Linking the Chemical and Physical Effects of CO2 Injection to

Geophysical Parameters

Investigators:

Stanford University: Gary Mavko, Professor (Research), Geophysics; Sally Benson, Professor (Research)

Energy Resources Engineering; Tiziana Vanorio, Sr. Research Scientist, Geophysics; Stephanie Vialle, Post-Doc, Geophysics.

Rice University: Andreas Luttge, Professor, Earth Science; Rolf Arvidson, Sr. Research Scientist,

Earth Science

Abstract

This project aims to demonstrate techniques for quantitatively predicting the combined seismic signatures of CO2 saturation, chemical changes to the rock frame, and pore pressure. This will be accomplished (i) by providing a better understanding the reaction kinetics of CO2-bearing reactive fluids with rock-forming minerals, and (ii) by quantifying how the resulting long-term, CO2-injection-induced changes to the rock pore space and frame (e.g. porosity, permeability, mineral dissolution, and cementation) affect seismic parameters in the reservoir.

This research involves laboratory, theoretical, and computational tasks in the fields of

both Rock Physics and Geochemistry. Samples have been selected based on mineralogy (carbonates, sandstones, and calcite-cemented sandstones), porosity, pore and cement type. Ultrasonic P- and S-wave velocities are being measured over a range of confining pressures while injecting CO2 and brine into the samples. Pore fluid pressure will be varied and monitored together with porosity during injection. The measurement of rock physics properties will be integrated and complemented by those obtained via geochemical experiments to link the physical (e.g. porosity enhancement, selective dissolution and change in microstructure) and chemical processes (e.g. reaction type and dissolution rates) underlying the mechanisms triggered by CO2 injection. We will also develop computational and analysis tools needed to simulate multi-fluid flow at the pore scale while including dissolution effects.

We previously reported laboratory results on carbonate samples showing that both P -

and S- wave (elastic) velocities under dry and saturated conditions decrease over time, due to dissolution during injection of CO2-bearing brine. However, for similar injected volumes of fluid, the magnitude of these changes differs from one sample to another. It appears that the different responses of the rocks to CO2 injection are related, in part, to their initial microstructures.

In this report, we begin by summarizing geochemical results from our team-members

at Rice University. A key finding is the loss of the classical assumption regarding a single-valued “mean” rate or rate constant as a function of environmental conditions. Reactivity predictions and reservoir long-term stability calculations based on laboratory measurements are thus not directly applicable to natural settings without a probabilistic approach. A comprehensive stochastic model of dissolution has been developed by applying kinetic Monte Carlo methods that treat the complex dissolution process. The model has been tested with laboratory observations. Another key result is demonstration of the decoupling of surface area and its related density of reactive sites such as steps and kinks, from the overall rate: e.g., as diameter decreases, the diffusion length of mobile surface components will eventually exceed the spacing of step and kink sites, leading to diffusion limitations that are not predicted in rate models parameterized simply by surface area. This success of this approach also suggests a powerful strategy for the simulation of complex reservoir frameworks, in which diverse grains reside in complex contact with one another.

We also summarize aspects of numerical simulation of flow in complex pore

microstructures, with particular emphasis on microporosity as might be in encountered in carbonate rocks or sandstones with clay. The flow is not only determined by pore connectivity and tortuosity, but also by the absolute size of pores relative to the molecular mean-free path of the pore fluids. The flow is also coupled with the stress-dependent deformation of pore walls and adsorption or chemical reaction of the pore fluid with the mineral phase.

Introduction:

Monitoring, verification, and accounting (MVA) of CO2 fate are three fundamental needs in geological sequestration. The primary objective of MVA protocols is to identify and quantify (1) the injected CO2 stream within the injection/storage horizon and (2) any leakage of sequestered gas from the injection horizon, providing public assurance. Thus, the success of MVA protocols based on seismic prospecting depends on having robust methodologies for detecting the amount of change in the elastic rock property, assessing the repeatability of measured changes, and interpreting and analyzing the detected changes to make quantitative predictions of the movement, presence, and permanence of CO2 storage, including leakage from the intended storage location. This project addresses the problem of how to interpret and analyze the detected seismic changes so that quantitative predictions of CO2 movement and saturation can be made. The main goals are:

(a) linking the chemical and the physical changes occurring in the rock samples upon

injection; (b) assessing the type and magnitude of reductions caused by rock-fluid interactions

at the grain/pore scale; (c) providing the basis for CO2-optimized physical-chemical models involving frame

substitution schemes.

Background:

Having the appropriate rock-physics model is generally a key element for time-lapse seismic monitoring of the subsurface, both to infer the significance of detectable changes (i.e. qualitative interpretation) and to convert them into actual properties of the reservoir rocks (i.e. quantitative interpretation). Nevertheless, because of the peculiar ability of CO2-rich water to promote physicochemical imbalances within the rock, we must address whether traditional rock-physics models can be used to invert the changes in geophysical measurements induced in porous reservoirs by the injection of CO2, making it possible to ascribe such changes to the presence or upward migration of CO2 plumes. Making this determination requires an understanding of the seismic response of CO2-water-rock systems. Seismic reservoir monitoring has traditionally treated the changes in the reservoir rock as a physical-mechanical problem—that is, changes in seismic signatures are mostly modeled as functions of saturation and stress variations (e.g. pore and overburden pressure) and/or intrinsic rock properties (e.g. mineralogy, clay content, cementation, diagenesis…). Specifically, modeling of fluid effects on seismic data has been based almost exclusively on Gassmann’s equations, which describe the interaction of fluid compressibility with the elastic rock frame to determine the overall elastic behavior of rock. Beginning with the bulk (Ks) and shear (µs) moduli of the mineral composing the rock, we use Gassmann’s fluid substitution scheme to compute the bulk modulus of the saturated rock (Ksat) from the bulk modulus of the fluid (Kfl) and from that of dry rock (Kdry). However, depending on the properties of the mineral composing the rock and the properties of the fluid, complex rock-fluid interactions may occur at the pore scale, leading to dissolution and formation of new mineralogical phases. All these physical modifications may compete in changing macroscopic rock properties such as permeability, porosity, and elastic velocities, which, in turn, can change the baseline properties for the Gassmann’s fluid substitution scheme. This entails two consequences: (a) a classical fluid-substitution scheme may underpredict time-lapse changes and thus mislead 4D monitoring studies; and (b) predictions of in situ velocity will compensate for the chemically softened velocities with erroneous estimates of saturations and/or pore pressure.

Results

How predictable are dissolution rates in natural and industrial systems? (Cornelius Fischer, Rolf S. Arvidson, and Andreas Lüttge, Rice University)

Summary: The large discrepancy between field and laboratory measurements of mineral reaction rates is a long-standing problem in earth sciences, often attributed to factors extrinsic to the mineral itself. Nevertheless, differences in reaction rate are also observed within laboratory measurements, raising the possibility of intrinsic variation as well. Critical insight is available from analysis of the relationship between the reaction rate and its distribution over the mineral surface. This analysis recognizes the fundamental variance of the rate. The resulting anisotropic rate distributions are completely obscured by the common practice of surface area normalization. In a simple experiment using a single crystal and its polycrystalline counterpart, we demonstrate the

sensitivity of dissolution rate to grain size, results that undermine the use of “classical” rate constants. Comparison of selected published crystal surface step retreat velocities (JORDAN & RAMMENSEE, 1998) as well as large single crystal dissolution data (BUSENBERG & PLUMMER, 1986) provide further evidence of fundamental variability. Our key finding is the loss of the classical assumption regarding a single-valued “mean” rate or rate constant as a function of environmental conditions. Reactivity predictions and reservoir long-term stability calculations based on laboratory measurements are thus not directly applicable to natural settings without a probabilistic approach. Such a probabilistic approach must incorporate both the variation of surface energy as a general range (intrinsic variation) as well as constraints to this variation owing to heterogeneity of complex material (density of domain borders etc.). [submitted to Geochimica et Cosmochimica Acta]

Does the stepwave model predict mica and clay mineral dissolution kinetics? (Inna Kurganskaya, Rolf S. Arvidson, and Andreas Luttge, Rice University)

Summary: Micas and clay minerals are a unique class of minerals because of their layered structure. A frequent question arising in mica dissolution studies is whether this layered structure fundamentally changes the dissolution mechanism. We addressed this critical question in this work, using VSI and AFM data from experiments using muscovite to evaluate crystallographic controls on mica dissolution. These data provide insight into the dissolution process, and reveal important links to patterns of dissolution observed in framework minerals. Under our experimental conditions (pH 9.4, 155°C), the minimal global rate of surface-normal retreat observed in VSI data was 1.42 × 10−10 mols ⁄ m2 ⁄ s (σ = 27%) while the local rate observed at deep etch pits reached 416 × 10−10 mols ⁄ m2 ⁄ s (σ = 49%). Complementary AFM data clearly show crystallographic control of mica dissolution, both in terms of step advance and the geometric influence of interlayer rotation (stacking periodicity). These observations indicate that basal/edge surface area ratios are highly variable and change continuously over the course of reaction, thus obviating their utility as characteristic parameters defining mica reactivity. Instead, these observations of overall dissolution rate and the influence of screw dislocations illustrate the link between atomic step movement and overall dissolution rate defined by surface retreat normal to the mica surface (Figure 1). Considered in light of similar observations available elsewhere in the literature, these relationships provide support for application of the stepwave model to mica dissolution kinetics. This approach provides a basic mechanistic link between the dissolution kinetics of phyllosilicates, framework silicates, and related minerals, and suggests a resolution to the general problem of mica and eventually clay mineral reactivity. [submitted to Geochimica et Cosmochimica Acta]

A Comprehensive Stochastic Model of Phyllosilicate Dissolution Kinetics

(Inna Kurganskaya and Andreas Luttge, Rice University)

Summary: Phyllosilicate dissolution is a complex process that consists of a large number of basic and/or elementary reactions operating on the mineral surface. Despite this fact, the common kinetic models reduce the dissolution process to only one reaction

that involves the breaking of a precursor complex. Here, we have developed a comprehensive model of phyllosilicate dissolution that is based on our original work on feldspars (e.g., Lasaga and Luttge, 2004a;b; Zhang and Luttge, 2007, 2008, 2009a;b). We applied kinetic Monte Carlo methods that treat dissolution as a many body problem and a sequence of randomly chosen reaction events (Figure 2A). We used muscovite as an example illustrating the capabilities of the model. The topographical data of dissolving muscovite surfaces were obtained from experiments and used as a comparison. In this way, we were able to test the model (Figure 2D). The results of the simulations with four different model versions are then used to explain specific features of the etch pits developing on the (001) face. The different versions of the model can be distinguished by a number of parameters and increasing complexity, i.e., chemistry and the number of first nearest neighbors, lattice resistance to hydrolysis (influence of the cluster size or site connectivity on bond hydrolysis), the role of second nearest neighbors and bond orientation (steric factors). The simplest version of the model is able to reproduce basic etch pit features, e.g. the overall shape, orientation and effects of the silicate stacking sequence on step superposition. The more complex versions of the model consider second coordination and other steric effects (Figure 2B). They can reproduce small scale features, such as detailed step morphology and anisotropy in dissolution rates as a function of crystallographic direction (Figures 2C,2D). The effects of the second coordination sphere on the stability of experimentally observed steps are explained by the structure and relative reactivity of the “ledge” and “kink” sites specific for micas. The new model in combination with experimental observations showed its unique effectiveness in providing a mechanistic insight into the dissolution kinetics.

Kinetic Monte Carlo Model of Sheet Silicate dissolution:

Parameterization of ab initio Calculations (Inna Kurganskaya and Andreas Luttge, Rice University)

Summary: Despite the intensive development of ab initio and Molecular Dynamics approaches to study reactivity of surface sites and species, most of the KMC models of mineral dissolution either remain at the oversimplified level of coarse-graining (e.g., Kossel crystal) or designed specifically to the given crystal structure or effect. In this paper we are discussing the role of the factors influencing reactivity of the surface sites inferred from the ab initio/MD studies and their implementation into the KMC models of sheet silicate dissolution. These factors include chemistry, structural position, coordination and charge of a surface site, size of the cluster etc. The lack of the ab initio data for a substantial number of surface reactions happening on mica surface creates a problem for a correct model parameterization. We made an attempt to solve this problem by making a series of assumptions about energetic effects of the factors described above and assigning them some energetic parameters. Systematic variation of values of these parameters in the KMC simulations produces a spectrum of different etch pit structures and surface morphologies (Figures 3A,3B, 3C) observed in a wide range of experiments conducted at different environmental conditions. The quantitative relationships between the rates or probabilities of the key surface reactions lead to the formation of the given surface structure. Thus, the structure of the steps and their relative velocities can be explained by the different probabilities to nucleate and dissolve a kink site (Figures 3D,

3E). The results of the simulations provided here illustrate the ultimate importance of the correct values of bond hydrolysis activation energies for modeling dissolution of complex silicate structures, such as phyllosilicates.

Figure 1. Schematic illustration of stepwave formation on muscovite {001} surface. A. Consecutive steps moving outwards from a dislocation center. B. Steps emanating from two major etch pits (linear defects) and surrounding monolayer pits (point defects) form a single stepwave of complex morphology. C. Multilayer etch pit formation, showing layer-by-layer dissolution in two principal directions.

A B

C D

Figure 2. An illustration to Kinetic Monte Carlo simulations of muscovite dissolution. A. Polyhedral model of a small cluster representing part of the muscovite surface. Dissolution probabilities are assigned to an each surface site according to its chemistry, structural position and coordination number. B. Size of a coordination sphere used to differentiate reactivity of the tetrahedral sites. C. An example of an etch pit on muscovite (001) face produced in the simulations. D. An etch pit formed on muscovite (001) face after reaction in water pH=9.4, T=155˚C.

A B C

D E

Figure 3. An illustration to the role of activation energy parameters on etch pit morphologies. A,B,C. Examples of the etch pits produced in the KMC simulations using different values of activation energies. D. A model of the reactive octahedral “ledge” and “kink” sites at the 3 primary {hk0} faces on muscovite surface. “Ledge” sites are labeled by light gray semitransparent squares, “kink” sites are labeled by yellow semitransparent squares. E. Structure of a (110) step reproduced in KMC simulations (only Si and Al cations are shown). Light gray spheres represent tetrahedral cations in the upper TOT layer, dark gray spheres represent octahedral cations, black spheres represent tetrahedral cations in the lower TOT layer.

Evaluation of the role of grain boundaries in mineral dissolution, with application to sequestration reservoirs

(Rolf S. Arvidson and Andreas Luttge, Rice University)

Summary: Despite the significant increase in the understanding of mineral dissolution and growth enabled by real-time, microscopic observations of reacting mineral surfaces, these observations typically involve relatively large grains or particles. The kinetics of such surfaces are dominated by the distribution of defects, e.g., screw dislocations, that provide a critical means of sustaining step movement, kink creation, and thus dissolution and growth processes. However, in natural settings such as sequestration reservoirs, the discontinuities created by physical grain boundaries and contacts may constitute a critical contribution to “whole rock” reactivity, a point that has been recognized in the past but in which there is no formalized approach. Because of the complex relationship between mineral surface characteristics such as area, roughness, and reactivity, this problem requires a modeling approach that permits examination of key aspects that distinguish the reactivity of grains of arbitrary size. Here, we use kinetic Monte Carlo techniques to explore the dynamics of dissolution involving a discrete, three dimensional crystal (Figure 4). This approach has several important motivations: for example, we may question if there is a critical grain size, governed by surfaces kinetics (and not simply thermodynamic properties), below which the reactivity of grains is dominated by contributions from boundary sites. A key result of this approach is the demonstration of the decoupling of surface area and its related density of reactive sites such as steps and kinks, from the overall rate: e.g., as diameter decreases, the diffusion length of mobile surface components will eventually exceed the spacing of step and kink sites, leading to diffusion limitations that are not predicted in rate models parameterized simply by surface area (Figure 5). This success of this approach also suggests a powerful strategy for the simulation of complex reservoir frameworks, in which diverse grains reside in complex contact with one another.

Figure 4. Detail of corner and monolayer pits of 3D Kossel-Stranski cubic crystal, showing terrace atoms in red-brown, step edge sites in tan, kink sites in yellow, and 2-bonded chains in red.

Figure 5. Dissolution time series. Left and right panels compare system stapshots at constant mass remaining (0.99 to 0.10) as a function of bond breaking activation energy: left panel,

Φ /kT = 4 ; right panel ,

Φ /kT = 11. Surface diffusion is prevented by setting ϵD ⁄ kT = 100.

The Effect of Diffusion, Adsorption, and Compaction on the Permeability of

Microporous Rocks (Adam Allan, Gary Mavko, Stanford University)

This work is aimed at improved understanding of flow and transport in microporous

rocks. Such rocks might include particularly tight portions of the reservoir or aquifer where CO2 sequestration is to occur, or perhaps even the cap rock. The results are not limited to tight rocks; however, tight rocks represent the most challenging of the range of rocks likely to be encountered.

Flow in porous media has, until recently, been modeled under the assumption that the

fluid is a continuous viscous fluid with negligible flow of molecules adjacent to the pore wall. This continuum approach is, however, only valid when the mean free path of the gas,

λ , is much smaller than the channel width, H. This stipulation can be quantified by the dimensionless Knudsen number, Kn: the ratio of the mean free path of the gas to the width of the pore the gas is contained by, Kn =

λ /H. The continuum approach and use of the no-slip Navier-Stokes equations is only valid for small values of the Knudsen number (Kn < 0.001) (Schaaf and Chambre, 1961). In confined pore spaces, as the pore width approaches the mean free path of the gas, the gas becomes rarefied resulting in an increase in the frequency of molecule-wall collisions with respect to intermolecular collisions, and the velocity of the molecules adjacent to the pore wall is no longer negligible. The slip flow regime, the subject of our previous work (Allan and Mukerji, 2011), is when Kn is small (< 0.1) but not negligible. In the transition flow regime, molecule-wall collisions and intermolecular collisions are both important and the analysis gets complicated. Flow regimes corresponding to Kn > 0.001, characterized by non-negligible velocities at the wall have been shown (Arkilic et al., 1997) to increase the mass transfer rate. Faithful prediction of mass transfer rates is requisite to correct computation of permeability from numerical simulation.

Attempts to measure gas permeability of tight rocks in the laboratory have been

made for over half a century. Klinkenberg (1941) proposed that gas permeability values are higher than water permeability values in tight rocks due to the ‘slip’ (Knudsen diffusion) of gas molecules adjacent to pore walls. However, the tight nature of shale in reservoir rocks and seals, with permeability on the order of nanoDarcies, prevents completion of standard steady-state experiments in a timely fashion. As a result, the transient pulse decay (TPD) method (Brace and Orange, 1968; Katsube, et al. 1991) and crushed sample methodologies (Vermylen, 2011 and others) have been applied to facilitate rapid, accurate permeability measurements on tight reservoir rocks (Kwon et al. 2004; Mallon and Swarbick, 2008; Billiote et al., 2008). While the relatively new TPD method has shown to be accurate thus far (Katsube et al., 1996; Mallon and Swarbick, 2008), it would be advantageous to be able to perform numerical permeability experiments at pressure-temperature conditions not trivially obtainable in the laboratory; hence, the attempt in this study and others to accurately represent the flow of gas in confined pore spaces numerically.

The significant internal surface area of these nanometer-scale pore networks can result in substantial adsorption of a high density, liquid-like phase to the walls of the pores. The high density of the adsorbed phase creates the potential for a significant storage mechanism, and the effect of storage on gas-in-place (GIP) calculations has been widely studied (Bustin et al., 2008; Cui et al., 2009; Ambrose et al., 2010); however, the effect of an adsorbed layer of gas on sample permeability has been less widely researched (Sakhaee-Pour and Bryant, 2011).

Several recent studies propose that the permeability of tight reservoir rocks will

be greater than the continuum prediction at reservoir temperature and pressure due to rarefaction of the pore-filling gas (Javadpour et al., 2007; Javadpour, 2009; Freeman et al., 2009; Sakhaee-Pour and Bryant, 2011; Michel et al., 2011). While others (Freeman et al., 2010; Shabro et al., 2011; Sakhaee-Pour and Bryant, 2011; Fathi and Akkutlu, 2011) add that the presence of an adsorbed gas phase may decrease in situ permeability values (Valkó and Lee, 2010). This recent research in tight rocks and shale has resulted in advances in understanding of the physics that govern gas flow in confined spaces; however, with the advent of advanced imaging techniques such as FIB-SEM, TEM, CLSM, and the ability to image nanometer-scale heterogeneity in shale samples (Loucks et al., 2009; Ambrose et al., 2010; Kale et al., 2010; Passey et al., 2010; Sondergeld et al., 2010; Curtis et al., 2011), it remains for these developments to be applied to complex, heterogeneous, real-world reservoir rocks. This study serves as a first attempt at the application of these new techniques and technologies to real-world rocks in order to better quantify the magnitude of the effect of nanometer-scale phenomena on reservoir rock transport properties.

The permeability of conventional reservoirs has long been accurately predicted via Darcy’s Law (Eqn. 1). Darcy’s Law is a phenomenological relation between the flux through the porous medium (in ms-1), q, to the permeability, k, through the viscosity, μ, and pressure gradient vector,

∇P (Darcy, 1856).

q = −µ∇P (1)

The formulation in Eqn. 1, however, is only valid in the continuum regime. As a result, an apparent permeability, kApp, must be defined by considering the effect of rarefaction of the pore-filling gas.

Previous work by Allan and Mukerji (2011) used the Lattice Boltzmann Method

(LBM) to implement a Maxwellian scattering kernel boundary condition (Maxwell, 1879; Zhang et al., 2005) in an attempt to recovery accurate wall velocities in a two-dimensional slit pore geometry. The Maxwellian methodology consists of weighting molecule-wall collisions between purely diffuse and purely specular reflections with a pressure-temperature-dependent tangential momentum accommodation coefficient (TMAC),

σV . The Maxwellian kernel is then integrated with the commonly implemented bounce-back scheme in LBM to represent Kn-dependent slip along the fluid-solid interface. The Maxwellian implementation of the slip boundary condition is accurate to first order in Kn and, as such, is only applicable to the slip regime (Kn < 0.01). As previously stated, the Knudsen number is the ratio of the mean free path of the gas to the

confining pore width, where the mean free path is defined by Eqn. 2 for Boltzmann constant, kB, temperature, T, gas pressure, P, and molecular diameter, d.

λ =kBT2πPd2 (2)

Figure 6 shows the Knudsen number for helium and carbon dioxide at 295 K in pores with widths of 1 micron and 5 nm, respectively, as calculated with Eqn. 2. High resolution SEM images of porous shale exhibit pore widths ranging from 5 nm to ~100 nm (Sondergeld et al., 2010); subsequently, these smallest pores will experience flow in the transition regime, especially at reservoir temperatures which will translate the curves to higher Knudsen numbers.

The ‘Unified Flow Model’ (UFM) of Karniadakis et al. (2005) is second-order

accurate in Kn, and is valid across the full range of Kn. The reader is directed to chapter 4 of Karniadakis et al. (2005) for a complete derivation of the model. The UFM prescribes a velocity profile given by

u x,y( ) = ∇PH 2

2µ−

yH

2

+yH

+

2 − σV

σV

Kn1− b Kn

(3)

, and Kn are defined as before, y is the distance from the center of pore, and b is a dimensionless slip coefficient defined as -1. Figure 7 shows a comparison of velocity profiles from LBM with slip boundary condition and UFM for Kn = 0.075 (upper slip regime) and demonstrates the expected over-prediction of slip velocities at the pore wall by the Maxwellian model. Subsequent work (Florence et al., 2007) manipulated the flux as defined in UFM (Eqn. 4) to an apparent permeability, kApp

, (Eqn. 5) that accounts for Knudsen diffusion at the pore walls.

Figure 6: Knudsen number as a function of pressure for helium (He) and carbon dioxide

(CO2) in 1 micron (open symbols) and 5 nm (filled symbols) wide pores, respectively. Flow regime boundaries are indicated by dashed red lines.

Figure 7: Velocity profile nondimensionalized by mean velocity across a 500 nm channel

at STP for Helium. The blue curve is the continuum Haagen-Poiseuille solution. The black curves are the first order Maxwellian solutions where stroke denotes TMAC, and the red symbols are the Karniakdakis and Beskok (2005) solution where shape denotes TMAC.

In Eqn. 4, L is the length of the sample and

α Kn( ) is the well-defined, dimensionless is the continuum permeability of the sample. In future plots rather than plot the apparent permeability as defined in Eqn. 5, the ratio of the apparent permeability to the continuum permeability will be plotted to better demonstrate the effect of Knudsen diffusion on permeability. Note that the term

1+ α kn( ) kn in Eqns. 4, 5 accounts for the transition between the slip flow and free molecular flow regimes and is well-defined although contracted here for space. Since the majority of pores in typical shale reservoirs will plot in the upper slip or transition flow regimes, Eqns. 3 and 5 corresponding to the second order solution will be used in the following simulations. However, it has been shown (Patil et al., 2005) that Knudsen diffusion (and other diffusive flow mechanisms) becomes a negligible mass transport mechanism when the pore-filling fluid is supercritical; subsequently, for pore pressures greater than the critical pressure Eqn. 5 is not implemented and slip velocities are neglected.

q =π8

H 4 ∇PµL

1+ α Kn( ) Kn[ ] 1+4Kn

1 = b Kn

(4)

kapp = k∞ 1+ α Kn( ) Kn[ ] 1+4Kn

1 = b Kn

(5)

Figure 8: (Left) Ratio of apparent to continuum permeability as a function of TMAC.

Black curves represent solutions for different Knudsen numbers while the red dashed lines indicate the experimental bounds of Grauv et al. (2009). (Right) Pore width as a function of pore pressure. Black curves represent the Knudsen number encountered at those conditions.

A great deal of research has been conducted into characterizing the pore-size

distribution and storage capacity of various sorbents in Earth sciences, material sciences, and chemical engineering (Lu et al., 1995; Vermesse et al., 1996; Lowell et al., 2004; Bustin et al., 2008; Ambrose et al., 2010; Thommes, 2010; Adesida et al., 2011; Vermylen, 2011). There has, however, been significantly less research conducted on the effect of an adsorbed layer of gas on the permeability of a sample. A recent study by Sakhaee-Pour and Bryant (2011) attempts to account for adsorption in flow simulation of a network model of cylinders, but make several assumptions which are incorrect for methane or CO2 adsorption in a shale sorbent and will be discussed further in the next section.

Adsorption is the adhesion of gas molecules to the surface of a solid resulting in a high-density, liquid-like gas film that is distinct from the bulk free gas phase. The adsorptive potential of a material is quantified by the amount of adsorption at many relative pressures (the ratio of the gas pressure to the saturation pressure) and constant temperature. The amount of adsorption as a function of relative pressure is then plotted as an isotherm. The adsorption isotherm is, most simply, a function of the pore size distribution, composition of the pore-filling gas (or mixture of gases), and the attraction energy between the surface (sorbent) and pore-filling gas (adsorbate) (Langmuir, 1916; Brunauer et al., 1938). The Langmuir isotherm is characterized by an immobile monolayer of molecules that adsorb homogeneously without adsorbate-adsorbate interactions and is illustrated in Figure 9 (Langmuir, 1916). It should also be noted that while Knudsen diffusion becomes negligible upon the transition from the gaseous phase to the supercritical phase, supercritical adsorption will continue at pore pressures greater than the critical pressure (Zhou et al., 2001).

Figure 9: Schematic representation of a Langmuir (Type I) isotherm. Where Vads is the

volume adsorbed, P/P0 is the relative pressure, and P0

is the saturation pressure. Adapted from Brunauer et al. (1940).

Adsorption will affect permeability in two manners: (1) the pore width will become geometrically narrower by 2n molecular diameters where n is the number of layers adsorbed, and (2) by changing the interactions of molecule and wall through the tangential momentum accommodation coefficient (TMAC). Affect (1) will be of little concern in larger pores, however, in narrow, e.g. 5 nm wide, pores a reduction in width by twice the molecular diameter of methane (0.38 nm) will have a non-negligible effect on the degree of Knudsen diffusion and must be accounted for in simulation. Effect (2) is addressed in Figure 8. Laboratory experiments by Grauv et al. (2009) find that, for Argon, Nitrogen, and Helium impinging on a Silicon surface, TMAC varies between bounds of 0.848±0.008 and 0.956±0.005 at atmospheric conditions. The left image in Figure 8 shows that for TMAC between ~0.75 and 1.00 (purely diffuse reflections) the change in apparent permeability is negligible for Knudsen numbers less than 5.0. The right image in Figure 8 shows that a Knudsen number of 5.0, at a characteristic reservoir temperature of 350 K and a pore pressure greater than ~2 MPa, requires a pore width of less than 1 nm which is below the resolution of current imaging technology. Subsequently, while the values presented in the Grauv et al. by no means constitute an exhaustive dataset, a range of TMAC well beyond the experimental bounds and error exhibits negligible variance in apparent permeability. Additionally, the values of Grauv et al. do agree well with ranges of values measured for hydrogen, helium, and air impingement on aluminium, iron, and bronze (Seidl and Steinheil, 1974; Lord, 1976). Subsequently, in all future simulations reflections are assumed to be completely diffuse (

σV = 1). This result gives extra confidence to the application of Eqns. 4, 5 as the assumption of purely diffuse reflections is fundamental to the derivation.

In order to satisfy the tenets of Langmuir adsorption theory in the simulation, the

following assumptions must be maintained: (1) the adsorbed gas is immobile, (2) all adsorption sites are equivalent, (3) each adsorption site can hold at most one molecule, and (4) no interactions exist between the adsorbed molecules. These assumptions result in a thoroughly homogeneous pore space in which adsorption occurs in all pores with the same probability regardless of the behavior of the free gas in the system. In order to better represent the heterogeneity of adsorption and flow within the pore space, the process of desorption is weighted so as to occur preferentially in the pores that transmit

the greatest fraction of the total flux. This is a physically valid assumption as low flux pores that are isolated from the main flow path or that ‘dead-end’ within the sample will not experience the same pore pressure depletion as high flux pores and will, subsequently, experience delayed desorption. This weighted desorption process is simulated by identifying each individual, isolated connected pore system that percolates the sample and treating each as its own Langmuir-satisfying space where global desorption occurs in a pseudo-Langmuir fashion in the highest flux pores first. Subsequently, adsorption is simulated by identifying the total number of gas molecules that will adsorb at a given pore pressure, f, the total number of adsorption sites in each connected pore, N, and the fraction of flux within each pore by randomly updating f adsorption sites to solid voxels beginning with the lowest flux pores.

In anticipation of nano-images from tight rocks, our flow simulation methodology is

initially tested on a geostatistically rendered proxy pore. A 2.9 x 3.6 μm SEM image (Figure 10) captured and provided by Arjun Kohli is thresholded and rendered in the third dimension by sequential indicator simulation (Journel and Gomez-Hernandez, 1993). In the interest of quick verification of the methods employed in this work, the three-dimensional volume is rendered from a smaller sample of the SEM image (red box in Figure 10). In order to fully resolve and model the adsorption process, the three-dimensional volume is scaled with a discrete cosine transform (DCT) such that the grid spacing, dx, is equal to the molecular diameter of methane (dx = 0.38 nm). The volume is then further sub-sampled (Figure 11) in the interest of computation time. A proprietary D3Q19 continuum LBM code (Keehm, 2003) is used to compute the intrinsic Darcy permeability of the sample at each step. The continuum permeability of the sample,

k∞ , for dx = 0.38 nm is 140 nD.

The effect of compaction on the structure of the pore is simulated through an image

processing dilatation algorithm in which each pore voxel adjacent to the fluid-solid interface is changed to a solid voxel. The relation between porosity and pressure used here is that described in McKee et al. (1988)

ϕ = ϕ0 exp −cP Pi − P( )[ ] (6)

under the assumption of constant confining pressure, where P is the pore pressure, Pi the pore pressure at initial reservoir conditions, cp the pore compliance, and φ is the porosity. The pore compliance for shale is assumed to be 6x10-6 1/psi (Jones and Owens, 1980) which results in sub-resolution variations in porosity with pore pressure. It is also worth noting that the value of Jones and Owens is between 8 and 10 times larger than values reported in Johnston (1987), adding confidence to the exclusion of compaction from the study under assumption of shale pore compliance. In order to quantify the effect of compaction on permeability the same geometry is studied with pore compliances of 1.44x10-4 1/psi (Post-Eocene clay: McKee et al., 1988) and an intermediate value of 7.5x10-5 1/psi. Figure 11 shows the unaltered sample at 40 MPa (29.8% porosity) and the sample at 1 MPa (13.2% porosity) for a clay skeleton. The intrinsic permeability of the compacted sample can then be computed for each pore compliance value with the LBM code. As pore pressure decreases, the effective stress experienced by the sample will increase (assuming constant confining pressure) causing the rock to compact and pores to

narrow. The narrowing of the pores results in a decrease in the intrinsic permeability of the sample as can be seen in Figure 12. The permeability as a function of compaction, thus, attains a maximum (the intrinsic permeability) at the highest pore pressure simulated. At low pressure (~ 1 MPa) the permeability of the clay sample approaches 1 pD (picoDarcy), i.e. the sample becomes nearly impermeable. Subsequently, for extremely compliant pores we find that the effect of compaction on permeability will dominate all other nanometer-scale effects.

Figure 10: SEM image of pores in a kerogen body in a producing shale reservoir imaged and provided by A. Kohli. The width of each pixel is 5.56 nm. The red box indicates the sub-sample of the image used to render the three-dimensional sample pore volume.

Figure 11: (Left) Initial geostatistically rendered kerogen pore (φ = 29.8%). Each dimension of the volume is 38 nm. White is solid matrix. (Right) Kerogen pore with uniformly scaled solid frame (φ = 13.2%).

Figure 12: Permeability (in nanoDarcies) as a function of pore pressure for each pore compliance value: solid line – shale (cp = 6x10-6 1/psi), triangles – intermediate value (cp = 7.5x10-5 1/psi), circles – clay (cp = 1.44x10-4 1/psi). In the work of Sakhaee-Pour and Bryant (2011), the isotherm for adsorption of

methane onto shale is assumed to increase linearly with pore pressure and attain a full double-layer at 28 MPa. The dissertation of Vermylen (2011) documents methane adsorption isotherms measured on crushed shale samples at room temperature and gas pressures up to ~10 MPa. The isotherms demonstrate Langmuir behavior (Type I), e.g. only a single monolayer is adsorbed and the isotherms exhibit a nonlinear relationship with pore pressure. The isotherm of sample 31vcde from Vermylen (2011) is sampled and fit with a logarithmic curve in Figure 13.

Since the magnitude of the effect of adsorption is a function of the pore width,

adsorption and compaction are inherently coupled – as a pore narrows via compaction, there are fewer adsorption sites available albeit that the presence of the monolayer is more significant. The coupled effect of compaction and adsorption on permeability is presented in Figure 14 along with the corresponding compaction trend from Figure 12 for comparison. A narrow pore will be more affected by the presence of an adsorbed layer as it constitutes a greater fraction of its cross-section as can be seen in the clay-frame case where the presence of the adsorbed layer causes an order of magnitude decrease in permeability at 5 MPa. Figure 15 attempts to explain the nonlinear effect of adsorption with pore pressure by comparing the effect of adsorption as the ratio of the intrinsic permeability to the permeability in the presence of adsorption with the percentage of total pore size reduction that is attributed to adsorption for a clay solid frame. The pore space narrows due to compaction more rapidly at elevated pore pressure than the adsorbed gas desorbs; subsequently, a large fraction of the adsorbed sites in a highly narrowed pore space remain filled and the adsorbed layer inhibits flow to a greater extent than if the pore

had not compacted (right image of Figure 14). Rapid desorption for pore pressures less than 5 MPa result in a steep decrease in the effect of the adsorbed layer.

The presence of an adsorbed layer will always result in permeability less than the

intrinsic value. In the absence of compaction this effect is at its greatest at high pore pressure when a full monolayer is adsorbed thereby maximizing the pore size reduction (Figure 14). When compaction is considered, the effect of adsorption on permeability is greatest at a pore pressure corresponding to ~70% adsorbate coverage. For lower pore pressures rapid desorption diminishes the effect, while at higher pore pressures wider pores are intrinsically more permeable and less sensitive to the presence of an adsorbed layer.

Figure 13: Type I adsorption isotherm for sample 31vcde from the Barnett shale (adapted from Vermylen, 2011).

Figure 14: Coupled effect of compaction and adsorption for (left) intermediate (red triangles) and clay (red circles) and shales (right). Black symbols correspond to effect of compaction alone.

Figure 15: Comparison of effect of adsorption on permeability and effect of adsorption on porosity decrease in a compacting system. Circles represent the ratio of intrinsic permeability to permeability in the presence of an adsorbed layer while the dashed line represents the percent of porosity reduction that is attributed to the presence of the adsorbed layer. The mean free path of methane (d = 0.38 nm) at a temperature of 350 K and pore

pressures ranging from atmospheric pressure (P = 0.1013 MPa) to 40 MPa is computed via Eqn. 2 and spans from 74.4 nm to 0.188 nm. Due to assumptions in the Unified Flow Model (UFM), however, in order to compute a global average Knudsen number for the sample, an accurate average pore width that honors the heterogeneity of the sample must be calculated. A significant, base assumption in the derivation of UFM is that flow occurs in a long channel (H/L << 1: L = length of channel) rather than a tortuous pore system (Karniadakis et al., 2005). As a result, application of Eqn. 5 assumes the pore network of the sample is a long channel. Since the correction for Knudsen diffusion in Eqn. 5 is only a function of Kn, we assume that accurate prediction of an average pore width for the tortuous pore system will minimize error in the estimation of gas permeability via UFM. This is achieved by weighting the three-dimensional minimum width of each pore in each two-dimensional slice cut in the plane perpendicular to the applied pressure gradient by the fraction of flux the pore transmits. In this manner, the pore widths with the most weight are those that the flow ‘sees’ to the greatest degree. As an example, the numerical average of pore widths in the study sample is 6.1 nm; however, upon consideration of the flux, the average pore width is 8.6 nm as the larger pores are more important to flow across the sample. The global average Knudsen numbers computed in this manner are summarized in Table 1. The correction permeability for Knudsen diffusion, however, is only applied for pore pressures below 4.60 MPa which is the critical pressure of methane (Lemmon et al., 2012). The effect of Knudsen diffusion on permeability in the absence of compaction and adsorption is calculated through Eqn. 5 and plotted in Figure 16 with the effect of Knudsen diffusion neglecting the supercritical phase (solid curve) for

comparison. The effect of temperature (not presented here) is observed to have a significantly smaller effect on permeability, over the range of expected reservoir pressure-temperature values, than pore pressure; subsequently, we assume flow is isothermal.

Table 1: Global average Knudsen number as a function of pore compliance. The label ‘Minimum’ corresponds to measurements at 40 MPa, while ‘Maximum’ corresponds to measurements at atmospheric conditions.

Frame

Composition In Situ

Pore Width [nm]

Minimum Kn Atmospheric Pore Width

[nm]

Maximum Kn

Shale 8.16 0.0231

8.64 8.60 Intermediate 5.76 12.90

Clay 4.78 15.55

Figure 16: Apparent permeability of the kerogen pore to methane as a function of pore pressure in the absence of compaction and adsorption. The pore-filling fluid transitions from gas to supercritical fluid (circles) at 4.60 MPa corresponding to the onset of deviation from the intrinsic permeability (red dashed line).

Figure 16 demonstrates, as required, that at pore pressures greater than the critical

pressure, the supercritical phase does not slip at the pore walls, and the permeability of the sample is equal to the intrinsic value. However, at a pore pressure of 1 MPa, the apparent permeability increases to 4.85 times, or 535 nD, greater than the intrinsic value. Subsequently, while Knudsen diffusion does not have any effect during initial production

or far from the borehole, as production induces pore pressure depletion in the reservoir or proximal to the borehole, Knudsen diffusion has the potential to cause a significant increase in the gas permeability of the reservoir. The effect of Knudsen diffusion in a real reservoir will be affected by the presence of an adsorbed layer and compaction of the pore space. As gas adsorbs or pores narrow, the intrinsic permeability,

k∞ decreases while the Knudsen number and thus the rarefaction correction in Eqn. 5 increase resulting in an increase in the effect of Knudsen diffusion.

Figure 17: Total permeability to methane of the sample kerogen pore. The dashed red line indicates a total permeability equal to the intrinsic permeability. The pore-filling fluid is gaseous below 4.60 MPa

Figure 18: Total permeability to methane of the sample kerogen pore assuming the pore compliance value for shale. The red dashed line indicates a total permeability equal to the intrinsic permeability, and the critical pressure is 4.60 MPa.

Ultimately, simulation can be conducted considering each perturbation to

permeability simultaneously. As previously stated, the effects of compaction, adsorption, and Knudsen diffusion are interrelated so all processes must be simulated at once to properly quantify permeability variation in situ as a function of pore pressure. The total permeability

kT of the sample is simulated for consistent pressure-temperature conditions using the logarithmic monolayer adsorption model and compaction model in Eqn. 6 and is plotted in Figure 17. The total permeability of the sample assuming pore compliance is also given in Figure 18 for clearer inspection.

From Figure 17, it is clear that for the intermediate and clay pore compliances the

permeability to gas will always be less than the intrinsic value due to the coupled effect of compaction and adsorption. In the absence of compaction (Figure 18) at supercritical pore pressures (> 4.60 MPa), the effect of adsorption governs the permeability to methane resulting in a permeability up to 30 nD (20%) less than the intrinsic permeability; however, for subcritical pore pressures, rarefaction and desorption of gas from the pore wall results in permeability values up to 50 times greater than the intrinsic permeability at atmospheric pressure. The extreme decrease in permeability documented for a clay pore skeleton is a result of extremely compliant pores. Additionally, the intermediate compliance is more than an order of magnitude greater than the compliance of shale pores documented by Jones and Owens (1980) which are themselves an order of magnitude more compliant than the values presented by Johnston (1987) indicating that ignoring compaction is not a deleterious assumption.

In summary, compaction, adsorption, and rarefaction-induced (Knudsen) diffusion in

a digital, statistically-rendered representation of the shaly microporous reservoir rock has been shown to cause permeability values to deviate from the continuum prediction by as much as a few microDarcies at atmospheric pore pressure and tens of nanoDarcies at 40 MPa..

(The authors thank Arjun Kohli for the SEM image, and Professors M.D. Zoback and

J. Wilcox for their discussions.)

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Progress

Sequestration of CO2 in geological formations is one of the carbon-management technologies having the potential to substantially reduce greenhouse gas emissions while achieving energy sustainability. To translate such potential into concrete development outcomes, realized benefits, and policy, common concerns (i.e. pore pressures reactivating faults or fractures, risk of leakages, and loss of storage capacity) need to be dispelled. These concerns are thus intimately associated with our ability to use geophysical techniques to monitor chemical processes and their effects on the rock properties. Exploring the links between geophysical observables, rock physical properties, and geochemical induced long-term changes upon CO2 sequestration and introducing them into standard monitoring programs are necessary steps to go towards quantitative predictions of pore pressure, saturation, and storage capacity.

This project is on track to meet the overall objectives. The laboratory measurements for ultrasonic monitoring of rock changes associated with CO2 injection have been completed. The team at Rice University is developing, via measurement and simulation, a modeling framework that can help to bridge the gap between laboratory and field geochemical observations.

Future Plans

The laboratory and modeling work at Rice University will continue. At Stanford, rock physics modeling will continue, aimed at providing a framework for understanding the seismic signatures of CO2 saturation, as well as chemical changes to the rock frame.

Contacts Gary Mavko: mavko@stanford.edu Sally Benson: smbenson@stanford.edu Tiziana Vanorio: tvanorio@stanford.edu Stephanie Vialle: svialle@stanford.edu Andreas Luttge: aluttge@rice.edu Rolf Arvidson: rolf.s.arvidson@rice.edu