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Matrix Permeability Measurements of Gas Shales: Gas Slippage and Adsorption as Sources of Systematic.. 2011
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Title | Matrix Permeability Measurements of Gas Shales: Gas Slippage and Adsorption as Sources of Systematic Error. |
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Letham, Eric Aiden |
Date Created | 2011-05-06T16:25:01Z |
Date Issued | 2011-05-06 |
Description | At pressures less than 4 MPa, matrix permeability of gas shales show a strong dependence upon the pore pressure at which they are measured. This is the result of gas slippage, a process that causes Darcy’s Law to break down during the low pressure flow of gas through porous media. In gas shales, the range of pore pressures where gas slippage is recognizable is larger than in conventional reservoir rocks. This is attributed to the smaller pore systems found in tighter rocks. The type of gas flowing through porous media also influences the matrix permeability, as the size of molecules is one of the factors that control the mean free path of a gas. For this reason, matrix permeability determined using helium as a probing gas will be different than when methane is used. Comparing this difference to the underestimation of matrix permeability resulting from neglecting to correct for adsorption, the latter is found to be negligible. Therefore, when trying to determine the matrix permeability of a rock to natural gas, no condition was identified where using helium as a probing gas would lead to a more accurate result than using methane. |
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Text |
Language | Eng |
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Earth, Ocean and Atmospheric Sciences Undergraduate Honours Theses |
Date Available | 2011-05-06T16:25:01Z |
DOI | 10.14288/1.0053604 |
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Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Letham, Eric Aidan. 2011. Matrix Permeability Measurements of Gas Shales: Gas Slippage and Adsorption as Sources of Systematic Error. Undergraduate Honours Thesis. Department of Earth and Ocean Sciences. University of British Columbia. http://hdl.handle.net/2429/34321 |
Peer Review Status | Unreviewed |
Scholarly Level | Undergraduate |
Copyright Holder | Letham, Eric Aiden |
URI | http://hdl.handle.net/2429/34321 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/1078/items/1.0053604/source |
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MATRIX PERMEABILITY MEASUREMENTS OF GAS SHALES: GAS SLIPPAGE AND ADSORPTION AS SOURCES OF SYSTEMATIC ERROR by ERIC AIDAN LETHAM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE (HONOURS) in THE FACULTY OF SCIENCE (Geological Science) This thesis conforms to the required standard ……………………………………… Supervisor THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) MARCH 2011 © Eric Aidan Letham, 2011 ii ABSTRACT At pressures less than 4 MPa, matrix permeability of gas shales show a strong dependence upon the pore pressure at which they are measured. This is the result of gas slippage, a process that causes Darcy’s Law to break down during the low pressure flow of gas through porous media. In gas shales, the range of pore pressures where gas slippage is recognizable is larger than in conventional reservoir rocks. This is attributed to the smaller pore systems found in tighter rocks. The type of gas flowing through porous media also influences the matrix permeability, as the size of molecules is one of the factors that control the mean free path of a gas. For this reason, matrix permeability determined using helium as a probing gas will be different than when methane is used. Comparing this difference to the underestimation of matrix permeability resulting from neglecting to correct for adsorption, the latter is found to be negligible. Therefore, when trying to determine the matrix permeability of a rock to natural gas, no condition was identified where using helium as a probing gas would lead to a more accurate result than using methane. iii TABLE OF CONTENTS TITLE PAGE………………………………………………….….…………….………..…....i ABSTRACT…………………………………………………….……………………...……..ii TABLE OF CONTENTS……………………………….…...…………….……….………...iii LIST OF FIGURES………………………………….………..…….……….…….………....iv LIST OF TABLES…………………………………………….………………...……….……v ACKNOWLEDGEMENTS……………………………………..…………………….…...…vi INTRODUCTION………………………...……………………….………………...…….…1 BACKGROUND AND THEORY………………………………………….………......……2 Permeability………………………………………………….…………………..……2 Gas Slippage …...……………..……………………………………………….……...3 Adsorption …...……………..……………………………………………….……….10 METHODS……………………………………………………………………………..……11 Measuring Matrix Permeability…………………………….……………..…………11 Measuring Porosity.…...……………..…………………………………..….…...…..16 RESULTS……………………………………………………………………...….……...….17 DISCUSSION………………….……………………………………………..……...…....…21 Implications for Reservoir Modeling.……………………….………..…………...…21 Implications for Permeability Measurement Techniques…………..………………..24 Implications for Probing Gas Selection……………………….……………………..25 CONCLUSION……………………………………………….…………………….….……26 REFERENCES CITED……………………………………….………………………...…...28 iv LIST OF FIGURES Figure 1. Velocity profile of fluid flow in a cylindrical pipe…………………….....................4 Figure 2. Schematic diagram of the influence of gas slippage on velocity profiles…………..5 Figure 3. Histogram of pore throat diameters for siliceous mudstones from the Barnett Shale, Texas…………………………………………….……………………………….……………7 Figure 4. Pore throat size distribution of the Fontainebleau Sandstone……..….…............….8 Figure 5. Relationship of mean free path to gas pressure……………………………………..9 Figure 6. Relationship of mean free path to inverse of gas pressure………………….……..10 Figure 7. Schematic diagram of the experimental setup used to measure matrix permeability…………………………………………………………………………….........13 Figure 8. Processing example of pulse decay data………………………………………….15 Figure 9. Schematic diagram of a Langmuir isotherm………………………………………16 Figure 10. Sample 1 matrix permeability as a function of pore pressure……………………19 Figure 11. Sample 1 matrix permeability as a function of inverse pore pressure……………20 Figure 12. Permeability as a function of pore pressure for Klinkenberg’s sample “A”……..22 Figure 13. Permeability as a function of inverse pore pressure for Klinkenberg’s sample “A”…………………………………………………………………………………………...22 Figure 14. Permeability as a function of pore pressure for Klinkenberg’s sample “L”……..23 Figure 15. Permeability as a function of inverse pore pressure for Klinkenberg’s sample “L”…………………………………………………………………………………………...23 v LIST OF TABLES Table 1. Percent underestimation of matrix permeability at different pore pressures….........20 vi ACKNOWLEDGEMENTS I would like to thank Dr. R. Marc Bustin for supervising the completion of this study. I appreciate the time you were able to give me and your willingness to answer all of my questions. It made for an enjoyable experience. I would also like to thank Oyeleye Adeboye, Dr. Gareth Chalmers, Dr. Amanda Bustin, and Kristal Li for their help in the lab, as well as Bryn Letham and Mary Lou Bevier for their constructive criticism while I was writing this thesis. 1 INTRODUCTION Natural gas resource estimates have grown significantly in the past decade (Energy Information Administration, 2010) due to the recognition of gas shale reservoirs and growing confidence in our ability to economically exploit them (Bustin, 2005; Natural Resources Canada, 2008). Successful exploitation of gas shales in the United States has led to an increase in North American supply. Coupled with the global economic recession beginning in 2008, this has led to a decrease in natural gas prices (NRC, 2010). Lower profit margins induced by this high supply-low demand (low price) situation create an environment wherein producers must keep their costs low in order to remain profitable. Aside from non-geological factors that determine the economics of a natural gas deposit (e.g. proximity to transportation networks), the characteristics of a deposit that govern its profitability are how much gas is in place and how easily it can be brought to the surface. Specific to gas shales, which North America is currently shifting towards as its dominant source of supply (Natural Resources Canada, 2009), the key parameters determining these characteristics are total organic carbon content (TOC), maturity, mineral matter, thickness, porosity, and permeability (Ross and Bustin, 2007a). Given that permeability is such an important parameter for predicting reservoir production, strides have been made to improve the accuracy of its measurement (Cui et al., 2009). Building upon this, the present study identifies systematic errors in gas shale permeability measurements, and provides suggestions for how they can be minimized. Measuring gas shale permeability is problematic. Gas shales typically have permeabilities much less than one millidarcy (Bustin, 2005), which are orders of magnitude lower than conventional reservoir rocks. Therefore, traditional steady-flow measurements require too much time to complete (Dicker and Smits, 1988; Cui et al., 2009). For this reason, specialized methods for measuring permeability of tight (low permeability) unconventional reservoir rocks - such as gas shales - have been developed that can be completed in a shorter time frame. These methods include pressure pulse decay, crushed sample dynamic pycnometry, desorption tests, and the analysis of Hg intrusion curves (Luffel et al., 1993; Cui et al., 2009). 2 The methods currently employed to measure the permeability of gas shales are by no means perfect, and refinement of these techniques is necessary before accurate results can be obtained. One problem is that desorption, crushed sample dynamic pycnometry, and Hg intrusion techniques do not take place under confining pressure (Cui et al., 2009), even though permeability varies markedly with effective stress (Katsube, 2000; Murthy, 2008; Dong et al., 2010). A second problem is that permeability is often assumed to be a property of the rock itself, and therefore that it does not vary with the pressure and type of gas flowing within it. However, this assumption breaks down under circumstances where the process of gas slippage is significant. A third problem is that many techniques are used without accounting for adsorption of gas onto the internal surfaces of the shale (Cui et al., 2009). This thesis addresses the latter two problems by exploring how permeability measurements change when different probing gases and pore pressures are used. By using different gases, the size and adsorption characteristics can be varied. Changing the pore pressure causes variation in the mean free path of the gas molecules. A comparison of permeability measurements calculated with and without a mathematical correction for adsorption will also be made. These experiments illuminate systematic errors that can be eliminated by refining measurement techniques, which will result in more accurate permeability measurements. This should lead to improved reservoir analysis, and therefore permit better economic decisions. BACKGROUND AND THEORY Permeability In 1856, Henry Darcy’s observations relating the discharge flux ( ) through porous media to the pressure gradient ( ) and a constant of proportionality dependent upon the permeability of the medium ( ) (see Equation 1) were published (Darcy, 1856). Shortly thereafter, the relationship became known as Darcy’s Law. However, by the 1950’s, it had been decided that permeability should be a property of the porous media only, meaning 3 it would be independent of the type of fluid flowing within it (Hubbert, 1957). Therefore, viscosity ( was separated from the proportionality constant resulting in a variation of Darcy’s Law where is the permeability. The permeability of shale gas reservoirs can be broken down into two distinct categories: fracture permeability and matrix permeability. Fracture permeability characterizes the ease with which fluid can flow through the natural fractures in the shale, as well as those fractures created through hydrofracturing. On the other hand, matrix permeability is the ease with which a fluid can flow within the intact portion of the shale. Fracture permeability usually falls within the millidarcy range (Bustin et al., 2008), whereas matrix permeability is in the microdarcy to nanodarcy range (Bustin, 2005). Although it is much lower than fracture permeability, under some circumstances matrix permeability controls the production of a shale gas reservoir (Luffel et al., 1993; Bustin et al., 2008). The present study is concerned with matrix permeability and the methods used to evaluate it. Compared to conventional reservoir rocks, gas shales generally have much smaller pores and pore throats. Therefore, in a gas shale, collisions between molecules and pore walls are more frequent in comparison to collisions between the molecules themselves. Under these circumstances, the process of gas slippage (discussed below) becomes important, resulting in matrix permeability that is dependent upon the type of gas flowing through the shale as well as the gas pressure. Thus, it has been suggested that the ease with which a gas can flow (permeability) within gas shales is better classed as diffusivity (Cui et al., 2009). To be consistent with the literature, the ease with which gas can flow through the matrix of gas shales shall be termed matrix permeability herein, even though the flow incorporates both advective (Darcy) flow and diffusion. Therefore, matrix permeability is not restricted to being a property of the rock, and may vary with the flow conditions. Gas Slippage Consider the idealized example where a pore throat is represented as a cylindrical pipe. The two dimensional velocity profile of a fluid flowing through the pore throat would 4 be a hyperbolic curve, as depicted in Figure 1 (John and Haberman, 1971). The velocity is zero at the pore wall, has a maximum in the middle of the pore throat, and an average velocity of half the maximum. The volume flow rate ( ) of fluid flowing through the pore throat can be quantified using Poiseuille’s equation for compressible fluids: where is the radius of the pore throat, is the viscosity, is the length of the pore throat, is the upstream pressure, is the downstream pressure, and is the average pressure (Klinkenberg, 1941). Figure 1. Two dimensional velocity profile for fluid flowing through a cylindrical pipe. Length of arrows from zero baseline indicate magnitude of velocity. Equation 3 holds true for the flow of liquids in reservoir rocks (Klinkenberg, 1941). However, for the low pressure flow of gas through microporous materials (i.e. low pressure gas shales) equation 3 breaks down. This is because, for a gas, the velocity directly adjacent to the pore wall is not zero, and varies with the mean free path of the gas. The mean free path of a gas is the average distance a molecule travels before it collides with another molecule (Knight, 2004). Therefore, at the pore wall there is a layer of gas the thickness of the mean free path where, on average, molecules do not experience any collisions with one another. Let us call this the MFP layer. Within the MFP layer, the average velocity will be dictated by the average velocity of the gas molecules immediately adjacent to the MFP layer (Klinkenberg, 1941). This is due to the fact that, on average, this is the last place a collision took place between the molecules in the MFP layer and other gas molecules. As can be seen in Figure 1, this means that the larger the mean free path, the faster the 5 average velocity in the immediate vicinity of the pore wall will be. This is depicted schematically in Figure 2. Figure 2. Schematic diagram depicting the impact of changing mean free path of a gas on its velocity profile when flowing through a cylindrical pipe. The equation for the mean free path of an ideal gas is as follows: √ where λ is the mean free path, is the number of gas molecules, is the volume occupied by the gas, and is the radius of the gas molecules (Knight, 2004). Equation 5 shows how Velocity 0 Pore Throat Wall Velocity Profile Zero Velocity Increasing Mean Free Path and Average Velocity 6 the molecular density term ( ) in equation 4 can be expressed in terms of pressure , temperature and Boltzmann’s constant . For an isothermal situation, the mean free path of a gas is therefore proportional to the inverse of pressure. Similarly, the mean free path is proportional to the inverse of the molecular radius squared (equation 4). So, the lower the pressure and smaller the molecules, the faster will be the average velocity in the pore throats. A higher average velocity leads to an increased volume flow rate, which translates to higher matrix permeability. The impact of mean free path on the volume flow rate of a compressible fluid was quantified by Klinkenberg (1941) as follows: in which is a proportionality factor and is the average mean free path over the pressure range to . Although not quantified exactly, the value of is slightly less than 1 (Klinkenberg, 1941). The rightmost term in equation 6 quantifies the amount of gas slippage taking place. If the mean free path ( ) approaches zero, the rightmost term becomes 1, and equation 6 is reduce to equation 3. This means that no gas slippage is taking place. Likewise, if the radius of the pore throat ( is much larger than the mean free path, the rightmost term also becomes 1, indicating no gas slippage. For this reason, the effects of gas slippage will be greater in rocks with smaller pores and pore throats. It is therefore important to compare the pore size distribution of gas shales to those of conventional reservoir rocks. Loucks et al. (2009) used capillary-pressure analysis to determine the distribution of pore throat sizes in siliceous mudstones from the Mississippian Barnett shale, a major gas shale play in the Fort Worth basin, Texas (Montgomery et al., 2005). Their results are shown in Figure 3. It can be seen that most of the pore throats lie in the nanometer size range. Lindquist et al. (2000) used synchroton X-ray tomographic images to characterize the pore throat size distribution of the Fontainebleau sandstone (see figure 4). These sandstones are fine grained, well sorted quartz arenites of Oligocene age located just south of Paris, France (Haddad et al., 2006). In contrast to the mudstones from the Barnett shale, most of the pore throats in the Fontainebleau sandstone are in the micrometer size 7 range. Assuming that this size relationship holds, it can be expected that gas shales, having smaller pore throat radii, will experience more gas slippage than conventional reservoir rocks (see equation 6). Consequently, matrix permeability of gas shales is more strongly dependent on the mean free path of gas molecules flowing within them than conventional reservoir rocks. Figure 3. Histogram of pore-throat diameters calculated from capillary-pressure sample analysis of four siliceous mudstone samples from the Barnett shale, Texas (Loucks et al., 2009). Two probing gases (gases used to experimentally evaluate the physical properties of a rock) commonly used when measuring permeability are methane and helium. Methane is the dominant component of natural gas, and therefore would naturally seem like a good selection for a probing gas. However, complications (discussed below) are introduced due to the fact that methane is not an inert gas (Cui et al., 2009). On the other hand, helium, a noble gas, is often chosen as a probing gas because it behaves more like an inert gas. However, the diameter of methane (0.38 nm) is 46% larger than that of helium (0.26 nm) (Cui et al., 2009). 8 Figure 4. Pore throat size distribution for four rocks from the Fontainebleau Sandstone (Lindquist et al., 2000). As can be seen in equation 4, all else held constant, this size discrepancy will result in a longer mean free path for helium than for methane (see Figure 5). A longer mean free path translates into a larger volume flow rate (equation 6), which results in higher matrix permeability. p ro b a b il it y 9 Figure 5. Relationship between the mean free path of a gas (calculated using equation 4) and pressure for both helium and methane. In Figure 5 it can be seen that the difference between the mean free path of helium and methane decreases as pressure increases. In fact, the mean free path converges on a common value of zero at infinite mean pressure (see Figure 6). Because the mean free path directly affects the volume flow rate of gas through pores and pore throats (equation 6), variations in matrix permeability with changing pore pressure follow the same trends found in Figure 5 and Figure 6. 0 5 10 15 20 25 0 2 4 6 8 M ea n F re e P at h (n m ) Pressure (MPa) Mean Free Path vs Pressure Helium Methane Power (Helium) Power (Methane) 10 Figure 6. Linear relationship between the mean free path of a gas and the inverse of pressure. Note the trends for helium and methane converge on a common mean free path of zero at infinite mean pressure. Adsorption It has been recognised that a significant portion of the natural gas contained within organic rich shales is held in the adsorbed state (Chalmers and Bustin, 2007). Adsorbed gas exists as a semi-liquid on the pore surfaces of the shale, and has a much higher density than free gas in the compressed state (Cui et al., 2009). Because the computation of matrix permeability for tight rocks is often based on mass balance equations, the presence of adsorbed gas is a complication; the amount of gas contained within the sample ( ) cannot be calculated using a simple formula involving the pore pressure ( ), temperature ( ), pore volume ( and the gas constant ( : The amount of gas contained within the sample when adsorption occurs will be greater than that calculated using equation 7. For this reason, a matrix permeability calculated using a mass balance equation without a correction for adsorption would be lower than the actual 0 5 10 15 20 25 0 0.5 1 1.5 Mean Free Path (nm) Inverse Pressure (MPa-1) Mean Free Path vs Inverse Pore Pressure Helium Methane 11 matrix permeability of the sample (Cui et al., 2009). The degree of error is dependent upon the pressure at which measurements are taken, as the ratio of gas held in the adsorbed state to gas held in the compressed state is not linearly proportional to pressure (Cui et al., 2009). Instead, at low pressures this ratio is high, leading to a larger underestimation of permeability than at higher pressures where the ratio is lower. The magnitude of permeability underestimation is also dependent upon the quantity and rank of organic matter in the sample, as adsorbed gas is mostly found within the microporosity of organic matter, and microporosity of organic matter is positively correlated with rank (Chalmers and Bustin, 2007). METHODS Measuring Matrix Permeability Measuring the matrix permeability of gas shales using traditional steady flow techniques is impractical due to the tight nature of these rocks; the flow is so slow that expensive, very precise equipment would be required (Dicker and Smits, 1988; Cui et al., 2009). Therefore, for this study matrix permeability measurements were made using the pulse decay technique on core samples. This process involves monitoring the decay with time of a pressure pulse imparted on the sample. Core samples were chosen over crushed samples so that a confining stress could be applied. This is important, as matrix permeability is known to vary with effective stress by more than an order of magnitude (Katsube, 2000; Murthy, 2008; Dong et al., 2010). Figure 7 is a schematic representation of the experimental setup. The sample is held in a Hoek cell between two platens, which also function as a pathway for gas to reach the sample. The platens and sample are surrounded on their curved cylindrical sides by a plastic membrane inside the cell. Confining pressure is applied using a hydraulic pump. The plastic membrane separates the sample from the hydraulic oil, and also functions as a seal to prevent the probing gas from escaping the system or traveling between the two reservoirs in a path other than through the sample. A 1.4 cm 3 upstream gas reservoir and a 1.2 cm 3 downstream gas reservoir are separated by the sample. The two reservoirs can also be directly connected 12 by opening valve 2. The upstream reservoir is adjoined to a pressurised tank containing the probing gas, which can be shut off from the system using valve 1. The downstream reservoir (and upstream reservoir if valve 2 is open) can be connected to or shut off from the atmosphere using valve 3. Samples were prepared by cutting 2.9 cm diameter cores from larger diameter core samples on a drill press. The sample core ends were then cut off using a low speed circular saw, leaving a roughly cylindrical sample of approximately 3 cm length and 2.9 cm diameter. The ends of the sample were then machined into flat parallel surfaces. The machining ensured that the samples were truly cylindrical, which was desirable to assure that the sample ends met flush with the ends of the platens. Pulse decay experiments were carried out using the following procedure. First, the sample was placed in the Hoek cell and the platens fastened tightly against the flat faces of the core. Second, a confining pressure equal to the desired effective stress of the measurement was applied. Next, valves two and three were opened, leaving the whole system open to the atmosphere. Once equilibrium was attained (i.e. the pressure everywhere in the system was atmospheric), both the differential and absolute pressure transducers were zeroed. Following this, valve three was shut. Both the gas pressure (which is also the pore pressure) and confining pressure were then incrementally increased to the test values, taking care not to exceed the desired effective stress. This was done to prevent fracturing of the sample, and to ensure that the pore structure didn’t collapse beyond what resulted from the desired effective stress. Next the sample was left to soak, which allowed the gas to migrate into the shale. Because gas shales have such low matrix permeability, the soaking period lasted up to ten hours. Once all pressure transducers stabilized at a constant value, meaning equilibrium had been attained and the soak period had finished, valve two was shut. Following this, valve three was opened long enough to decrease the pressure in the downstream reservoir by 0.28 MPa. The above steps generated a pressure gradient across the core that drove flow of the probing gas from the upstream to the downstream reservoir. The differential pressure between the two reservoirs was then measured with time and recorded by a computer program at intervals of 0.0007 MPa. The rate of differential pressure decay is related to the matrix permeability of the rock; differential pressure will decay more rapidly for higher 13 F ig u re 7 . S ch em at ic d ia g ra m o f th e ex p er im en ta l se tu p u se d t o m ea su re m at ri x p er m ea b il it y. 14 matrix permeability samples than lower matrix permeability samples. An exact value of matrix permeability was obtained using equation 8 (Cui et al., 2009): where is the matrix permeability, is the viscosity of the probing gas, is the compressibility of the probing gas, is the length of the sample, is the cross sectional area of the sample, and and are the volumes of the upstream and downstream reservoirs respectively. is a quantity that reflects the rate at which differential pressure decays. To obtain , the log of dimensionless differential pressure is plotted against time; is the slope of this graph (see Figure 8). is defined as: where is the first solution of the transcendental equation: The values and are the ratio of the amount gas that can be held in the sample compared to the upstream and downstream reservoirs respectively: In equations 10 and 11, is the pore volume and is a factor quantifying the amount of gas held in the adsorbed state in comparison to the amount of gas in the free state, defined as: where is the effective adsorption porosity and is the porosity (Cui et al., 2009). To ascertain , Langmuir isotherms (Langmuir, 1918) were determined. Figure 9 shows the typical shape and important parameters of a Langmuir isotherm, which is a curve representing adsorbed gas content as a function of pressure with temperature held 15 constant. The Langmuir volume ( ) is the maximum amount of adsorbed gas the rock can hold. The Langmuir pressure ( ) is the pressure at which the amount of adsorbed gas is one half of the Langmuir volume. Assuming the adsorbed gas exists as a monolayer of molecules, the two parameters and can be used to generate the Langmuir isotherm using the equation: where is the quantity of gas adsorbed and is the pore pressure. This Langmuir isotherm can then be used to determine the effective adsorption porosity of the sample at the test pressure. Dividing this by the porosity of the sample yields . Figure 8. An example of how was calculated from the experimental data. The log of differential pressure is plotted as a function of time. is the slope of this plot. Langmuir isotherms for the sample used in this study were not measured directly. Instead, the TOC content of the sample was measured, and by comparison with samples from the same area and with similar TOC content and known Langmuir parameters, the 𝑥 𝑦 𝑠 𝑦 𝑥 16 Langmuir volume and pressure of the sample used in this study was estimated. These parameters were then used to correct the matrix permeability measurements for adsorption. Matrix permeabilities were also calculated without correcting for adsorption (i.e. calculated with equal to zero, see equations 8 through 13) and with a very high Langmuir volume (8 cm 3 /g) for means of comparison. Figure 9. Schematic diagram of a Langmuir isotherm. The Langmuir volume is the maximum amount of adsorbed gas the rock can contain and the Langmuir pressure is the pressure at which half of the Langmuir volume is adsorbed. As discussed above, both the process of gas slippage and the underestimation of matrix permeabiltiy due to adsorption show a dependence on pore pressure as well as the type of probing gas used. For that reason, matrix permeability measurements were made at a variety of pore pressures and with different gases while holding all other variables known to effect matrix permeability (e.g. effective stress) as constant as possible. Through doing so, the effects of gas slippage and adsorption on matrix permeability were isolated. Measuring Porosity To determine porosity, both the skeletal density and bulk density were first measured. The skeletal density is the mass per unit volume of rock, excluding the porosity. It is the Pressure Amount of Adsorbed Gas Langmuir Volume (𝑉𝐿 Langmuir Pressure 𝑉𝐿 17 density of the rock if it were to have zero porosity. The bulk density is the mass per unit volume of the rock as a whole. It therefore lies in between the skeletal density and the void volume (pore) density. The difference between skeletal density and bulk density was then used to determine porosity as follows: To measure skeletal density, pycnometry was employed. This process involved the expansion of helium between two previously separated cells with known volume, one of which contained the sample. Samples were crushed and sieved to a size range of 0.84 to 0.59 mm so that measurement could be completed in a reasonable time frame (approximately 0.5 hours per sample). By comparison of the initial pressures of the two cells and the final pressure of the two connected cells once equilibrium was established, Boyle’s Law was used to determine the total volume occupied by the gas. Subtracting this volume from the total volume of the two cells leaves the volume of the shale sample. Dividing the mass of the sample by this volume gave the skeletal density. The bulk density of the shale was determined using mercury immersion. Whole shale samples were first weighed, and then submerged in a beaker of mercury. The very high surface tension of mercury prevents it from entering the pores of the shale. The displacement of mercury was therefore taken to be the volume of the shale sample. Bulk density was then calculated by dividing the mass of the shale sample by its measured volume. RESULTS The sample used in this study was a green-grey parallel laminated siltstone from the Horseshoe Canyon Formation in south central Alberta. The sample was found to have 13% porosity and a TOC content of 0.77%. Comparison of the TOC content of this sample to other rocks from the Horseshoe Canyon Formation with known Langmuir parameters led to an estimated Langmuir pressure of 1.82 MPa and Langmuir volume of 0.11 cm 3 /g. 18 Figure 10 shows the results of matrix permeability measurements made on Sample 1 using both helium and methane. Measurements were made over a range of pore pressures from 1 to 8 MPa. Correction for adsorption of methane using the estimated Langmuir pressure and volume resulted in matrix permeability that was at most underestimated by 0.2%. This difference is so small that it was impossible to graph on this scale; the two values overlapped. They are therefore both represented as red squares. Correction for adsorption using a hypothetical Langmuir volume and pressure of 8 cm 3 /g and 2 MPa respectively is represented as green triangles. A Langmuir volume of 8 cm 3 /g is high for a gas shale, and would be characteristic of more organic rich materials such as coal. That being true, in this hypothetical case, error induced by not correcting for adsorption represents the maximum that would be expected for a gas shale. The percent of underestimation is tabulated in Table 1, and ranges from 10.0% at a pore pressure of 0.93 MPa to 1.2% at a pore pressure of 7.9 MPa. The trends for both helium and methane are a decrease in matrix permeability with increasing pore pressure (Figure 10). The matrix permeability to helium drops by 54% and the matrix permeability to methane 33% over the measured interval. These trends become more gradual at higher pressure, and closely resemble those for the mean free path of helium and methane over the same interval of pore pressures (see Figure 5). Figure 11 shows the relationship of inverse pore pressure to permeability, which is characterized by linear trends for both helium and methane that converge upon a common matrix permeability at infinite mean pressure of approximately 3.5 10-4 md. Based on this evidence, it is concluded that the trends seen in matrix permeability as a function of pore pressure are the result of gas slippage. At a common pore pressure, the matrix permeability to helium is always greater than the matrix permeability to methane. Again, this observation can be attributed to gas slippage; the smaller diameter of helium compared to methane results in a larger mean free path, and therefore more slippage. 19 Figure 10. Results from matrix permeability measurements on Sample 1 over a range of pore pressures and using different probing gases. 0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 0 1 2 3 4 5 6 7 8 M at ri x P er m ea b ili ty ( m d ) Pore Pressure (MPa) Relationship Between Permeability and Pore Pressure - Sample 1 Helium Methane Methane - Langmuir Volume 8cc/g 20 Pore Pressure (MPa) Percent underestimation of permeability – no correction compared to Langmuir Volume of 8 cm 3 /g and Langmuir Pressure of 2 MPa 0.93 10.0 1.78 6.7 2.71 4.6 3.60 3.4 4.44 2.7 6.23 1.7 7.90 1.2 Table 1. Percent underestimation of matrix permeability when adsorption is ignored for Sample 1 using a hypothetical Langmuir volume of 8 cm 3 /g. Figure 11. Matrix permeability as a function of inverse pore pressure for Sample 1. Note that the trends for both helium and methane converge on a common matrix permeability at infinite pore pressure. 0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 0 0.2 0.4 0.6 0.8 1 1.2 M at ri x P er m ea b ili ty ( m d ) Inverse Pore Pressure (MPa-1) Permeability as a Function of Inverse Pore Pressure - Sample 1 Helium Methane Linear (Helium) Linear (Methane) 21 DISCUSSION Implications for Reservoir Modeling By analyzing the influence of pore pressure on permeability, Klinkenberg (1941) was able to recognise the process of gas slippage in petroleum reservoir rocks. However, he concluded that in formations of moderately high permeability, gas slippage was “not of first importance for practical purposes” (Klinkenberg, 1941). He went on to state that for formations of lower permeability (2-10 millidarcies in Klinkenberg’s experiments), the percent difference of permeability measured at atmospheric pressure compared to that indicated by extrapolation to infinite mean pressure was considerable (up to 100%). However, this discrepancy was only significant at atmospheric pressure, and therefore is of little applicability to modeling a natural gas reservoir, where pressures are much higher than atmospheric. Contrastingly, the results of the present study indicate that for rocks of much lower permeability (e.g. gas shales with matrix permeability in the nanodarcy range), gas slippage results in significant variation of matrix permeability over a larger range of pressures, including those experienced in natural gas reservoirs. Under these circumstances, gas slippage is important for practical purposes, as is discussed below. For means of comparison, the results from two of Klinkenberg’s 1941 permeability experiments on core samples are shown in Figure 12 and Figure 14. When compared with Figure 10, the expanded pressure range where gas slippage is important becomes obvious (note Figure 10 has the same pressure scale as figures 12 and 14). In comparing the results of Klinkenberg’s study, of notable significance is that the decay of matrix permeability with increasing pore pressure is more gradual for the lesser permeable rock (Figure 14) than the more permeable rock (Figure 12). The results of this study are particularly important for low pressure gas shale reservoirs, such as the Antrim and New Albany, where typical initial reservoir pressures are 2.8 MPa and 2.1 - 4.1 MPa respectively (Curtis, 2002). During production of these reservoirs, the reservoir pressures will drop from their initial values, which will result in increased gas slippage and therefore higher matrix permeabilities. For instance, typical drawdown curves 22 Figure 12. Permeability to air as a function of pore pressure for sample “A” in Klinkenberg’s experiments. Data from Klinkenberg, 1941. Figure 13. Permeability to air as a function of inverse pore pressure for sample “A” in Klinkenberg’s experiments. Data from Klinkenberg, 1941. 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 7 8 P er m ea b ili ty ( m d ) Pressure (MPa) Relationship of Permeability to Pore Pressure - Klinkenberg's Sample "A" 0 50 100 150 200 250 300 350 0 200 400 600 800 1000 1200 P er m ea b ili ty ( m d ) Inverse Pore Pressure (MPa-1) Relationship of Permeability to Inverse Pore Pressure - Klinkenberg's Sample "A" 23 Figure 14. Permeability to hydrogen as a function of pore pressure for sample “L” in Klinkenberg’s experiments. Data from Klinkenberg, 1941. Figure 15. Permeability to hydrogen as a function of inverse pore pressure for sample “L” in Klinkenberg’s experiments. Data from Klinkenberg, 1941. 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 5 6 7 8 P er m ea b ili ty ( m d ) Pressure (MPa) Relationship of Permeability to Pore Pressure - Klinkenberg's Sample "L" 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 5 6 7 P er m ea b ili ty ( m d ) Inverse Pore Pressure (MPa-1) Relationship of Permeability to Inverse Pore Pressure - Klinkenberg's Sample "L" 24 for the Antrim Shale indicate that reservoir pressures as low as 1.0 MPa are reached during the final stages of production (Martini et al., 2003). Even though most gas shales have initial pressures higher than those of the Antrim and New Albany (Curtis, 2002), the effects of gas slippage are still important for these reservoirs. To produce natural gas from the subsurface, a pressure gradient is established between the wellbore and the surrounding reservoir. Pressure in the well bore is kept low in order to maintain flow. Therefore, near the well bore, it is expected that the gas pressure is within the pressure range where gas slippage significantly influences the matrix permeability of the shale. In a reservoir model, the accuracy of the input parameters will dictate how accurate the results are. For this reason alone, the implications of this study are important; care should be taken to accurately determine how matrix permeability will evolve over the pressure range of production. However, it is important to recognise that not all gas shale reservoir production is controlled by the matrix permeability (Bustin et al., 2008). Only under certain circumstances will matrix permeability have strong control on the outcomes of a model. Nonetheless, for these cases, the trends observed in this study should be considered. Implications for Permeability Measurement Techniques Aside from pressure pulse decay experiments on core samples, two alternate methods commonly used to determine the matrix permeability of gas shales are crushed sample dynamic pycnometry and canister desorption tests. Crushed sample dynamic pycnometry involves a similar setup to that used when determining the skeleton density of samples in this study. However, of interest is the rate of change of pressure upon allowing gas to expand between the two cells of known volume, one containing the sample. In a theoretical discussion of this method, Cui et al. (2009) suggested that matrix permeability would show variation depending on the pressures used. The analytical results of the present study support this conclusion. This being the case, when trying to determine the matrix permeability of a high pressure gas shale reservoir, crushed sample dynamic pycnometry measurements performed at low pressures will yield an overestimation of matrix permeability. Adding the fact that this method does not take place under confining pressure, the overestimation could be orders of magnitude in size. 25 Canister desorption tests involve retrieving core from the subsurface and immediately placing it in a sealed canister that is maintained at reservoir temperature. The cumulative volume of gas released from the sample is then measured as a function of time. From this data, matrix permeability can be calculated (Cui et al., 2009). However, during the transition from the subsurface to the canister, it is inevitable that the pore pressure in the sample will drop by some amount. Additionally, once the sample is sealed in the canister, pore pressure will continue to drop throughout the duration of the experiment. If the pressure drops down into the range where matrix permeability is sensitive to pore pressure, the result would be a measured matrix permeability higher than that of interest. Implications for Probing Gas Selection When measuring matrix permeability to natural gas, intuitively it would be a good idea to use helium as a probing gas so that the systematic error induced by using methane without a correction for adsorption can be avoided. This would save the time and money required to determine the Langmuir isotherm of the sample. However, using helium, a gas significantly smaller than the average molecules in natural gas, also leads to systematic error; the smaller diameter of helium leads to a larger mean free path, which translates to higher volume flow rates through pore throats, and therefore higher matrix permeability. As is seen in Figure 10, even for a hypothetical unrealistically high Langmuir volume for a shale, the systematic error caused by not correcting for adsorption is dwarfed by the systematic error resulting from the increased gas slippage taking place when using helium as a probing gas. Both the systematic errors induced by not correcting for adsorption when using methane as a probing gas and by increased levels of gas slippage when using helium as a probing gas decay as pressure is increased; at high pressures, the mean free path of either gas is so small that the effects of gas slippage become negligible, and the ratio of gas held in the free state to gas held in the adsorbed state becomes so high that the effects of adsorption on matrix permeability can be ignored. Under this circumstance, it might be concluded that matrix permeabilities determined using either helium or methane as a probing gas will be similar. In this study, not enough data was taken at high pressures to assess whether or not this is true. However, some studies have indicated that, because of its smaller size, helium will be able to access pores and percolation pathways that methane cannot (Ross and Bustin, 26 2007b). This could conceivably influence the matrix permeability. It is therefore concluded that when trying to determine the matrix permeability of a rock to natural gas, no situation has been identified in this study where using helium as a probing gas will give a more accurate result than methane. CONCLUSION Increased scarcity of easily retrievable natural gas has resulted in the exploitation of tight unconventional resources. These include gas shales, which in comparison to conventional reservoir rocks are characterized by much smaller pores and pore throats, and therefore considerably lower matrix permeability. Due to gas slippage, the smaller pore structure also results in dependence of matrix permeability on pore pressure over a wider range of pore pressures than in conventional reservoir rocks. This became evident when comparing permeability trends from Klinkenberg’s 1941 study of conventional core samples to those measured for gas shales in this study. Even with an unrealistically large Langmuir volume, underestimation of matrix permeability due to not correcting for adsorption was at most 10%. However, using helium as a probing gas (as is common practice to avoid correcting for adsorption) resulted in overestimation of methane matrix permeability by as much as 65%. This is attributed to helium, a smaller molecule, experiencing more gas slippage. Therefore, when determining the matrix permeability to natural gas (of which methane is the dominant component) no circumstance was identified in this study where using helium as a probing gas gave more accurate results than using methane. By measuring matrix permeability using methane as a probing gas and taking care to complete measurements at the correct pore pressure, systematic error can be avoided. This will result in more accurate model parameters. In turn, this should permit better economic decisions in a time when profit margins on natural gas are thin. The sample used in this study had a measured matrix permeability in the 10 -4 md range. Gas shales can have matrix permeabilities orders of magnitude smaller than this. 27 Further study should therefore be completed using tighter rocks than that used in this study. It would be expected that tighter rocks would have even smaller pore structures, and therefore would show an even stronger dependence of matrix permeability on pore pressure. 28 REFERENCES Bustin, R.M. 2005. Gas Shales Tapped for Big Play: AAPG Explorer, February. Bustin, A.M.M., Bustin, R. 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