within the range 0.28-0.3g/cc was found using Molecular Simulation. It is found that the

    value is sensitive to changes in temperature, pressure and pore size. Typically, at 3,043 psi and

    176oF, the adsorbed phase density of methane is 0.3 g/cc for a pore width of 2.3nm.

    Using scanning electron microscopy, it has been repeatedly shown that a significant

    portion of the total pore volume is associated with pores in the organics (Wang and Reed, 2009;

    Loucks et al., 2009; Sondergeld et al., 2010). Hence, the organic pores contribute significantly to

    the total free gas pore volume. Although it might appear as a rudimentary discussion in shale

    petrophysics, the observation is important to the gas-in-place calculations. The organic pores

    ( )

    +

    =

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    are typically in a micro size level, thus a portion of the total pore volume could be taken up by

    finite-size adsorption layer and not available for the free gas molecules.

    In addition, in a porous medium characterized by the presence of small pores and

    occurrences of formation water (e.g., free and inherent moisture), such as a coal and shale, gas

    can be stored as dissolved gas in the interstitial water. For gas in the large pores, it is known that

    the dissolved gas is only a small fraction of the amount of gas pore could hold. The amount of

    dissolved gas in the water could increase in smaller pores, however. The amount dissolved then

    may represent a significant portion of the total gas reserves in unconventional reservoirs. The

    latter argument suggests that we should consider gas solubility in water confined to micropores.

    Best quantification of this solubility and insight into this solution chemistry problem ideally

    comes from theoretical studies at the pore-scale, particularly from molecular-level simulations

    and the subsequent free energy change calculations accompanying dissolution. In this thesis,

    natural gas solubility enhancement in water residing in a graphene slit-pore with a length-scale

    on the order of nanometers is investigated using molecular dynamics simulations. The solubility

    of methane in water is estimated under controlled temperature condition using the test particle

    insertion method with the excluded volume map sampling (Stapleton and Panagiotopoulus,

    1990). The results indicate that the methane solubility in nano-scale pores is increased

    significantly (typically, one order of magnitude larger), compared with that in the bulk phase,

    i.e., in the absence of pore walls. The enhancement is further investigated under varying force-

    field conditions due to pore-wall surface potential, namely hydrophobicity, and it is found that

    the solubility is extremely sensitive to pore wall wettability. The work is a fundamental approach

    to better understand the unconventional gas reservoirs, and important for the estimation of gas

    reserves.

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    1.2Purpose of studyIt is not possible with the current technology to carry out direct experimental measurement and

    analysis of the gas behavior and sorption in small pores under extreme conditions of

    temperature, pressure, for example at typical Barnett initial conditions of 4000 psi and 176oF.

    Computer simulations, however, provide a direct connection between the microscopic

    descriptions of the system (potential interaction between molecules, molecular mass, charges,

    etc.) and macroscopic properties of interest, such as adsorption profile, solubility, and vapor-

    liquid coexistence of hydrocarbons etc.

    In this work, based on computer simulation analysis, we argue that the total gas storage

    capacity measurements and the resulting gas-in-place values are inherently uncertain. The

    source of the uncertainty involves the improper accounting of the pore volumes occupied by the

    adsorbed and free gas phases. The work investigates the nature of the uncertainty using

    Equilibrium Molecular Dynamic (EMD) and Monte Carlo simulation (MC) of methane, carbon

    dioxide and water in pores with sizes varying between 1-4 nanometers (nm).

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    CHAPTER 2

    MOLECULAR MODELING AND SIMULATION

    Molecular Dynamics and Monte Carlo Simulations have made enormous strides in recent years

    and are gradually becoming a common tool in science and engineering. This is due to the

    development of new methods for description of complex fluids and materials, and to the

    availability of high-speed and large-capacity computers. Today, molecular models of, for

    example, multi-component mixtures are common (Nath et al., 1998), and predictions of

    structural, mechanical and electronic properties for large molecules are considered to be

    reliable (Karayiannis et al., 2002). Furthermore, molecular simulations are being used more

    frequently to construct virtual experiments in cases where controlled laboratory experiments

    are difficult and too costly, if not impossible, to perform. Molecular investigations of the phase

    transformation of materials under extreme pressures are good examples of the latter

    application (Lacks, 2000).

    The main objective in this study is to predict accurately the thermodynamic state of

    natural gas confined to micro- and nanopores using molecular simulation techniques. The

    output obtained from a molecular simulation is then used, with appropriate time correlation

    functions, to calculate certain macroscopic quantities of interest such as simple thermodynamic

    averages (e.g. kinetic, potential and internal energies, temperature and pressure). The most

    important macroscopic quantity during this investigation is mass density, to determine the

    adsorption profile of methane in a modeled organic wall, and the Ostwald coefficient which can

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    be related to Henrys law constant of solubility. A detailed description of the numerical

    approach is summarized below.

    Regarding the sorption processes in small pores, IUPAC introduced a classification of

    pores as follows (E. Robens et al., 2005). The pores of internal width greater than 50 nm are

    classified as macropores, less than 2 nm as micropores, and between 2 and 50 nm as

    mesopores. Since the term micro is misleading for the petroleum community, we are going to

    refer to the pores less than 2nm as nanopores, to those with sizes in between 2-5nm as

    micropores.

    2.1Molecular Dynamics SimulationMolecular Dynamics (MD) simulation theories lie in classical mechanics, meaning that every

    system in nature is in constant evolution. Of all the physics laws, only those which define a

    trajectory of a system along the time, without losing information from the past, are the

    accepted ones to describe its molecular dynamics.

    One of the most important laws in physics is that the total energy and the total

    momentum of a system is conserved, i.e., / / 0 or / 0 and, / 0,respectively. Total energy, a sum of kinetic plus potential energies ( ), and totalmomentum are conserved as the consequence of Newtons law. Hence, MD is just the numeric

    solution of the classical Newtons equation.

    MD uses the basic principles of Newtonian mechanics to study thermodynamic and

    transport properties of a many-particle system. A system could be a free fluid (bulk) or solid

    material, or fluid confined, fluid in a pore or porous medium. The system is modeled as particles

    (atoms or molecules) with an initial and instantaneously-predicted position and momentum. The

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    particles are allowed to interact for long periods of time to allow the system to be ergodic, i.e., it

    is reached the time when it holds no other conserved quantity besides energy. Hence, the

    system reaches the equilibrium.

    Once the equilibrium has been reached, given a time interval, the entire motion of the

    particles in space and time can be defined and it satisfies the deterministic law of physics.

    Deterministic laws mean if a property is known at one instant of time in the phase space of the

    system, then it is known at any time in future. It is completely deterministic in the sense that not

    only the property can be predicted into the future but also into the past.

    The trajectory of each particle in space and time represents the evolution, the dynamics,

    of the system. Once the trajectory of a system is defined, the information in the microscopic

    level is converted into macroscopic terms such as virial pressure, density, internal energy, etc.

    All these thermodynamic properties, averaged over a long finite period of time, are derived

    from the equations of state and the fundamental thermodynamic equations, taking into account

    that a system is constrained to a conserved quantity. Such quantities are the energy (E), density

    (), number of particles (N), temperature (T), pressure (p) and chemical potential (). Notice

    that the most important conserved quantity is energy. The rule is to study the system subject to

    the constraint of some conserved quantity by fixing it to a known value. The value of the

    conserved quantity can be taken from any experimental result or a previous computer

    simulation work. This means for example we need to know what is the bulk methane density at

    4000psi and 176o

    F, we can take that value from a previous simulation or a previous

    experimental work. Knowing the density of the system we initialize positioning each particle in a

    known volume. Temperature and pressure are fixed and now we run it to find out what it is the

    chemical potential. Hence we start the simulation with a bulk density profile and run the system

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    until it reaches the equilibrium. Once the equilibrium has been reached we know now the

    chemical potential and that value will be used for the next simulation when the bulk system is

    now confined in a porous system.

    A collection of various conserved quantities, usually three of them, makes up an

    ensemble. If an ensemble is defined by NPT it is referred as an isobaric-isothermal ensemble. If

    it is defined by NVE, micro-canonical; NVT, canonical and finally VT is referred to as the grand-

    canonical ensemble. The latter ensemble is used in MC simulation.

    Since numerical integration of the Newtons equations of motion make up the core of

    the simulation, special algorithms (e.g., Verlet, Leap Frog, Beeman algorithms) have been

    developed and commonly utilized during a molecular simulation study. Others exist (Smith, et al.

    2008) like the deflection algorithm (Ercolessi, 1997). In the present study both Verlet and Leap

    Frog algorithms have been used to estimate and analyze the fluid density under typical

    reservoir pressure and temperature conditions: (i) runs involving bulk-phase (i.e., in the absence

    pore-walls) methane density measurements using a fixed number of methane molecules at fixed

    pressure and temperature (NPT), and (ii) runs involving measurements of density of methane at

    the same temperature but confined to a small pore with organic (graphite) walls using a

    canonical ensemble (NVT); (3) runs involving methane solubility in water confined to small pore

    with graphite walls by mean of a canonical ensemble as well.

    2.2Monte Carlo SimulationThere exists an exhaustive literature studying the equilibrium thermodynamics of fluids using

    MC simulation involving phase change of bulk fluids (Harris, 1995), characterization of porous

    materials using gas adsorption (Aukett, 1992; Sweatman and Quirke, 2001), and multi-

    component gas separation (Ghoufi, 2009). Monte Carlo (MC) simulation is basically a stochastic

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    method for sampling points in multidimensional space, according to a prescribed probability

    distribution defined on that space. It allows us predict thermodynamic properties such as: Gibbs

    free energy (specifically in grand canonical MC), enthalpy, virial pressure, compressibility gas

    factor, particle density, etc.; based on the principles of statistical mechanics. There is a

    trajectory in phase space generated by MC which samples from a chosen statistical ensemble.

    Note that this trajectory is not a dynamic trajectory, there is no time involved in the calculation.

    It is solely a trajectory along the phase space. Since in MC it is not important the dynamics of the

    system, it is a simpler technique, less computer intensive hence faster technique than MD. The

    Monte Carlo method enables the use of certain ensembles specifically designed for computing

    phase equilibria (particularly the Gibbs ensemble) that are very difficult to simulate using

    molecular dynamics (Marcus Martin, 2008).

    As in the case of MD, there are various algorithms used to perform MC simulation. One

    of the most useful is the Metropolis algorithm (uses Markov chain) in which first it has to be

    designed some rate at which it has to be chosen new states jfrom an initial state iand then it

    should have an acceptance criteria. A typical acceptance criterion is that when the new state has

    lower energy than the previous one, then the new state is accepted at 100%. If the new state

    has a higher energy than the previous state then the acceptance of the new state is related with

    a certain thermal-like probability (Boltzmann factor), such that the detailed balance is

    satisfied. Detailed balance states that the rate of going out of a state iinto a statejdivided over

    the rate of going out of a state jinto a state iequal the probability of being in a statejover the

    probability of being in a state i. If the detailed balance is satisfied then we end up with a steady

    state distribution; hence the probability distribution does not change anymore. The multi-

    dimensional space sampled is the configuration space spanned by the coordinates of the

    molecules constituting the system. The probability density is set by an equilibrium ensemble,

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    i.e., collection of a large number of replications of the system. Sampled configuration-space

    points (or states) form a Markov chain (Metropolis algorithm), with each state being formed

    from the previous one in a Monte Carlo (MC) step. Each MC step is executed in two stages. The

    first stage attempts an elementary move from the current state i into a new state j with

    probability (ij), where the stochastic matrix of attempt probabilities is symmetric. The

    second stage then accepts (or rejects) the attempted move with a certain probability which is a

    function of the probability density set by the equilibrium ensemble. If the move is rejected, state

    iis retained for the current step.

    As the consequence of this iterative process, the generated Markov chain of states

    asymptotically samples the probability distribution; thus, all the configuration-dependent

    properties fluctuate around constant average values representative of thermal-equilibrium. It is

    said that the equilibrium is reached when the variance in the estimate of the average ensemble

    property, , say in NVT ensemble; is minimized 2 (Allen and Tildesley, 2007). In

    other words the Markov chain has converged. The system will continue in such way that the

    energy will always fluctuate around the minimum value. Those states with low energy are the

    ones that are sampled in order to find the average thermodynamic quantities.

    In this study the Grand Canonical Monte Carlo (GCMC) simulations are used to

    investigate the fluid density profile, when the fluid, methane, is confined in small organic pores

    under typical shale reservoir pressure and temperature conditions.

    2.3Numerical Procedures for Gas Adsorption2.3.1 Molecular Dynamics Approach

    For the molecular-level investigation, methane, which is the main component in natural

    gas reservoirs, is considered at some supercritical condition under thermodynamic equilibrium

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    in three-dimensional periodic orthorhombic pore geometry consisting of upper and lower pore-

    walls made of graphene (carbon) layers, see Fig. 2.1. For simplicity it is considered the shape of a

    pore as a slit-like pore.

    Using MD, for comparison, we take into account three slit-like pores with a pore width

    equal to H=3.9 nm (micro pore) and H=2.3 nm (micropore) and 1.1nm (nanopore), respectively.

    Figure 2.1 Molecular simulation cell consisting of graphite walls and OPLS-UA methane

    Pore width is an important length-scale of the study, which is defined as the distance

    between the inner most graphene planes. The pores maintain fixed dimension in the y-

    coordinate: Ly=3.93nm; however, the dimension is changed in the x-coordinate such that both

    pores host approx the same number of methane molecules (400-450) during the simulations

    and save some computational time as well. Hence, the dimensions in the x direction become Lx

    =4.26 nm and Lx =7.67 nm for the large and small pores, respectively. The fluid density profile

    was calculated for the 2.3 nm pore. A total of 400-600 methane molecules are typically used

    during the simulations. 100 up to 144 processors have been used for the simulation and the

    zy

    x

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    approximate parallel computation time was typically between 48 min and 25 seconds for the

    corresponding number of processors respectively.

    A united-atom carbon-centered Lennard-Jones potential (based on the optimized

    potentials for liquid simulations, OPLS-UA force field), is used as the model of a methane

    molecule, Table 2.1 shows the energy and distance Lennard Jones parameters used for fluid-

    fluid and solid-solid interactions (Nath et al. 1998). Notice that has energy units (Temperature

    multiplied by the Boltzman constant, kB, has energy units)

    The methane-solid and methane-methane interactions are of the Lennard-Jones (LJ)

    type (Frenkel et al., 2002). A Lorentz-Berthelot mixing rule [ij=(ii + jj)/2 and ij=(iijj)1/2] is set

    up in order to describe solid-fluid (i.e., pore wall-methane) interactions. Here, ij and ij are the

    Lennard-Jones (LJ) parameters accounting for interactions between a molecular site of methane

    species and a carbon atom of the organic wall. LJ were cut off at 4.1 for the 3.9 pore width, at

    3 for the 1.1 and 2.3 nm pore width respectively. and are Lennard Jones parameter where

    , in units of energy (kJ/mol or K), and in units of distance () are the depth of the potential

    well and the distance between the two molecules when its potential energy is cero.

    The molecular simulation package, DL-POLY (version 2.20), was used to perform the

    molecular dynamics simulations (Todorov and Smit, 2008). Methane bulk density computations

    are done considering isobaric and isothermic conditions with a constant number of atoms,

    pressure and temperature (NPT) at three constant temperatures (T), 176 oF, 212oF, and 266oF,

    using Nos-Hoover thermostat and barostat ensemble (Frenkel and Smit, 2002). In order to

    keep pressure and temperature constant it is used an algorithm which will measure those

    parameters at a certain interval of time, this time is known as relaxation time, which is less than

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    the time step in the simulation. The relaxation time used in the ensembles was between 8 to12

    picoseconds (ps) for the thermostat and 10 to 15 ps for the barostat.

    In the case of simulations involving methane in carbon slit-pore, initially the fluid system

    was equilibrated at a constant temperature, with a canonical ensemble (constant number of

    atoms, constant volume and constant temperature, i.e., NVT) using Berendsen thermostat

    algorithm (Frenkel and Smit, 2002). The equilibrium is assumed to be achieved if no drift as a

    function of time was observed in the time-independent quantity, such as the total energy of the

    system. After the equilibrium is reached, a new simulation is carried out in the canonical (NVT)

    ensemble using the Leapfrog algorithm (Frenkel and Smit, 2002; Allen and Tildesley, 2007) and

    Nos-Hoover thermostat. Although the Berendsen algorithm smoothly forces the fluid system to

    reach certain positions and velocity in which the temperature fluctuation is minimized, it is time-

    irreversible and the computations are not considered in the canonical ensemble; the Nos-

    Hoover algorithm needs to be used to control the system temperature while the real canonical

    ensemble computations are carried out for the density computations (Frenkel and Smit, 2002).

    The relaxation time used for the Nos-Hoover thermostat is optimized to 13, 15 and 20 ps for

    the pore widths of 1.1, 2.3 and 3.9 nm respectively. The values of the relaxation time needed to

    be optimized because very small values are known to cause high frequency temperature

    oscillations on the thermostat, whereas values that are too large lead to drift in temperature

    (Hnenberger, 2005). Typically, the total run time for a simulation was from 1.2 to 1.5

    nanoseconds for the nano and micropores respectively and the time step used was 0.003 ps.

    At the end of each simulation run, number density Number (number of molecules per a

    certain volume, in this case the volume is Lx=76.7 , Ly=30.3 and Lz=0.2) for methane across

    the pore space was computed. Notice that Number is computed at every z =0.2 interval (for

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    the continuous density profile) and CH4 (discrete density profile) at every volume segment

    Lz=3.8 in the z-direction. 3.8 corresponds to the diameter of methane particle. The number

    density for each volume segment is estimated by summing the number of molecules counted in

    each z-interval within the volume segment and dividing the total by the number of intervals.

    Following, the number density is converted to the local mass density of methane using the

    molecular weight of methane MCH4 and Avogadros number as follows:

    ... (7)

    As explained earlier, it is expected that the local mass density for methane across the

    pore will be different from its mean bulk density due to changing levels of interactions between

    the methane-methane and methane-carbon bodies. The purpose of our numerical investigation

    was to predict a precise density profile across the pore as an indication of the presence of these

    interactions and to determine an average adsorbed gas density value.

    Table 2.1- Lennard-Jones Potential Parameters for Methane-Graphite System.

    2.3.2 Monte Carlo Approach

    NPT-Gibbs Ensemble Monte Carlo (NPT-GEMC) simulations are used in this work. The

    Gibbs ensemble is defined by introducing a system in which the temperature, T, pressure, p, and

    number of fluid particles, N are fixed a priori. T = 176 F, p = 3043 psi and N = 2000. This system is

    divided into two (separate) sub-systems labeled 1 and 2. The (variable) volumes of these sub-

    =

    atom (nm) /kB (K)carbon 0.340 28.0

    methane 0.373 147.9

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    systems are V1 and V2 and (variable) number of particles N1 and N2 respectively. Note that N = N1

    + N2. System 1 contains the porous medium; system 2 contains methane in ideal gas phase.

    System 2 behaves as a reservoir source. During the adsorption process carried out within the

    porous medium. The thermodynamic equilibrium is achieved when the Gibbs free energy (G)

    reaches a minimum value and fluctuates around that minimum point (Gibbs free energy is kept

    constant). This means that the chemical potentials in the adsorbed phase (system 1) and in the

    bulk phase (system 2) are equal, which allows one to directly relate the adsorption information

    to the bulk phase properties. The simulations were performed in cycles, each cycle having four

    steps: a biased volume displacement step, attempt to transfer molecules between the 2 sub

    systems, reinsertion moves, translation moves and attempt to rotate a molecule (adsorbate) in

    both systems. Notice that since methane is modeled as a neutral sphere LJ particle, the rotation

    of the same will not produce any different result than when it is not rotated.

    In the volume displacement step system 1, which contains the porous medium is

    constrained to have zero displacement probability and system 2 will have a random volume

    displacement of maximum 0.13. The maximum displacement was chosen in such a way that the

    acceptance ratio was approximately 50 per cent. This is a biased displacement because we want

    to avoid change in volume of the porous medium. The next step was to transfer molecules

    between both sub-systems, at initial state there is no adsorbate in system 1 and the probability

    of transferring it from system 2, is set up to 1. Once system 1 accumulates a certain number of

    adsorbates then the probability of placing one particle from one to another system is set up to

    5%. The third step is the re-insertion move, set up to 10%; this move takes a molecule out of

    one system and tries to place it back into the same system. The last step is the attempt of

    performing a center-of-mass translation move on a molecule without regard to which system

    the molecule is currently located in. This move chooses a molecule of the appropriate type at

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    random, chooses a vector on a unit sphere at random, and then attempts to displace the entire

    molecule a random distance. The target acceptance rate for the center-of-mass translation

    move was 50%.

    Three pore width systems were modeled, in order to compare it with the results

    obtained with Molecular Dynamics. In this case it was elaborated a pore width of 3.8 nm which

    differs a little with the pore in MD, however it was considered better the 3.8nm because it will

    contain along the width 10 methane molecules and the analysis of the density profile was

    simpler. The Lennard-Jones potential was truncated at 4.5CH4 for the 3.8nm width pore, 3CH4

    for the 2.31nm pore width and at ~1.9CH4 for the 1.14nm pore width. For each simulation we

    performed 20000 cycles.

    At the end of each simulation run, number density of methane is calculated from the

    last configuration space obtained in system 1. It is counted, on z direction (where the pore width

    extends to), the number of particles encountered at every z=3.8 and divided this amount by

    the volume in which the correspondingly number of methane fall within. Once the particle

    density is known, Number, local mass density is calculated using methane molecular weight MCH4

    and Avogadros number as it is shown in equation 7.

    The purpose of our numerical MC investigation was to confirm the values of adsorbed

    density phase for methane obtained by MD. In that way we can validate both of our results,

    since there is no experimental value at the moment to compare with.

    2.4Numerical Procedure for Methane Solubility in NanoporesThe main objective is to accurately predict the solubility of methane in bulk water and in water

    confined to nanopore using Test Particle Method (Widom, 1963) combined with the Excluded

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    Volume Map Sampling (EVMS). The output obtained from molecular simulation is then used,

    with appropriate time correlation functions, to calculate certain macroscopic quantities of

    interest such as simple thermodynamic averages (e.g. kinetic, potential and internal energies,

    temperature and pressure) and transport coefficients related to the dynamics of fluid (e.g. shear

    and bulk viscosities, diffusive heat and mass transfer coefficients). In our case this macroscopic

    quantity is the Ostwald coefficient which can be related to Henrys law constant of solubility. A

    detailed description of the numerical method can be found in classical books by Allen and

    Tildesley (1987), and Frenkel and Smit (2002).

    2.4.1 Fluid Model and Potential Function.Simulations of fluid in the canonical (NVT) ensemble are carried out using the Leapfrog

    algorithm and Nos-Hoover Thermostat at 75.5oC.Typically, the bulk fluid system is composed of

    891 water molecules in a 3-D parallelepiped computation cell of 3.85x3.46x2.08 nm furnished

    with periodic boundary conditions, resulting in 0.961g/cm3 water density at 75.5oC.

    There are a number of atomistic models for water (Chaplin, 2004). The so-called

    extended simple-point charge water model (SPC/E) characterized by three point masses with a

    O-H bond of 0.1 nm, and H-O-H angle equal to 109.47 o was used in this study. The centered

    electron charges on the oxygen and hydrogen are -0.8476e and +0.4238e, respectively. O-H

    bond is kept rigid during the simulations using SHAKE algorithm. A time step of 0.2 fs was

    assigned to maintain good energy conservation.

    Lennard-Jones interaction is used to represent the van der Waals forces between the

    oxygen-oxygen atoms of the two SPC/E water:

    4

    . (8)

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    where 0.6504 kJ/mol and & 3.1656 are the depth of the potential well and thedistance between the two water molecules respectively (when its potential energy is at

    minimum); r is the inter-particle distance at real time during the simulation. Calculations of long-

    range water interactions were performed using Ewald sum technique. The total interaction

    energy between the water molecules consists of the Lennard-Jones potential and the Coulombic

    potential based on the classical electrostatics: UTOTAL = ULJ +UC, where UC between two molecules

    iand j is represented as the sum of Coulomb interactions acting among the charged points in

    the following way:

    * ,-,- .. (9)

    where is the distance between site L of the molecule i and site J of the molecule j.

    A united-atom carbon-centered Lennard-Jones potential (based on the optimized

    potentials for liquid simulations, OPLS-UA) force field, =1.2309 kJ/mol and =3.73 , has been

    used in the simulations of methane. Methane is considered as a spherical molecule and the

    water-methane interaction was described by the optimized Konrad-Lankau (2005) potential, see

    Table 2.2, the methane-methane interaction is not required since methane is the test particle.

    Fluid-fluid and fluid-solid interactions are also of the Lennard Jones type and act between

    methane-water, water-water, graphite-water and graphite-methane respectively. The LJ

    interactions were cut off at 0.9nm.

    2.4.2 Small Pore Model.Graphite is modeled as a slit-like pore, using the Lennard Jones conventional potential

    where each plane of the graphitic surface, i.e., graphene, consists of stacked planes of carbon

    atoms separated by 3.4 distance. The carbon-carbon separation on the plane along a hexagon

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    side is 0.142nm.The distance between two graphene layers which confine the fluids is

    considered as the width of the slit-pore system. The solubility study is numerically performed at

    four separate confined systems with pore width values of 6.9, 11.6, 16.2 and 20.8. As the pore

    width is increased, the number N of water molecules varies from 297 until 891. Standard

    Lorentz-Berthelot combination rule is used to obtain the corresponding LJ parameters and

    describe the interactions of graphite-methane and graphite-water:

    & & &7 and 8 ..... (10)

    where , and jj, jj correspond to the distance and energy parameters of the pure

    chemical species case. The distance and energy parameter for describing graphite-methane

    interaction are modified by a constant of 0.97 and 1.66 respectively. The total length of each

    MD run was 400 picoseconds.

    interacting molecules (kJ/mol) ()

    SPC/E water- SPC/E water 0.6504 3.1656

    OPLS-UA methane-SPC/E water 1.0131 3.56

    graphite-graphite 0.2353 3.4

    graphite-OPLS-UA methane 0.8947 3.4478

    graphite-SPC/E water 0.3912 3.2828

    Table 2.2- Lennard-Jones potential parameters for methane-water-graphite system.

    2.4.3 Infinite Methane Dilution and Widom Test Particle with EVMS.Usually the solubility of a solute (2) in a liquid solvent (1) is measured by the Ostwald

    coefficient, L ; /

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    L e>?ABC>ABDE/FGH..... (11)

    where B; and B< are the excess chemical potentials of the solute in the liquid and gas phases,respectively. Hence, in general, the evaluation of the solubility requires knowledge of the excess

    chemical potential of the solute in both phases.

    When the system density in the gas phase is negligible, Ostwald coefficient becomes

    identical to the liquid phase activity coefficient ; and the only required quantity for itsevaluation is the excess chemical potential of the solute in the liquid phase alone (Guillot and

    Guissani, 1993):

    L ; e>ABC/FGH...... (12)

    In statistical mechanics the excess chemical potential B of one molecule of solute in a fluidsolvent, i.e., the infinite dilution limit, can be given in terms of the Helmholtz free energy

    difference between the system composed of (N+1) molecules and the one composed of N

    solvent molecules:

    B MNln ?PQRSTUPQR E ... (13)

    Here, ZT,N is the configurational partition function at temperature T and volume V. Using test

    particle method described by Widom (1963) Eqn. 10 can be re-written as

    B

    MNln Ve

    WXYZQ

    [\ MN]..... (14)

    where X is the potential energy difference between a system composed of (N+1) molecules andthat containing only the N solvent molecules. Consequently, it is possible in practice to

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    introduce methane, as a test particle, at a fixed vacant location into the solvent and evaluate the

    Boltzman factor exp(-/kBT), while moving the whole system using the MD simulation.

    In addition to the Ostwald coefficient, the solubility of gases in liquids can be expressed

    in terms of Henrys law constant, kH. In the liquid phase kH is simply related to the activity

    coefficient l with the expression:

    =

    ......... (15)

    where is the number density of the pure solvent.

    Before calculating the potential energy of the interaction between the test particle that

    has to be inserted and the solvent, a map is made of the previous solvent system in equilibrium,

    this map is done by dividing the system into small boxes that can contain at most one molecule

    of solvent. Each box is labeled depending on whether it contains a particle or it is empty. This

    map is used to check whether there is an empty cavity or space for the particle to be inserted. In

    our case, we divided each configuration into a series of cubic sub-cells with length of 3.1656

    and inserted in an empty cavity the methane molecule. See Figure 2.2.

    Prior to the insertion of methane, the system was equilibrated at 75.5oC using Berendsen

    thermostat algorithm. After the equilibrium has reached, Nos-Hoover thermostat algorithm is

    used to control the system temperature and carry out the real canonical ensemble on the N and

    N+ 1 particle fluid systems. The equilibrium is assumed to be achieved if no drift was observed

    in the time-independent quantity, such as the total energy of the system. Figure 2.3 shows the

    total energy of water system under equilibrium conditions in a nano slit-pore at 75.5oC.

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    At frequent intervals during the simulation we generate, by using the excluded map

    sampling method, a coordinate with N+1 particles, we then compute the Boltzmann factor exp(-

    /kBT), sampling the last 360 canonical configurations for each confined system. The interval

    for the insertion was every 50ps. In the case of the bulk model a trajectory of 120ps was

    sampled for analyzing the Boltzmann factor.

    Figure 2.2 Methane insertion at a vacant spot.

    Figure 2.3 Total energy (left) and temperature (right) fluctuation over the time for a pure water

    system in thermodynamic equilibrium in a nanopore of 1.5nm at 75.5oC

    2.4.4 Pore Walls with Controlled Wettability.When a small amount of two immiscible fluids are put in contact with a flat solid

    surface, three phases are delimited by a certain triple contact line. Since each interface has its

    y = 7E-05x - 31287

    -31,294

    -31,292

    -31,290

    -31,288

    -31,286

    -31,284

    -31,282

    -31,280

    0 100 200 300 400

    TOTAL ENERGY EVALUATION

    Time, ps

    0.2fs timestep, 348.7K, Nos-Hoover the rmostat

    y = 5E-06x + 348.7

    348.0

    348.2

    348.4

    348.6

    348.8

    349.0

    349.2

    349.4

    0 100 200 300 400 500 600

    TEMPERATURE EVALUATION

    Time, ps

    0.2fs timestep, 348.7K, Nos-Hoover thermostat

    Temperature,K

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    own free energy per unit area, the fluid-solid interaction energy will determine whether the

    solid surface is non-, partial- or totally wet by the correspondent fluid. The partial or total fluid-

    solid wetting can be defined as surface hydrophilicity/hydrophobicity depending on whether the

    fluid is water or methane. For low-energy surfaces such as graphite, its hydrophobicity can be

    driven by a strong methane-wall interaction as well as by a weak water-wall interaction,

    promoting an easier sliding for the water particles across the solid (Roselman, 1976).

    In principle a given surface is wetted completely if the solid-fluid attraction (i.e., van der

    Waals interaction near the surface) is much higher than the fluid-fluid attraction. Using MD

    simulation Cao et al. (2006) have earlier shown that a small slit-like pore structure can have an

    enhanced surface hydrophobicity due to changes in the liquid-solid interaction. In this study, we

    varied the LJ energy parameter for the potential interaction between water-graphite as follows:

    = ........................................................................... (16)

    where the coefficient c takes values of 0.4, 0.6 and 1.0, in order to see the hydrophobic (when c

    = 0.4 and 0.6) of the wall over the solubility.

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    CHAPTER 3

    NUMERICAL RESULTS AND DISCUSSION

    3.1Methane Adsorption in Organic Pores using Molecular Simulations.3.1.1 Free Methane Density Calculations at Reservoir Conditions.

    Table 3.1 shows that the predicted free gas density at the center of the pore

    corresponds to the bulk methane density predicted independently by National Institute of

    Standards and Technology (NIST) thermo-physical properties of hydrocarbon mixtures database,

    SUPERTRAPP. The comparison shows that our numerical simulations based on molecular

    modeling are accurate.

    3.1.2 Methane Adsorption Confined to Micro- and Nano-scale Pores.Figures 3.1 and 3.2 show density profiles for methane confined to the large pore and the

    small pore. It is clear that the predicted methane density is not uniform across the pores: its

    value is significantly greater near the wall, where adsorption takes place, and decreases with

    damped oscillations as the distance from the pore wall increases. The oscillations are due to

    presence of adsorption in the pores and involve structured distribution of molecules, i.e.,

    molecular layers. The layers indicate the existence of thermodynamic equilibrium in the pore.

    The number of molecules is largest in the first layer near the wall indicating physical adsorption.

    The wall effect becomes significantly less in the second layer, indicating that desorption of some

    methane molecules is allowed due to equilibrium adsorption. The molecules in the second layer

    are still under the influence of pore walls although intermolecular interactions among the

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    methane molecules begin dominating, not allowing locally high methane densities. In this layer,

    the density of methane is slightly larger than the bulk gas density of methane (at the center of

    the pore). The pore pressures are around 3,043psia which is a quantity predicted using the free

    gas molecules and the volume at the center of the pore.

    pore

    temperature

    oF

    pore

    pressure

    psi

    free gas density

    at the center of

    pore,

    g/cm3

    methane bulk

    density**,

    g/cm3

    176 2206 0.090 0.089

    176 3043 0.124 0.122

    176 3226 0.133 0.129

    176 3676 0.147 0.144

    176 3994 0.156 0.154

    176 4141 0.160 0.159

    176 4404 0.168 0.167

    176 4586 0.176 0.172

    176 4800 0.178 0.175

    176 4878 0.181 0.179

    176 6272 0.214 0.211

    176 7300 0.231 0.229

    176 7550 0.235 0.233

    176 8707 0.253 0.250

    ** from NIST-SUPERTRAPP

    Table 3.1- Methane density at different pressures. Comparison with the bulk density values

    obtained using SUPERTRAPP under the same pressure and temperature conditions.

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    Pore width = 3.93nm

    Figure 3.1- Number density (left) and discrete density (right) profiles for methane in 3.9 nm

    graphite slit-pore at 176oF (80oC). Density values are estimated at each 0.2 interval for the

    continuous density profile. Discrete density corresponds to molecular layer density for methane

    across the pore. The estimated pore pressure at the center of the pores is 3,043 psi.

    41

    7

    5

    0

    8

    16

    24

    32

    40

    0.0 3.8 7.6 11.4 15.2 19.0

    CH4,

    (1000A-3)

    pore half length,

    0.281

    0.163

    0.133 0.126 0.124

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.0 3.8 7.6 11.4 15.2 19.0

    CH4,

    g/cc

    pore half length,

    0.331

    0.124

    0.0

    0.1

    0.2

    0.3

    0.4

    0.0 3.8 7.6 11.4 15.2 19.0

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    Pore width = 2.31nm

    Figure 3.2 Number density (top) and discrete density (bottom) profiles for methane in 2.31 nm

    graphite slit-pore at 176oF (80oC). Density values are estimated at each 0.2 interval for the

    continuous density profile. Discrete density corresponds to molecular layer density for methane

    across the pore. The estimated pore pressure at the center of the pores is 3,043 psi.

    43

    8 8

    43

    0

    8

    16

    24

    32

    40

    48

    0.0 3.8 7.6 11.4 15.2 19.0 22.8

    CH4,1

    000

    -3

    pore length,

    4-5 (bulk)

    0.292

    0.171

    0.135

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.35

    0.0 3.8 7.6 11.4

    CH4,g/cc

    pore half length,

    0.350

    0.124

    0.0

    0.1

    0.2

    0.3

    0.4

    0.0 3.8 7.6 11.4

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    The observed density profiles show that the assumption of Langmuir adsorption theory

    with monolayer is reasonable to describe the equilibrium adsorption dynamics of natural gases

    in the organic pores.

    3.1.3 Pore Size Effects on Methane Adsorption.In essence, supercritical methane in small organic pores is structured due to pore wall

    effects and shows layers of graded density across the pore. Depending on the pore size, a bulk

    fluid region may exist at the central portion of the pore, where the influence of molecular

    interactions with the pore walls is very weak. In pores with sizes up to 50 nm a combination of

    molecule-molecule and molecule-wall interactions dictates thermodynamic states of the gas and

    its mass transport in the pore (Krishna, 2009). On the other hand, within a small pore, see Figure

    3.2 and 3.3, methane molecules are always under the influence of the force field exerted by the

    walls; consequently, no bulk fluid region can be observed in the pore, hence, no pore pressure

    can be measured, therefore, behavior of the adsorbed molecules should be considered rather

    than motion of the free gas molecules. Further, the adsorbed-phase density at the first layer is

    significantly large value. The smaller the pore the greater the adsorbed phase density. In the

    limit, Langmuir theory is not suitable for description of physical interactions between gas and

    solid.

    3.1.4 Effect of Temperature on Methane Adsorption.The effect of temperature on methane density is shown in figure 3. 4. The estimated

    average adsorbed methane density is around 0.3 g/cm3 although it decreases with temperature,

    for example to 0.259 g/cm3 at 266oF. These values show variations due to changing levels of

    kinetic energy at the microscopic scale.

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    Pore width = 1.14nm

    Figure 3.3 Number density (left) and discrete density (right) profiles for methane in 1.14 nm

    graphite slit-pore at 176oF (80oC). Density values are estimated at each 0.2 interval for the

    continuous density profile. Discrete density corresponds to molecular layer density for methane

    across the pore.

    47

    16

    46

    0

    10

    20

    30

    40

    50

    0.0 3.8 7.6 11.4

    *CH4,(1000-3)

    pore length,

    4-5(bulk)

    0.307

    0.234

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.35

    0.0 3.8

    CH4,g/cc

    pore half length,

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    Figure 3.4 Number density (top) and discrete density (bottom) profiles for methane in 3.93nm

    slit-pore as a function of temperature.

    0

    8

    16

    24

    32

    40

    0.0 3.8 7.6 11.4 15.2 19.0

    *

    CH4,

    (1000A-3)

    pore half length,

    176 F

    212 F

    266 F

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.0 3.8 7.6 11.4 15.2 19.0

    CH4,

    g/cc

    pore half length,

    176 F

    212 F

    266 F

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    3.1.5 Methane adsorption confined to nano-scale pores using Monte Carlo SimulationsGibbs ensemble NPT Monte Carlo results are shown in Figure 3.5. Figure 3.5 confirms

    the result obtained by Molecular Dynamic simulation using canonical ensemble. When the

    system gets a large pore size the monolayer adsorption phase decreases and there is a uniform

    profile along the center of the pore. See dashed distribution along 1.14 and 1.9nm. While in

    small pore size widths the uniform behavior at the center of the pore disappears.

    Figure 3.5 Density profiles for methane in three different-size slit-pore 1.14, 2.3 and 3.8nm

    using MC simulation. Estimated pressure at the center of the pore is 3043psi.

    0.300

    0.286

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.35

    0.0 3.8 7.6 11.4 15.2 19.0

    CH4,g/cc

    pore half length,

    1.14 nm

    2.3 nm

    3.8 nm

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    Carbon Dioxide

    Figure 3.6 Number density (top) and discrete density (bottom) profile for carbon dioxide in

    3.8nm slit-pore as function of distance from the wall to the center of the pore, at p = 1980psi.

    100

    25

    15

    25

    97

    0

    20

    40

    60

    80

    100

    0.0 3.2 6.4 9.6 12.8 16.0 19.2 22.4

    CO2,(1000-3)

    pore length,

    0.910

    0.564

    0.4190.372

    0.0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.60.7

    0.8

    0.9

    1.0

    0.0 3.2 6.4 9.6

    CO2,

    g/cc

    pore half length,

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    3.2Carbon Dioxide adsorption confined to nano-scale pores using Molecular Dynamics SimulationsMolecular modeling and simulations using Carbon Dioxide as supercritical gas is important for

    the development of new technologies in subsurface sequestration of greenhouse gases. In this

    study carbon dioxide was modeled as a linear rigid triatomic molecule with three charged

    Lennard Jones centers having point partial charges at the center of the oxygen (-0.332e) and at

    the center of the carbon (+0.664e) to represent the strong quadrupole moment that has CO2.

    The oxygen-oxygen distance was 0.2324 nm. The pore width modeled for this system was

    2.3nm. In total 553 CO2 molecules were used and the density profile results are shown in figure

    3.6. The approximate pressure at the center of the pore is about 1980 psi. It can be observed

    that even though the pressure, at the center of the pore, is low; the adsorption selectivity of the

    organic wall is 3 to 1 in favor of CO2 when compared with CH4 adsorption confined to 3043 psi.

    Unless the phase of carbon dioxide at these conditions of pressure and temperature is

    uncompressible liquid, it is fairly easy to convince ourselves that at higher pressures, say 3043

    psi, CO2 adsorbed phase should increase at higher values than 0.91g/cc.

    3.3Methane Solubility Enhancement in Water Confined to Nano-scale Pores using MolecularDynamics Simulation

    For each system modeled, values of the solubility parameters: l, L, and the Henrys law

    constant, kH, of the methane activity in water are estimated at 75.5oC and reported in Table 3.2.

    The results also include the estimated water density values at the same temperature. At 75.5oC

    the density of the water vapor coexisting with the liquid phase is considered negligible (Guillot

    and Guissani, 1993), therefore methane activity coefficient in the water vapor phase, g, is

    assumed to be ~1 and the Ostwald Coefficient L ~ l. The corresponding experimental data for

    the bulk (i.e., without pore) is from Economou et al. (1998). It is clear that the numerically-

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    obtained density and kH values of this study compare quite well with the experimental

    measurement. If the authors of the experimental work were to calculate the liquid phase

    activity coefficient l, they would have predicted 25x103 which is also a value quite close to that

    obtained in this study using MD simulation. Due to the lack of experimental and simulation

    results of methane solubility in water under confinement and since the methodology we carried

    out for each confined system follows the same process than the bulk one, we consider valid our

    results even though they may carry out large statistical errors. Evaluation of Henrys law

    constant for methane in water by Economou et al. (1998) illustrates that inserting a molecule by

    using the Widom process in an energetically favorable position carries out a larger statistical

    error, 10-22%.

    Figure 3.7 shows the estimated methane solubility (L ~ ) in water for varying slit-pore

    widths and compares with the bulk case. Notice the enhancement in solubility when the fluid

    system is confined to smaller pores. When the pore-length is in 1-2 nm range, the estimated

    solubility is approximately 2-8 times more. The solubility confined to a slit pore of 0.7nm, a size

    comparable to macro-molecular openings in coal and kerogen, is enhanced with a factor of 13.7.

    Pore width

    ()6.9 11.6 16.2 20.8

    Bulk

    (numerical)

    Bulk*

    (empirical)

    l~ L, (x103) 357 54 198 34 135 14 62 13 26 4

    kH ( kbars) 4.3 7.8 11.5 25.1 60.2 63

    (g/cm

    3

    ) 0.960 0.960 0.960 0.961 0.961 0.961

    N 297 495 693 891 891

    Table 3.2- Calculated values of the solubility parameters l, L and Henrys law constant kH at75.5oC; N is the number of water particles per each slit pore and bulk system.

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    Figure 3.7 Estimated Ostwald coefficient, L (x103), of methane in water confined to slit-

    pores with varying sizes compared with the solubility in bulk water (top). Corresponding SCF

    Methane/BBl water (down). Calculated values at 200psi and 168F.

    Comparing our simulation results with the study of the temperature dependence of

    methane solubility in water (Guillot and Guissani 1993) in a bulk system; methane solubility of a

    357

    198

    135

    62

    260

    50

    100

    150

    200

    250

    300

    350

    400

    6.9 11.6 16.2 20.8 BULK

    OstwaldCoefficient,L(*103)

    Pore width,

    Methane Solubility in Water

    22.2

    12.3

    8.4

    3.8

    1.60

    5

    10

    15

    20

    25

    6.9 11.6 16.2 20.8 BULK

    SCFMethane/BBLwater

    Pore width,

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    confined system by a slit pore of 0.7nm at 75.5oC equals to the bulk methane solubility that

    Guillot and Guissani (1993) have found at 336.8oC.

    Table 3.3 and Figure 3.8 show the effect of varying pore wall wettability on the

    solubility. Here, we control the wall wettability by changing the parameter c in equation (16).

    When c takes values smaller than 1, we can say that it represents a hydrophobic wall because

    the water-wall interaction has been minimized. In the same way when c is larger than 1, we can

    say that this represents a hydrophilic wall. When the coefficient c is 0.4 we observe a factor of

    1764/135=13.1 increase with respect to the c=1.0 case and a factor of 1764/26=67.9 increase

    with respect to the solubility in bulk water; on the other side when c increases up to 1.5, then

    methane solubility increases in a factor of 24.3 with respect to when c=1. Indeed, the pore wall

    wettability has an influence on the enhancement of methane solubility in water either way it is

    hydrophilic or hydrophobic. This enhancement is more dramatic when the wall-water

    interaction increases, in other words when the wall is more water wet than methane wet. P. G.

    de Gennes in his review of Wetting: Statics and Dynamics, explains that most molecular liquids

    achieve complete wetting with higher surface-liquid than liquid-liquid interactions near the

    surface (USf>Uff); meaning that if we want to wet completely any surface then the surface-fluid

    interaction must be higher than the fluid-fluid interaction. Surface-fluid interaction in our case

    corresponds to 0.3912kJmol-1 while fluid-fluid to 0.6504kJmol-1 (water-water). An increment in c

    yields to a higher surface-water interaction and, therefore we can say that the wall is more

    water wet or more hydrophilic. Correlating methane solubility of 1.76x10 -3 g/cm3 [McCain,

    1990], the enhancement produced by confining methane-water solution at 16.2 of slit pore

    and having a more hydrophilic pore wall than the organic graphitic one, is two orders of

    magnitude (0.22g/cm3). The enhancement on methane solubility, when decreasing water

    wettability of the wall - having c less than 1, is lower because the interaction difference between

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    methane-methane and methane-wall interaction is higher than water-water and water wall

    interaction.

    c 0.4 0.6 1.0 1.2 1.5

    l ~ L, (x103) 1,764 304 135 2,025 3,283

    kH ( bars) 876 5090 11479 763 471

    Table 3.3- Effect on the solubility parameters l, L at 75.5oC by changing the wall-waterinteraction. The pore width = 1.62nm.

    Figure 3.8 Estimated Ostwald coefficient, L (x103), of methane in water confined to 1.62nmpore with varying pore-wall wettability.

    1,764

    304

    135

    2,025

    3,283

    0

    500

    1,000

    1,500

    2,000

    2,500

    3,000

    3,500

    0.4 0.6 1.0 1.2 1.5

    OstwaldCoefficient,L(*103)

    c, (final = cinitial)

    Wettability effect on Methane Solubility

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    3.4 Importance of Molecular Simulation FindingsIn order to show the practical significance of our numerical investigation in shale gas

    reservoir engineering, a shale gas in-place study is performed below. The study involves

    estimation of free and adsorbed gas volumes of two shales with high and low sorption

    capacities. The formula used for the volumes Gf and Gst are from Ambrose et al. (2010). The

    adsorbed densities chosen are:

    1. 0.29g/cc, corresponds to the value predicted in this study at 3043 psi and 176oF,we are considering the average density between two pore widths, larger ones.

    2. 0.37g/cc, corresponds to the value estimated by Kobayashi and Haydel and,3. 0.42g/cc, that corresponds to the liquid methane density, at atmospheric boiling

    point condition.

    Both shale samples, low and high adsorbed gas volume see table 3.4, considered in

    Ambroses et al. research, are taken into account for the comparison.

    Shale A:(low sorption capacity)

    Shale B:(high sorption capacity)

    0.06 0.06

    0.35 0.35

    0.0 0.0

    0.0046 0.0046

    16 lb/lb-mol 16 lb/lb-mol

    50 scf/ton 120 scf/ton

    4000 psia 4000 psia

    180oF 180

    oF

    1150 psia 1800 psia2.5 g/cm

    32.5 g/cm

    3

    s1 0.29 g/cm3

    0.29g/cm3

    s2 0.37 g/cm3

    0.37 g/cm3

    s3 0.42 g/cm3

    0.42 g/cm3

    Table 3.4- Shale properties, low and high sorption capacity, taken from (Ambrose et al., 2010),including three different density phases for comparison.

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    Free gas storage capacity, Gf, for Shale A is obtained by replacing s for s1,s2 ands3.

    For example, when the adsorbed density corresponds to s1=0.29g/cc; the free and total storage

    capacities for shale A are 89scf/ton and 128scf/ton respectively.

    =

    +

    =

    +

    =

    Notice that Langmuir storage capacity, GsL, and Langmuir pressure pL, are different for

    Shale A and B. Hence Ga for Shale A is 38.8scf/ton.

    =

    +=

    =+=

    Using the same procedure and replacing the corresponding parameters for Shale B, its

    free and total storage capacity are 66.7scf/ton and 149scf/ton respectively.

    Table 3.5 summarizes the obtained Gf and Gst values for Shale A and B at different

    adsorbed phase densities as mentioned before. When shale gas is of high sorption capacity, the

    variance in the estimated Gf and Gst values is larger. It is evident that when the inappropriate s

    value is used, the free and total storage capacity calculated for shale A, is increased in 4.2 and

    6.1scf/ton. For Shale B it is increased in 9.1scf/ton and 13scf/ton. In percentage, the free storage

    capacity is increased in 4% and 7%, for Shale A and in 13.6% and 19% for Shale B. The total

    storage capacity is increased in 3% and 4.7%, for Shale A and in 6.7% and 8.7% for Shale B.

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    In our calculation of the gas in-place we predict dramatic change in the estimated gas

    in-place values due to volume adjustments. This change in gas in-place persists even when the

    sorbed density values reported by other researchers are used, such as the liquid methane

    density of 0.4223g/cc corresponds to 1atm and -161oC (Mavor et al., 2004) and of 0.374g/cc

    (Kobayashi et al., 1967).

    Reference

    s Shale A Shale B Shale A Shale B

    0.29 89.0 66.7 128 149

    0.37 93.2 75.8 132 159!!!!""""

    #$%$%$%$%& 0.42 95.1 79.7 134 162

    Table 3.5- Free and total storage capacity for Shale A and B, at different adsorbed density phasevalues.

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    CHAPTER 4

    CONCLUSIONS AND RECOMMENDATIONS

    1.1ConclusionsThe following can be concluded in regards to the gas behavior in organic shales:

    1. The adsorbed phase follows Langmuir theory in pores larger than approximately 2 nm and thatit takes up a nearly one-molecule thick portion of a pore, although there is a damped oscillation

    density profile with an increased density in the second layer. For a 100 nm pore, the volume is

    fairly insignificant; however, for pores on the order of a 10 nm, it is quite large.

    2. The shale gas producers should not disregard the volume occupied by the adsorbed phaseduring their reserve estimation calculations, because they inadvertently overestimate the pore-

    volume available for the free-gas storage.

    3. Through MD simulation and Langmuir theory we showed that the density for methane at thepore wall typically equals to 0.29 g/cm3. This value now corresponds to numerical work which

    has two limitations: (i) simulations have been done on flat organic pore wall surfaces and in

    small pores the adsorbed layer thickness may be affected by the roughness and curvature of the

    solid surface, (ii) molecular dynamics simulations in the canonical ensemble in the presence of

    multiple fluid phases need to further supported by simulations in the Gibbs ensemble. One way

    of approaching the latter problem is using PT Monte Carlo simulations. Investigations using

    GCMC shows that the predictions based on MD are reasonable.

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    4. It is found that the level of confinement in pores affects the thermodynamics of fluidssignificantly. Consequently, as the pore size reaches nano-scale sorption becomes considerably

    larger. The solubility is extremely sensitive to the changes in the small pore wall wettability.

    5. By using molecular modeling and simulations it is feasible to measure density of the adsorptionphase, which can be used during gas storage capacity calculation and minimize its uncertainty.

    6. Molecular simulations provide the capability to avoid significant errors in gas storage capacitymeasurements that results from laboratory methodologies, because of the overwhelming and

    expensive procedure to follow up during sample preparation and handling. Such as: minimize

    sample exposure to air, light, excessive temperatures; removal of extraneous materials like

    drilling fluid; minimize desiccation; samples with inert gas Argon, etc.

    7. Carbon dioxide adsorption calculation shows selectivity ranging from 3 to 1 in favor of CO2 whencompared to methane adsorption at about 1980 psi.

    8. Shale gas depleted reservoirs may be suitable for storing carbon dioxide because of the organicnature of the porous medium and selectivity for adsorbing three times more CO2 than CH4. In

    the same context, it can be inferred that methane production can be increased by injecting CO2.

    9. In our calculation of the gas in-place we predict dramatic change in the estimated gas in-placevalues due to volume adjustments. This change in gas in-place persists even when the adsorbed

    density values reported by other researchers are used.

    10.The enhancement produced by confining methane-water solution at 16.2 hydrophilic slit pore,is two orders of magnitude (0.22g/cm3) compared to the bulk methane solubility of 1.76x10-3

    g/cm3.

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    1.2Recommendations1. Future work includes re-evaluation of the concepts and approaches that are presented and

    discussed in the presence of multi-component gases with varying pore sizes. It is expected that

    there will be added uncertainties in the gas-in-place estimation due to (1) multi-component

    nature of gas and its adsorbed phase, and (2) volume fraction of organic nanopores that are

    currently inaccessible for visual observation and measurement. The former requires MD

    simulation studies on the adsorbed phase thickness and density using binary, ternary, and

    quaternary mixtures of gases with components such as ethane, propane, and carbon dioxide.

    The latter uncertainty, on the other hand, can be reduced by measuring the absolute area of the

    organic surfaces using isotherm data.

    2. In addition, a new study on transport properties of natural gases in nanoporous organicmaterials is necessary using molecular modeling and simulations in the light of the new pore-

    scale findings.

    3. Solubility studies using hydrocarbon liquids is also recommended for the gas-in-placecalculations.

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    REFERENCES

    [1] Ray J. Ambrose, Robert C. Hartman, Mery Diaz-Campos, I. Yucel Akkutlu, and Carl H.Sondergeld. Shale Gas-in-place Calculations Part I - New Pore-scale Considerations.

    SPE 131772, submitted December 2010.

    [2] Annual Energy Outlook 2010 with Projections to 2035. Energy InformationAdministration 2010, Report Number: DOE/EIA-0383(2010), available at:

    http://www.eia.doe.gov/oiaf/aeo/gas.html.

    [3] Long, G. U.S. Crude Oil, Natural Gas, and Natural Gas Liquids Reserves. E.I.A. 2009,available at:

    http://www.eia.doe.gov/oil_gas/natural_gas/data_publications/crude_oil_natural_gas_

    reserves/cr.html.

    [4] Allen, M.P. and Tildesley, D.J. "Computer Simulation of Liquids". Oxford University Press,London, pp 4-16, 20-28, 33-36, 52-65, 71-108, 110-139, 156-159; 2007.

    [5] Cao, B.Y., Chen, M. and Guo, Z. Y. "Liquid flow in surface-nanostructured channelsstudied by molecular dynamics simulation", Physical Review E. Vol: 74 pp 66311, 2006.

    [6] Chaplin, M. "Water Structure and Science" 2009. available at:http://www.lsbu.ac.uk/water

    [7] Comer, J. B. and Hinch, H. H. "Recognizing and Quantifying Expulsion of Oil from theWoodford Formation and Age-Equivalent Rocks in Oklahoma and Arkansas", AAPG

    Bulletin 71 (7) pp 844-858, 1987.

    [8] de Gennes, P.G. " Wetting: Statics and Dynamics" Reviews of Modern Physics Vol: 57 pp827-863, 1985.

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    [9] Economou, I.G., Boulougouris, G.C., Errington, J.R., Panagiotopoulos, A.Z. andTheodorou, D.N., " Molecular Simulation of Phase Equilibria for Water-Methane and

    Water-Ethane Mixtures" J. Phys. Chem., B. Vol: 102 pp 8865-8873, 1998.

    [10]E. Robens, P. Staszczuk, A. Dabrowski and M. Barczak. "The Origin of Nanopores" Journalof Thermal Analysis and Calorimetry, Vol: 79 pp 499507, 2005.

    [11]Ercolessi, F., "To Molecular Dynamics" at http://www.fisica.uniud.it/~ercolessi/md/,1997.

    [12]Frenkel, D. and Smit, B., "Understanding Molecular Simulation: From Algorithms toApplications", 2nd. ed. (Academic Press, 2001).

    [13]Guillot, B. and Guissani, Y. "A Computer Simulation Study of the TemperatureDependence of the Hydrophobic Hydration" J. Chemical Physics, Vol: 99:10 pp 8075-

    8094, 1993.

    [14]Haydel, J., and Kobayashi, R. "Adsorption Equilibria in the Methane-Propane-Silica GelSystem at High Pressures" Industrial and Engineering Chemistry Fundamentals, 6: 564-

    554, 1967.

    [15]Hnenberger, P.H. "Thermostat Algorithms for Molecular Dynamics Simulations". Adv.Polym. Sci. Vol: 173 pp 105-149, 2005.

    [16]Jenden, P. D., Drazan, D. J.,Kaplan, I. R. "Mixing of Thermogenic Natural Gases inNorthern Appalachian Basin", AAPG Bulletin 77 (6) pp 980, 1993.

    [17]Karayiannis, N.C., Giannousaki, A.E., Mavrantzas, V.G., and Theodorou, D.N. "AtomisticMonte Carlo Simulation of Strictly Mono-disperse Long Polyethylene Melts through a

    Generalized Chain Bridging Algorithm". J. Chemical Physics Vol: 117pp 5465-5472, 2002.

    [18]Konrad, O., Lankau, T. "Solubility of Methane in Water: The Significance of the Methane-Water Interaction Potential" J. Phys. Chem. B, Vol: 109 pp 23596-23604, 2005.

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    [19]Lacks, D.J. "First-order Amorphous-amorphous Transformation in Silica" Physical ReviewE, Vol: 84:20 pp 4629-4632, 2000.

    [20]Loucks, R.G., Reed, R.M.

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