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1 The Effect of Adsorption and Diffusion on the Gas Permeability of Kerogen Adam M. Allan* and Gary Mavko, Stanford Rock Physics Laboratory, Stanford University, USA Summary The main purpose of this study is to determine the respective and coupled effects of adsorption and Knudsen ( slip ) diffusion on the permeability of a kerogen body to methane gas while honoring the heterogeneity intrinsic to the pore space. Adsorption is modeled at the molecular scale in accordance with Langmuir adsorption theory in order to preserve induced topological variations in the pseudo-pore wall structure. Knudsen diffusion is incorporated through application of the Unified Flow Model with a novel pore width averaging technique that better represents the tortuosity of flow paths and thus the global Knudsen number of the kerogen body. The coupled effect of adsorption and Knudsen diffusion is modeled for a wide range of pore pressures on a digital kerogen body that is rendered in three dimensions from a 2D scanning electron microscope (SEM) image by thresholding and sequential indicator simulation (SISIM). Introduction Declining production from national conventional natural gas resources over the past 15 years has created a significant increase in the demand for unconventional resources. In this vein, it is projected (EIA, 2010a) that gas shale reservoirs will supply 6.0 TCF of natural gas in 2035 as compared to 2.4 TCF produced in 2009, potentially accounting for 6% of all energy consumed in the US. As of 2009, 60.6 TCF of total gas shale resources in the US are believed to be proven reserves 21.3% of all economically recoverable natural gas reserves in the US (EIA, 2010b). Subsequently, it is important that we develop a fundamental understanding of the transport properties of gas shale resources. It is now routinely reported that the majority of porosity in gas shale is contributed by nanometer-scale pores in the organic content of the rock (Kale et al., 2010; Loucks et al., 2010; Sondergeld et al., 2010). As the width of these small pores approaches the mean free path of the pore-filling gas, Knudsen diffusion of gas molecules adjacent to the pore wall becomes a non-negligibly contributing flow mechanism that must be accounted for in simulation (Schaaf and Chambre, 1961; Arkilic et al., 1997). Additionally, the high internal surface area of these nanometer-scale pores can result in substantial adsorption of a high density, liquid-like phase (Lu et al, 1995; Vermylen, 2011) that can perturb flow paths and block the smallest pores. Several recent studies have considered the effect of Knudsen diffusion on permeability (Javadpour et al., 2009; Michel et al., 2011) and of adsorption on gas-inplace (GIP) calculations (Bustin et al., 2008; Ambrose et al., 2010); however, the coupled effect of an adsorbed layer of gas and Knudsen diffusion on sample permeability has been less widely studied (Sakhaee-Pour and Bryant, 2011). In addition to limited study of the coupled effect of adsorption and Knudsen diffusion on gas permeability, previous research has been conducted on idealized geometries or under misleading assumptions as to the nature of adsorption for a given adsorbate-sorbent combination. The geometry studied in this work is derived from an SEM image of a kerogen body in a producing shale reservoir and as such contains additional structural heterogeneity than a simplified geometry such as a network of cylinders. Continuum Lattice Boltzmann (LB) simulation (Keehm, 2003) of the pore network is manipulated to include the effect of adsorption by invocation of Langmuir adsorption theory and modeling of the adsorption of gas molecules in the pore space at the molecular level. Additionally, adsorption site selection and Knudsen diffusion are weighted in the simulation as a function of the flux within each pore in order to honor the complexity of flow paths due to the tortuous nature of the pore network. Single phase flow of rarefied methane through a kerogen pore in the presence of an adsorbed layer of variable thickness is simulated in order to quantify changes in the gas permeability of kerogen as a function of pore pressure. At all points in this study, the heterogeneity associated with the evolving tortuous pore space is honored to better capture the magnitude of the effect of each phenomenon. Theory and Method Flow in conventional porous media is modeled under the assumption that the fluid is a continuous viscous fluid with negligible flow of molecules adjacent to the pore wall. However, this continuum assumption becomes invalid in the nanometer-scale pores present in the organic content of shale. In these nano-pores, the rarefaction of the porefilling gas results in an increase in the frequency of molecule-wall collisions with respect to intermolecular collisions and a non-negligble increase in the velocity of the molecules adjacent to the pore wall. The Unified Flow Model (UFM) of Karniadakis et al. (2005), and its subsequent adaption for permeability estimation (Florence et al., 2007), accounts for the non-negligible wall velocities and associated mass transfer across the entire Knudsen regime. This non-negligible slip of gas molecules adjacent to the pore wall is termed Knudsen diffusion, and its SEG Las Vegas 2012 Annual Meeting Page 1

2 Gas Permeability of Kerogen importance is measured by the non-dimensionless Knudsen number Kn H (1) where λ is the mean free path of the gas and H is the characteristic width of the confining pore. Knudsen diffusion becomes a non-negligible transport mechanism when Kn which corresponds to a pore width of approximately 1 micron at representative reservoir pore pressure and room temperature (Figure 1). The characteristic width of the confining pore is taken to be the average pore width of the sample; however, in order to appropriately account for isolated pores that do not contribute to flow, the pore widths are weighted by the mean flux transmitted by each pore respectively. The effect of Knudsen diffusion on gas permeability, k App, is computed via Eqn. 2 as presented in Florence et al. (2007) where k is the intrinsic permeability of the sample. 4Kn k App k 1 Kn Kn 1 (2) 1 Kn occur preferentially in pores that transmit the greatest fraction of the total flux. Each isolated pore is treated as its own Langmuir-space in which adsorption occurs at any site with equal probability while, globally, each isolated pore has a different probability of containing an adsorbed layer dependent upon the flux within that pore. In this manner, the adsorbed layer within disconnected pores or those that terminate within the sample does not desorb as quickly as the adsorbed layer within the high flux pores that are much more sensitive to pore pressure depletion. This pseudo- Langmuir methodology shows minimal (~2%) deviation from a true Langmuir implementation when all pores traverse the entire sample, e.g. a bundle of cylinders, but is better able to account for heterogeneity in the flow paths of more complicated pore spaces by reproducing spatial variations in pore pressure depletion. The effect of the adsorbed layer on permeability is to narrow the pore space due to the immobility of the adsorbed gas molecules thus reducing the intrinsic permeability of the sample. Since the apparent permeability in the presence of Knudsen diffusion (Eqn. 2) is a function of the intrinsic permeability and the average pore width, it is of the utmost importance to accurately model which pores are most affected by adsorption as a function of pore pressure. By weighting the average pore width (and, thus, the effect of Knudsen diffusion) and preferential location of adsorption by the flux within each pore, our methodology is better suited to identify which pores within the system contribute most to flow and more accurately model bulk flow across the sample. Additionally, by modeling adsorption at the molecular scale the effect of pseudotopological variations in the pore structure due to the inclusion of immobile, randomly located gas molecules on gas permeability can be more directly quantified. Application Figure 1: Knudsen number as a function of pressure for helium (triangles) and carbon dioxide (squares) in 1 micron (open symbols) and 5 nm (filled symbols) wide pores, respectively. Flow regime boundaries are indicated by dashed red lines. The assumptions intrinsic to Langmuir adsorption theory require that (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 requirements are fulfilled by selecting random adsorption sites (pore voxels at the fluid-solid interface) and updating the pore voxel to a solid voxel. However, in so doing, adsorption sites in all pores are subject to adsorption with the same probability regardless of the behavior of the bulk gas in each pore. In order to better represent the heterogeneity of adsorption and flow within the pore space, desorption is weighted so as to Our methodology is applied to a 3D kerogen body generated by sequential indicator simulation (Journel and Gomez-Hernandez, 1993) from a thresholded 2.9 x 3.6 micron secondary electron SEM image (Figure 2) captured and provided by Arjun Kohli (Stanford University). In the interest of computational efficiency for this preliminary investigation, the 3D pore geometry is sub-sampled and full analysis is conducted upon a cubic pore structure of dimension 38 nm. The pore structure used in this study is presented in Figure 3 and clearly represents a more heterogeneous structure than a bundle of cylinders or other idealized representative geometry. All proceeding simulations are performed at a temperature of 350 K for pore pressures ranging from atmospheric conditions (P = kpa) to 40 MPa at constant confining pressure with methane (molecular diameter = 0.38 nm) as the pore-filling SEG Las Vegas 2012 Annual Meeting Page 2

3 Gas Permeability of Kerogen gas. LB simulation of the sample in Figure 3 gives an intrinsic permeability of 140 nd ( m 2 ). affected (30 nd) at high pore pressure where adsorptioninduced constriction is at a maximum. The sharp decrease in the reduction of permeability for pore pressures less than ~5 MPa correlates with rapid desorption of the adsorbate until, as expected, at atmospheric pressure the absence of adsorbed gas results in a permeability equal to the intrinsic value. Figure 2: Secondary electron SEM image of a nanoporous kerogen body. Imaged and provided by Arjun Kohli (Stanford University). The SEM image is thresholded such that black areas are pores image porosity is 4.6%. Figure 4: Logarithmic type I isotherm (solid line) fit to sample values (circles) from Vermylen (2011). Figure 3: SISIM rendered pore structure from SEM image shown in Figure 2. The pore space is black while grain voxels are white porosity is 29.8%. The volume fraction of adsorbed gas at a given pore pressure is estimated by a logarithmic fit (Figure 4) to values sampled from a published isotherm for methane adsorption onto a crushed Barnett shale sample (Vermylen, 2011). The type I isotherm in Figure 4 indicates that the application of Langmuir adsorption theory and the adsorption of a full monolayer at ~40 MPa in our methodology is appropriate. Permeability is then simulated as a function of adsorbed volume fraction across the entire range of pore pressure values (Figure 5). The presence of an adsorbed layer decreases permeability by constricting the pore space with an immobile layer of gas molecules; subsequently, it is sensible that permeability would be most Figure 5: Effect of adsorption on gas permeability as a function pore pressure, e.g. volume adsorbed. The black dashed line is the pressure-invariant intrinsic permeability, the red circles are the permeability in the presence of adsorption. The increase in gas permeability due to Knudsen diffusion can be simulated independently of adsorption through application of Eqn. 2 at pressure-temperature conditions consistent with previous simulations (Figure 6). The numerical average pore width of the sample is 6.1 nm; however, when weighted by the contribution to total flux of each pore, the average pore width is 8.6 nm. As such, the larger pores are more significant to the flow paths through the sample. As indicated by Figure 1, Knudsen diffusion will be non-negligible at all pore pressures for nanometer- SEG Las Vegas 2012 Annual Meeting Page 3

4 Gas Permeability of Kerogen scale pores; although as seen in Figure 6, the increase in permeability due to Knudsen diffusion is significantly greater at lower pore pressures (< 20 MPa) as the porefilling methane becomes increasingly rarefied. monolayer adsorption and Knudsen diffusion in this 3D sample is, therefore, to subdue the apparent permeability as seen in Figure 6 for pore pressures greater than ~5 MPa. Due to the diminishing effect of Knudsen diffusion at high pore pressure, the gas permeability of this sample kerogen pore is approximately 10% less than the intrinsic permeability for pore pressures greater than 17.5 MPa. Production-induced pore pressure depletion and subsequent rarefaction of the pore-filling gas results in gas permeability values that exceed the intrinsic value by as much as 400% at 1 MPa. Conclusions Figure 6: Effect of Knudsen diffusion on gas permeability (black curve) as a function of pore pressure. The red dashed line is the pressure-invariant intrinsic permeability. We have presented a novel methodology for modeling single phase flow through a tortuous pore network in the presence of an adsorbed layer of gas with consideration of Knudsen diffusion. The weighting of the average pore width and preferential location of adsorption by the flux within each isolated pore allows more accurate modeling of in situ gas permeability of complex porous media. This methodology benefits from its use of an efficient, robust, and parallelizable continuum Lattice Boltzmann code enabling rapid simulation of relevant scale complex pore structures directly from FIB-SEM experiments. The application of this methodology to a geostatistically generated kerogen pore shows that in situ gas permeability varies widely as a function of pore pressure and must be better quantified in future reservoir simulation. Acknowledgements We acknowledge the Stanford Rock Physics and Borehole Geophysics Project for their support. Adam M. Allan is supported by The William R. and Sara Hart Kimball Stanford Graduate Fellowship Fund. The authors thank Arjun Kohli for capturing and permitting use of the SEM image, and M.D. Zoback for his discussion. Figure 7: The coupled effect of adsorption and Knudsen diffusion on gas permeability (black curve) as a function of pore pressure. The red dashed line is the pressure-invariant intrinsic permeability. At high pore pressure, as adsorption constricts the pore space the Knudsen number increases and the effect of Knudsen diffusion is greater, thereby offsetting part of the permeability reduction associated with adsorption (Figure 7). At 40 MPa, the permeability to gas is 16 nd, rather than the previously simulated 30 nd, less than the intrinsic value. The decrease in pore pressure has a greater effect on the Knudsen number than the widening of the pore space associated with desorption; as a result, the pore-filling methane becomes further rarefied. This rarefaction results in gas permeability values that can exceed the intrinsic permeability of the sample by as much as 5.7 times (550 nd) at 1 MPa pore pressure. The coupled effect of SEG Las Vegas 2012 Annual Meeting Page 4

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Ambrose, R. J., R. C. Hartman, M. Diaz-Campos, I. Y. Akkutlu, and C. H. Sondergeld, 2010, New porescale considerations for shale gas in place calculations: SPE , doi: / MS. Arkilic, E. B., M. A. Schmidt, and K. S. Breuer, 1997, Gaseous slip flow in long micro-channels: Journal of Microelectromechanical Systems, 6, , doi: / Bustin, R. M., A. M. M. Bustin, X. Cui, D. J. K. Ross, and V. S. Murthy Pathi, 2008, Impact of shale properties on pore structure and storage characteristics: SPE EIA, 2010a, Annual energy outlook 2010: U.S. Energy Information Administration. EIA, 2010b, U.S. crude oil, natural gas, and natural gas liquids proved reserves, 2009: U.S. Energy Information Administration. Florence, F. A., J. A. Rushing, K. E. Newsham, and T. A. Blasingame, 2007, Improved permeability prediction relations for low permeability sands: SPE , doi: / MS. Javadpour, F., 2009, Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone): Journal of Canadian Petroleum Technology, 48, 16 21, doi: / DA. Journel, A. G., and J. J. Gomez-Hernandez, 1993, Stochastic imaging of the Wilmington clastic sequence: SPE Formation Evaluation, 8, 33 40, doi: /19857-PA. Kale, S. V., C. S. Rai, and C. H. Sondergeld, 2010, Petrophysical characterization of Barnett shale: SPE , doi: / MS. Karniadakis, G., A. Beskok, and N. Aluru, 2005, Microflows and nanoflows: Fundamentals and simulation: Springer. Keehm, Y., 2003, Computational rock physics: Transport properties in porous media and applications: Ph.D. thesis, Stanford University. Loucks, R. G., R. M. Reed, S. C. Ruppel, and D. M. Jarvie, 2009, Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett shale: Journal of Sedimentary Research, 79, , doi: /jsr Lu, X. C., F. C. Li, and A. T. Watson, 1995, Adsorption studies of natural gas storage in Devonian shales: SPE 26632, , doi: /26632-PA. Michel, G. G., R. F. Sigal, F. Civan, and D. Devegowda, 2011, Parametric investigation of shale gas production considering nano-scale pore size distribution, formation factor, and non-darcy flow mechanisms: SPE , doi: / MS. Sakhaee-Pour, A., and S. L. Bryant, 2011, Gas permeability of shale: SPE , doi: / MS. Schaaf, S. A., and P. L. Chambre, 1961, Flow of rarefied gases: Princeton University Press. SEG Las Vegas 2012 Annual Meeting Page 5

6 Sondergeld, C. H., R. J. Ambrose, C. S. Rai, and J. Moncrieff, 2010, Micro-structural studies of gas shales: SPE , doi: / MS. Vermylen, J. P., 2011, Geomechanical studies of the Barnett shale, Texas, USA: Ph.D. thesis, Stanford University. SEG Las Vegas 2012 Annual Meeting Page 6