**Table of Contents**

Abstract ……………………………………………………………………………… (2)

- Introduction ……………………………………………………………………. (3)

- Statement of problem ………….………………………………………………. (5)

- Methodology ………….……….………………………………………………. (5)

- Summary and Discussion of Selected Papers …………………………………. (6)
- Article 1 – Adsorption Effect on Permeability (Cao et al 2016) …………….. (6)
- Article 2 – Surface Diffusion Effect on Permeability (Zhang et al, 2015)…. (6)
- Article 3: Anisotropy Effect on Permeability (Pan et al, 2015) ……………. (6)

- Comparison Between the Proposed Methods – Discussion and Results ……… (7)

- Conclusions ……………………………………………………………………. (8)

- References ……………………………………………………………………… (8)

- Appendix ………………………………………………………………………. (8)

*Abstract*

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*Unconventional reservoirs, such as shale gas, are an interesting solution for natural gas exploration with low carbon emission, as the world makes the transition from fossil fuels to renewable and cleaner sources. Recently, there have been many advances in the prospection and extraction of fuel gas from these deep rock formations, such as hydraulic fracturing, new coring methods and horizontal drilling, which have made the exploration commercially viable. In regions of the world like North America, there are considerable reserves of shale (O’Reilly et al, 2011), which make a great contribution to the production of natural gas and reduce the amount of oil and other types of fuel that the countries need to import.*

*Since shale gas reservoirs play an important role as a rich source of methane, it is very important to accurately determine key parameters that characterize this type of rock formations. One of the most important ones is the permeability, which reveals the degree that a rock can be penetrated and release gas (Pathi 2008).*

*The purpose of this project is to perform a literature review on proposed permeability measurement methods for shale formations, effectively making a summary of three selected papers from the Journal of Natural Gas Science and Engineering, observing the differences in the experiments, results, discussions and conclusions. This project will also focus on stating the advantages and disadvantages of each proposed method and, finally, determine a best method of all three presented for calculations of permeability for shale gas. The methods discussed by the researched articles focus on different effects on permeability measurements, such as the adsorption/desorption, surface diffusion and anisotropic effects.*

*Conclusions*

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*Keywords: Adsorption, surface diffusion, anisotropy, shale, permeability*

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- Introduction

Natural gas reserves estimation has grown considerably is the last years because of discoveries of shale gas reservoirs and the evolution in the technology involved to extract gas from these deep formations. Moreover, shale gas reservoirs play an important role on energy supplies in countries in North America, Europe, Latin America and Asian countries, like China (National Petroleum Council, 2011).

For any reservoir that consists of a rock formation, like conventional reservoirs, tight gas, coal bed methane and shale gas, to name a few examples, the permeability is one of the most important parameters. The monetary value of a specific reservoir depends strongly on the permeability and several other factors, like transport properties, amount of original gas in place, the ease of bringing the gas to the surface level, proximity to logistics networks, total organic carbon (TOC), porosity, etc (Pathi 2008). The permeability of any rock formation is basically a variable that quantifies the amount of resistance that it imposes to the passage or flow of a specific gas or liquid (fluid). The bigger the value of the pressure to allow flow of the fluid through a rock, the lower the permeability of this rock. The importance on having accuracy when measuring permeability for shale formations is significant. This is one of the main reasons why there have been several papers published on different journals that focus on new methods and models to determine permeability of tight formations (Zhang et al 2012), with the ultimate goal of evaluating the impact of several physical effects (desorption, diffusion, anisotropy to name a few) on the permeability measurement and the amount of error when calculating this parameter by conventional methods (i.e. Darcy’s equation).

For conventional reservoirs, several techniques have been developed to measure the permeability of rock formations, like the steady-state technique, unsteady state technique and mercury intrusion. However, for shale formations, the permeability is considerably harder to measure due to the fact that the pore sizes are in the nanoscale (in consequence, the permeability values will be significantly lower than one millidarcy) and the heterogeneity of the rock formation (the composition and physical properties can vary considerably in the reservoir). There is significant complexity when modeling the fluid flow through pore throats in the scale of nanometers (Letham 2011) and there isn’t a universal method for determining permeability in shale.

That’s why this project will focus on comparing recently proposed permeability methods for shale gas, each one focusing on a specific effect on this important parameter.

- Statement of Problem

There have been several methods proposed to measure and calculate permeability of shale formations, each one with its own advantages and disadvantages. Due to the unconventional nature of shale gas reservoirs and the fact that its pores are in the nanoscale, several new techniques have been proposed focusing on reducing the error measurements and the time taken for the experiments. The big issue is: what is the major effect that must be considered when determining permeability of shale gas reservoirs and what is an efficient method to correctly measure it? Hence, this project will summarize, discuss the results and conclusions of three selected papers that focus each on a different physical effect on this important parameter, the permeability, and, finally, elect a best method of all three.

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- Methodology

This present work’s methodology consists of three parts:

- Perform an accurate summary of selected papers that discuss permeability methods for shale gas, discussing their methodologies, techniques applied, mathematical models, results and conclusions.
- Compare the methods proposed by the selected papers in terms of permeability measurement, stating the advantages and disadvantages of each method.
- Determine the best method of all three, proposing, where necessary, changes and combinations of mathematical models.

- Summary and Discussion of Selected Papers

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*4.1.** **Article 1 – Adsorption Effect on Permeability (Cao et Al 2016)*

This article focuses on the effect of the adsorption/desorption of gas in the pores of shale in permeability measurements. The paper proposes some modifications in the traditional pulse decay method by creating an improved apparatus to measure permeability of shale samples. The main additions to the conventional apparatus for this method were:

- Upstream and downstream reservoirs’ volumes can be changed to alter the ratio between the volume of the pores of the sample and the volume of the reservoir (aand
b), which has an impact on the error of the final calculated permeability;

- A bypass conductor that enables the determination of shale permeability in both directions (forward and reverse) almost instantaneously, thereby reducing the time taken for the tests.

The main reason why this paper uses the unsteady-state technique to perform the permeability measurements is because in the steady-state technique, there is a necessity of stabilization of variables, such as pressure, in respect to time, having thus, a low efficiency compared to the unsteady-state one. Since shale is a very heterogeneous rock, the total time that is going to be elapsed in the steady-state technique can be sometimes weeks or even months, which makes the whole experiment very impractical. In the unsteady state technique, there is not such necessity of constancy of variables in respect to time and the time elapsed for the tests is much smaller (Cao et al 2016).

In spite of the applications of the techniques mentioned above, these conventional methods have a few problematic areas when it comes to shale permeability measurements:

- The traditional pressure decay method uses fixed and unchangeable upstream and downstream reservoir volumes. This can cause larger errors in the permeability measurement.
- The permeability is only measured according to one direction of flow of the gas used in the experiment. The article presents results of measurement of permeability in both directions (forward and reverse) and the values are in the same order of magnitude but different.
- The conventional method uses helium or nitrogen as the gases for the tests. They have significantly different transport properties than methane, the most abundant gas in shale reservoirs.

In face of these problems, the authors propose an improved method to measure permeability in shale using the pulse decay technique, by focusing on the desorption effect and evaluating important parameters in the phenomenon of adsorption, like the volume and pressure of Langmuir.

According to a paper published by Jones, 1997 (Cao et al, 2016), if the volumetric ratio of the upstream and downstream reservoirs (

a

and

b

values) are in the range from 0.25 to 0.5, the lower the values of reservoir volumetric ratio, the better. This is due to the fact that when Jones tested a 0.25 volumetric ratio of the reservoirs, there was room for more error in the pore volume that will cause the same error in the permeability than when he tested a 0.5 ratio. In the article, the volumes of the upstream and downstream reservoirs were doubled from 5 cm^{3 }to 10 cm^{3}. They observed that when they committed 50% error on the pore volumes, the error on permeability with 10 cm^{3} was half of the error with 5 cm^{3}.

Since there is considerable time taken to even pressure and temperature of the upstream and downstream reservoirs in the traditional pulse decay method, the improved apparatus described by the authors in this paper introduced a bypass line between the reservoirs. This not only allowed for measurement of permeability in both directions (forward and reverse) but also practically eliminated the source of human error related to the placement of the sample in the core.

In addition to the modifications done, the gas that was used for the experiment was methane instead of the traditional helium or nitrogen, because this gas makes roughly more than 90% of the total gas stored in shale reservoirs (O’Reilly et al, 2017). Moreover, the transport properties of methane are significantly different than helium or nitrogen (effective diameters of the molecules of helium and nitrogen are lower than the diameter of methane – table 1), and the adsorption energy is larger for methane than it is for helium and nitrogen. Shale has pores with size in the nanoscale, but high value of pore surface area, allowing a great deal of adsorption. This is due to the fact that the ratio of area divided by volume of each pore in the matrix is inversely proportional to the size of the pores. Since helium and nitrogen have smaller values of effective diameter than methane, it is harder for methane to be transported from one pore to another pore. So, as we can clearly see, the permeability will certainly be affected by the type of gas used in the experiment. Therefore, methane is more representative to the reservoir’s characteristics than the conventional gases.

Figure 1 shows the sketch of the improved apparatus constructed by the authors.

Regarding the mathematical model used, the authors added a factor of

1-∅*∂q∂ρ

, in which

∅

is the porosity,

q

is the amount of gas adsorbed and

ρ

is the density of the gas, and called it the “effective adsorption porosity”. For conventional reservoirs (i.e. sandstones), this term can be neglected, because it is often very low. If this term is not considered in the derivation of the permeability expression for shale reservoirs, though, then the proposed model would be nothing more than a conventional model (i.e. Darcy’s law). In the final equation for the porosity considering the adsorption effect (equation number 6 in the article), we can clearly see that the porosity depends on certain parameters, like the volume and pressure of Langmuir, pressure in the pores of the sample and gas compressibility.

The authors made then a comparison between two values of Langmuir volume – 0.01 m^{3}/kg (figure 2A) and 0.001 m^{3}/kg (figure 2B) – for the samples they analyzed. Their intent here was to see the effect of both Langmuir pressure and volume in porosity. The first value refers to a considerably adsorptive sample, while the second one refers to a low adsorption sample (or with weak adsorption).

What they found out was that the bigger the Langmuir pressure, the bigger the effective porosity for both values of Langmuir volume for pore pressure bigger than 6 MPa. When the pressure of the pores is in the range of 5 to 10 MPa, the bigger the Langmuir pressure, the smaller the effective porosity. The most important conclusion is: the effective porosity for adsorptive samples is approximately 10 times the porosity for samples that are weakly adsorptive. Hence, the effect of adsorption and desorption implies errors in the determination of permeability if not considered.

The article then analyzes errors of testing permeability by adsorption: for shale samples with strong adsorption, figure 3A, it has been observed a permeability deviation of up to 90% (relative error in percentage form that is equal to the absolute value of the difference between permeability measurement not considering the adsorption effect and permeability value considering this effect divided by the permeability value not considering the adsorption effect). For shale samples with weak adsorption, figure 3B, the maximum value observed was 50%, still high. This corroborates that the effect of adsorption is considerable.

There are some errors in the experiments that can impact on the final values of permeability. The article lists a few, like gas leaking from the sample, instability of temperature, presence of bubbles of gas residually and rescission of the pressure measuring device. In spite of these, the maximum error identified was due to instability in temperature, that accounted for almost 3%. The other ones accounted for smaller errors than this and can thus be disregarded.

The authors do a final comparison between the conventional apparatus (PoroPDP-200) for the pulse decay method and their improved apparatus, when it comes to the time taken for the tests and the permeability measured in two different directions. Table 2 summarizes their results.

The first conclusion that they make is that for the same sample (i.e. sample 1, pure shale), the permeability measured in the forward direction is slightly different than the parameter calculated in the reverse direction. Hence, the permeability cannot be considered as a single value measured in one flow direction. The second conclusion that they arrive is that the time taken for the test is significantly smaller for the improved apparatus than it is for the conventional one, especially for the first sample, pure shale (7 hours). The third conclusion is that the error on the permeability measurement for the pure shale samples is considerable comparing the results in which the adsorption was considered and the ones in which it was not. The minimum values are over 70%. This trend doesn’t happen for the sandstones, though. Since they are conventional reservoirs and the effect of adsorption isn’t as significant as in shales, the error was one order of magnitude lower than in the shale samples (approximately 6%).

*4.2.** **Article 2 – Surface Diffusion Effect on Permeability (Zhang et Al, 2015)*

This article proposes a new approach to calculate the permeability of shale samples considering surface diffusion effect. The latest papers that consider the effect of diffusion in permeability measurements only take into account the slip flow and Knudsen diffusion. This paper focuses on the surface diffusion effect to evaluate its impact on the permeability.

Since shale is a rock formation with permeability in the nanoscale, some of the assumptions made in Darcy’s equation to express the gas flux in terms of the concentration difference are not valid for shale. The size of the pores of this type of reservoir is basically in the same order of magnitude as the gas molecules the flow through the pores. When this happens, two major mechanisms have important roles: the slip flow and Knudsen diffusion.

Like discussed previously, the surface area of shale formations is considerably high, allowing for significant adsorption of gas on its surface. Surface diffusion happens when molecules of adsorbed gas are transported due to concentration difference on the surface of the pores. Roughly, the quantity of adsorbed gas in shale ranges from 20 to 80% of the original gas that is in the reservoir (Zhang et al, 2015). Hence, the article affirms that the surface diffusion must have a considerable impact in the gas flow through the porous media.

Surface diffusion is a considerably complex subject. Several mathematical models have been proposed. This article’s work on an expression to evaluate permeability is based on the work of Javadpour (2009). On Javadpour’s work, the original gas flux predicted by Darcy’s equation is transformed into a broader analysis considering the gas flux due to pressure forces or advective flux and the flux due to Knudsen diffusion. Since this work describes the gas flow in only one tube with the shape of a cylinder, the others consider some parameters, like porosity, tortuosity and surface roughness of the pores and derive a coefficient for Knudsen diffusion.

The apparent permeability equation (equation 17 in the article) is reached by comparing the final expression of the total flux of gas in the sample with the traditional Darcy’s law. The authors combined the flux due to pressure forces, Knudsen diffusion and surface diffusion into one expression for the total flux of the gas. Finally, they arrived at an expression for the apparent permeability considering all of the diffusion effects. The major difficulty in calculating permeability by the model proposed by the authors is that the expression to calculate the permeability is dependent on several different parameters, including the Knudsen diffusion coefficient and corrected diffusivity, and the latter is not easy to obtain. There is another problem with the approach proposed by the article. The final equation for calculation of apparent permeability considers that the contribution of the flux due to pressure forces, Knudsen diffusion and surface diffusion can be considered separately and summed up to evaluate the overall contribution. This may be not always true, since each flux mechanism can have effects on the other mechanism in the details of the influence of one to another are not yet clear.

The authors validated their model by referring to an experiment done by Keizer et al. (1988). The experiment consisted of making nitrogen and carbon dioxide flow through a microporous γ-alumina membrane (two layers). The two gases have considerably different properties when it comes to adsorption. Roughly there is no nitrogen adsorption in the alumina membrane, so the surface diffusion effect of nitrogen is neglected. This doesn’t happen for the carbon dioxide, which is considerably adsorptive. Moreover, the fractal dimension of the surface of the membrane is not given in the research of Keizer et al (1988). Hence, the article referred to another research by Ahmad et al (2006), in which the average value for the fractal dimension is 2.4 approximately.

Figure 4 shows the experimental data for the nitrogen tests, along with the model the article considered. We can clearly see that the data fits nicely into the model.

In figure 5, the article shows the experimental data and model data for the CO_{2} flow through the alumina membrane. If the surface diffusion is not considered, the values of calculated apparent permeability are considerably lower than the experimental data.

The reported error for the nitrogen test between the permeability value if the surface diffusion is disregarded and the value considering surface diffusion is approximately 2.5%, considerably lower than the reported error for the CO_{2} test, which is 25%. Hence, in rock formations like shale, if there is considerable amount of gas adsorbed, the surface diffusion will have a considerable impact in the permeability determination.

The authors then calculate and show in figure 6 the surface permeability (ks) calculated by the proposed model and the intrinsic permeability (Darcy’s equation – ko). The behavior of the ratio ks/ko is that if the pressure and temperature decrease, there will be a decrease of this ratio and the surface diffusion effect will be less intense. If the pressure increases, this ratio approaches the value 0 and the surface diffusion basically becomes negligible.

The article then conducts a series of parametric studies of the proposed method, more specifically, the sensitivities and effects of each variable on the permeability. The first two variables studied are the temperature and pressure. An increase of pressure causes a significant decrease in the ratio of surface permeability calculated by the model divided by the permeability predicted by Darcy’s equation at a constant temperature (figure 6,). The effect of the temperature is the same of the pressure, but less intense. An increase of temperature will cause the decrease of the ratio apparent permeability divided by Darcy’s permeability, but with less severity (figure 7). Hence, like stated above, with more temperature and more pressure, the surface diffusion effect becomes more trivial. With an increase of pressure, the molecules of the gas come closer and the mean free path or lambda becomes smaller. Hence, the Knudsen number becomes smaller and the flow approaches the condition of no slip flow.

The third variable analyzed is the porosity (figure 8). This parameter has very little effect on the ratio of the apparent permeability. For lower pressures, we can see a slight decrease of the ratio when the porosity increases. For higher pressures this doesn’t happen.

The fourth variable analyzed is the pore radius (figure 9). Opposed to the porosity, the increase of the pore radius makes the ratio of the permeability decrease significantly. For lower values of pressure, this decrease is more intense. This is already expected, due to the fact that for bigger pore sizes, the effect of diffusion on gas transport is less intense.

Up until now, we could see that the ratio between the apparent permeability and intrinsic permeability becomes larger and away from the unity at lower pressures. Hence, the surface diffusion effect on the permeability is more intense at smaller values of pressure.

The last variable analyzed is the surface fractal dimension (figure 10). This one has little effect on the permeability ratio. For lower pressures (0.5 MPa), the permeability ratio decreases more significantly with an increase of the surface fractal dimension, which doesn’t happen with higher pressures (10 MPa). This happens because the bigger the surface fractal dimension, the bigger the surface roughness of the pores and the effect of the Knudsen diffusion is less intense.

The article`s main conclusion is that if the surface diffusion effect is ignored, the gas production rate in shale is underestimated by up to 25%. Moreover, the effect of the surface diffusion on the permeability is more intense for lower pressures and becomes almost negligible for high values of pressure.

*4.3.** **Article 3: **Anisotropy Effect on Permeability (Pan et al, 2015)*

In this article, the authors focus on the anisotropic effect of the permeability measurement. Since reservoir rocks like shale are very heterogeneous, due to heterogeneous surface and pore behaviors on horizontal and vertical directions, the anisotropic effect on the permeability measurement can play a big role.

Many measurement methods of permeability for shale use samples that came from vertically drilling the rock formation and, hence, can review only the vertical permeability of the reservoir, which is often significantly smaller than the permeability measured in the same direction of the beddings of the sample. Moreover, the most common methods or traditional methods to measure the anisotropic effect use cylindrical samples. The biggest problem with that is that the sample is very heterogeneous and this can cause significant errors in the permeability measurement. More modern methods use cubic samples to eliminate this problem, but there is another issue, the sealing of the sample. This is especially hard on the edges of the cube, because there is less area and more pressure that the gas exerts and this can lead to leakage.

To correct for that effect, the authors used a 3-D printed membrane (figure 11) to hold the cubic shale sample, and the membrane and sample were installed in a rubber sleeve for the permeability measurement in a triaxial cell. The chosen technique was the pulse decay method again, with helium as the gas to fill the pores of the cubic shale sample.

The whole system (figure 12) was placed initially under a vacuum and then the upstream reservoir was filled with helium gas, as it began to flow in the direction of the downstream reservoir. The authors kept the volumes of the upstream and downstream reservoirs considerably low, because, as discussed previously, for high values of volumes of these reservoirs, implementing change in pressure in the core is more difficult and the time taken for the test is significantly bigger. To avoid the problem of the mean pressure fluctuating considerably in the core, the authors used constant and equal volumes for the upstream and downstream reservoirs, with an assumption of ideal gas for helium and low compressibility.

Since the applied method for permeability determination was the pulse decay method, the authors performed a test of the sealing of the rubber sleeve by monitoring the pressure values of the upstream and downstream reservoirs with time (15 days). Figure 13 shows the results. We can clearly see that there is no variation of pressure for both reservoirs and, therefore, the sealing is very effective. The shale sample was taken from the Longmaxi Formation in Sichuan basin, China.

One important factor that needs to be mentioned is that the equation used in this article to calculate the permeability is considerably simpler then the expression of apparent permeability of the previous article, since it involves fewer parameters that are easy to be obtained.. If the volumes of the upstream and downstream reservoirs, the cross-sectional area of the core, the length of the core, the viscosity of the gas used and the pressure of the upstream and downstream reservoirs are known, the permeability of the sample can be easily calculated.

When measuring the permeability, the authors charged helium in the upstream reservoir to a pressure of about 1.6 MPa. It was observed that the values of pressure in the upstream and downstream reservoirs already started to equalize one day after the beginning of the experiment for the horizontal directions. However, for the vertical directions it took more than 13 days for the difference in pressure of the upstream and downstream reservoirs to be less than 0.3 MPa. This means that the effect of anisotropy on the permeability is considerable.

Figures 14 to 16 show the graphs of the difference in pressure between the upstream and downstream reservoirs versus time. By fitting the experimental points into an exponential function, the authors could calculate the permeability. What they found out is that the permeability in both directions X and Y (horizontal directions) value is significantly higher (200 nD) than in the vertical direction (8 nD).

The second part of the article consisted in using reservoir simulators to estimate the gas production with the values that were found of permeability. The authors considered three cases for the simulations: in the first case, they used permeability values of 200 nD for all three directions. In the second case, they used permeability values of 200 nD for the horizontal directions and for the vertical directions, 8 nD. Finally, in the third case, they used the permeability values in the lowest, of only 8 nD in all directions**. **The results are shown in figure 17. We can clearly see that the total gas production rate is higher when considering the highest value of permeability, 200 nD. For the cases in which the anisotropic effect was considered to calculate permeability, the total gas production rate is approximately 33% lower than in the case of the highest permeability, which shows that if this effect is not considered, the gas production will probably be overestimated. The article makes clear that the calculations made with the reservoir simulators are only illustrative. They will depend on several parameters, like hydraulic stimulation results and operating conditions. If these vary, the impact of the anisotropic effect of permeability on gas production will alter, but the authors conclude that if only a value of permeability is used, then the gas production rate will be underestimated (using vertical permeability only) or overestimated (using horizontal permeability only).

- Comparison Between The Proposed Methods – Discussion and Results

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In this session, the project will focus on comparing the proposed methods in terms of the permeability measurements, techniques, experiments on shale and results. It will focus on simple differences between the papers in terms of improvements in the apparatus to measure permeability, total test time, nature of the gas used in the experiments, parametric and sensitivity analysis, and mathematical models to calculate permeability.

We will define:

- Article 1 – the first article summarized in this project (section 4.1), that focuses on the adsorption/desorption effect on the permeability measurements;
- Article 2 – the second article commented in this project (section 4.2), that focuses on the surface diffusion effect on the permeability;
- Article 3 – the third article mentioned in this project (section 4.3), which focuses on the anisotropic effect on the permeability.

The first subject to be discussed is the improved apparatus that article 1 constructs. Article 1 was the only one that performed their experiments in an improved apparatus, with some modifications in the traditional pulse decay method technique. Article 2 doesn’t even perform an experiment of their own, it is only based on the work and experiments of another author that didn’t use shale as the sample for the experiment (γ-alumina) and article 3 performs a traditional pulse decay method technique in a cubic shale sample, with no improvements on the apparatus used. Moreover, article 3 has the disadvantage of repositioning the cubic sample to measure the permeability and the three directions (two horizontal and one vertical) and, therefore, has to load and unload the core several times, which can lead to human error and compromise permeability results. Since article 2 based their work on another author’s experiments, it doesn’t even use an apparatus constructed by the own authors, and the sample involved is not shale. Moreover, it is difficult to affirm that the γ-alumina test results that article 2 bases its work on imitate the same transport properties and fluid flow characteristics of actual shale reservoir conditions. Therefore, this casts some doubt on the results of apparent permeability that article 2 finds. In the pulse decay method applied by article 1, the volumes of the upstream and downstream reservoirs can be changed, which implies in better control of the pressure (because the volumetric ratios of these reservoirs can be lowered the way the authors need in order to be able to change the pressure to reduce the time taken for the tests) and less error in the calculation of the permeability, like discussed previously. The authors in article 1 found that for 50% error in pore volume determination, if you have lower volumetric ratios of the upstream and downstream reservoirs, the error in the permeability is half of the error considering higher volumetric ratios. This allows more room for error in the determination of the volume of the pores of the sample (and therefore, porosity). Article 3 performs the pulse decay method, but using constant volumes of upstream and downstream reservoirs (the gas cylinders representing the upstream and downstream reservoirs for the anisotropic effect article have about 7 mL, while in the article 1, these volumes can be changed from 10 mL to 5 mL). Therefore, article 3 is unable to check the impact of the volumetric ratios of the upstream and downstream reservoirs in the final value of permeability and article 2 doesn’t even perform an experiment on shale. In face of the subject, we can conclude that article 1 does a better job in evaluating the impact that changes in volumes of the upstream and downstream reservoirs can cause in the final determination of permeability for shale.

Still regarding the improved apparatus that article 1 creates, the authors use a bypass line, which reduces the total test time compared to the conventional apparatus and this is the only article that measures instantaneously the permeability of shale (without having to reposition the sample in different directions) in the forward and reverse directions. Since article 2 doesn’t consider an experiment on shale, the authors in this article don’t consider that the permeability may have different values depending on the direction. In article 3, the authors have to load and unload the core several times to change to reposition the sample in different directions, therefore, taking more time for the tests than the time elapsed for article 1 (the biggest value of total test time for pure shale sample in article 1 was 28 hours, while in article 3 the biggest value of time elapsed is 13 daysfor the vertical direction permeability, which is a huge difference in testing time). Hence, we can see that article 1 is more time efficient than article 3, because it considers permeability in different directions (forward and reverse) and takes a significant smaller amount of test time than article 3.

Regarding the type of gas that the articles use for the experiments, article 1 is the only one that uses the gas which is predominant in shale gas reservoirs, methane. Article 2 bases its work on an experiment that used nitrogen and carbon dioxide, while article 3 used helium for the pulse decay method. As discussed previously, we already know that methane has considerable different transport properties than nitrogen, carbon dioxide and helium (table 1) and the permeability that actually exists in the reservoir is affected by the type of gas that predominates. Moreover, we know that the effective diameter of the molecules of methane is bigger than nitrogen and helium, and that the adsorption energy and affinity of the molecules of methane are bigger than nitrogen and helium. Since we have really small pores in shale (nanoscale), the specific area of the pores (area divided by the volume, which is inversely proportional to the pore radius) is significant and, therefore, the number of active sites available for adsorption is considerably high, higher than conventional reservoirs. Therefore, if methane is used, the experiment is being more representative to the transport properties and flow characteristics of the actual reservoir. Hence, because the transfer properties are different (viscosity, density, molar mass, and interactions of the molecules of the gas with the walls of the pores, adsorption and desorption mechanisms, slip flow condition and other parameters), probably the nature of the gas will affect the permeability results.

Regarding the experimental data of the papers, article 1 excluded some data points of differential pressure, in both forward and reverse directions, in terms of time. The authors only considered the data points to perform the linear regression to obtain the slope to calculate the permeability from the third experimental point, in order to reach a value of the square of the correlation coefficient (r^{2}) closer to one. The methods applied in article 3 didn’t do that, considering all the experimental points.

Article 1 is the only one that considers different samples for their tests, including pure shale, sand shale and sand interlayer, in order to evaluate and perform a comparison of the impact of the adsorption effect in unconventional reservoirs (in this case, shale) to its effect in conventional reservoirs (in this case, the sand interlayer or sandstone). Article 2 doesn’t even perform an experiment in shale and article 3 only considers their cubic shale sample, without comparing the results of anisotropic effect on other types of rock formations.

Article 2 derives the expression of apparent permeability considering the gas flux due to three terms: gas flux due to pressure forces, due to Knudsen diffusion and surface diffusion. The problem with this approach is that it makes an assumption that these three mechanisms that contribute to the total gas flux in the reservoir have the same importance and “weight” to the total flux. However, one mechanism can affect and influence the other, and this influence is not yet quite understood. This certainly casts doubt on the final expression that article 2 finds for the apparent permeability, since it considers the three mechanisms individually with the same “weight”.

The mathematical expression of apparent permeability that article number 2 arrives at has an advantage over the expressions for apparent permeability used in articles 1 and 3. It considers the effect of pressure forces, Knudsen diffusion and surface diffusion in the total gas flux. The Knudsen diffusion is a very important mechanism of gas transport in shale reservoirs, due to the fact that the pore sizes of shale are roughly in the same order of magnitude as the molecule sizes of the gas that flows through the pores. Article 1 adds a correction factor to account for the effect of adsorption on permeability and article 3 considers the conventional mathematical expression for the traditional pulse decay method, without accounting for the diffusion mechanisms and adsorption in the shale sample. Therefore, articles 1 and 3 maintain a constant value of mean pressure in the upstream and downstream reservoirs and, to study the effect of Knudsen diffusion on the permeability, this value of mean pressure must be changed and a sensitivity analysis must be performed.

However, the surface diffusion coefficient (

DSo

) that article 2 uses is considerably higher in terms of order of magnitude than average values for shale. According to a research done by Yang et al (2016), the order of magnitude of surface diffusion coefficient for shale in mesopores and micropores of this rock formation is 10^{-20} m^{2}/s. The value that article number two considers for the surface diffusion coefficient is in the order of magnitude of 10^{-9} m^{2}/s, significantly bigger than the value of what other researchers find. Moreover, the value of the coefficient for surface diffusion that article number 2 considers is a fitted value. Hence, this casts doubt on the actual effect of surface diffusion on the permeability measurement, since using a much higher value for surface diffusion coefficient assigns more impact and “weight” to the gas flux due to surface diffusion on the expression for the total gas flux. The authors of article 2 state that the surface diffusion effect is significant at low pressures, when the reservoir is more depleted. For higher values of pressure, this ratio is significantly close to one, which suggests that the surface diffusion’s effect is much more significant at low pressures and, at earlier stages of the reservoir, doesn’t play a role as important as the effect of adsorption.

Article number 3 discusses an important effect on the permeability, the anisotropy. Because of the presence of sediment and clay deposited vertically on shale reservoirs, that are very deep, the vertical permeability of these rock formations is considerably lower than the horizontal permeability. With article number three, we can know the numeric difference, which is significant. The authors in this article are bold in using cubic samples for shale to eliminate the problem of heterogeneity and facing the problem of the sealing of the sample. As already discussed previously, the sealing of the sample is flawless and the experiment was conducted in a very professional way. As already discussed previously, the sealing of the sample is flawless and the experiment was conducted in a very professional way. One detail that is very important to mention about the shale sample used in article number three is the total organic carbon or TOC. The total organic content is more than 4% for article 3 and, as we have bigger values of TOC, we automatically have bigger values of adsorbed gas, neglecting the effect of adsorption of gas molecules on the surface of the pores of the shale sample in the calculation method for the permeability seems radical. If this experiment was conducted with nitrogen, then probably be effect of adsorption could be neglected, since the adsorption affinity of the nitrogen gas is considerably low, but the same trend is not observed for helium, the gas used in the experiments in article 3.

In article number 3, the mathematical model used to calculate permeability is from the pulse decay transient method. However, the authors use the values of initial pressure decay data from the experiment, because they affirm that there is no need for equalizing the upstream and downstream pressures. It could be that article 1 has more trustworthy results of permeability, because it doesn’t use only the early pressure decay data, it uses all of the data.

In the last part of article number 3, in the reservoir simulations using computer programs, the authors make clear that the calculations performed are only illustrative, because the simulation results will depend on operating conditions and height of the stimulated reservoir volume. Hence, we cannot affirm with certainty that the impact of the anisotropic effect of the permeability in gas production rate is necessarily the observed in figure 17 for shale reservoirs.

This part of the comparison will focus on comparing the three methods of permeability measurement in terms of the relative error between the permeability result when not considering the specific effect that each researched paper focuses on and the permeability result considering this effect. For article 1, we can extract some data of relative error of permeability to evaluate the impact of the adsorption effect on the calculations for shale samples (table 2 – samples 1, 2 and 5, comparing the forward permeability tested in the PoroPDP conventional apparatus and the improved apparatus considering adsorption). For article 2, we can evaluate and extract data of the permeability term (permeability (

k

) multiplied by the gas density (

ρ

) divided by the product between gas viscosity and sample length (

μ.L

)) for the permeability tests with CO_{2} (figure 5) and calculate a relative error between the result not considering the surface diffusion effect and the one considering it (three experimental points compared with three modeled points). For article 3, we can extract relative error data regarding the estimated gas production rate (figure 17) for the cubic shale sample in reservoir simulators in three different time values (graph of total rate vs time). In this graph, we can read the values of total rate, in m^{3}/day, for time values of 2, 5 and 20 years, considering the permeability of 200 nD (horizontal direction) and comparing the points with the anisotropic permeability points. Table 3 shows these results.

We can clearly see that the biggest values of error happen when we focus on the adsorption effect (up to 97%) and the lowest values of error happen when we focus on the surface diffusion effect (up to 27%). When we focus on the anisotropic effect on permeability, the relative errors on the gas production rate considering and not considering this effect is still lower (up to 57%) than the relative errors found for article 1. Considering that the permeability is the main factor for calculating total gas rate in article 3, maintaining other operating conditions and parameters constant, the values of error found for the three papers suggest that the impact of the adsorption effect on the permeability for shale reservoirs is higher than the surface diffusion and anisotropic effects, because when not considering the adsorption effect, the relative error on the permeability is higher than when not considering the surface diffusion or anisotropic mechanisms. Certainly, we cannot affirm that the adsorption effect has a bigger impact than the surface diffusion and anisotropic effect on permeability for all cases of shale reservoirs, no matter the operating conditions or locations or composition of the gas that lies in the formation. However, for the cases studied in this project, comparing the error values of table 3, we can certainly state that the adsorption effect is the most important one of all three.

Table 4 presents a summary of the advantages and disadvantages for the three proposed methods of permeability determination for shale reservoirs studied in this project, along with a short description of each model.

To sum up, the best method to determine permeability in shale gas reservoirs, of all three studied in this project, is the one described by article 1 (adsorption effect), because, as stated in this section, blablabla

- Conclusions

** **

- References

Cao, C., Li, T., Shi, J., Zhang, L., Fu, S. Wang B., Wang, H. ** A new approach for measuring the permeability of shale featuring adsorption and ultra-low permeability**. Elsevier, J. Nat. Gas. Sci. Eng. 30 548-556 (2016)

Letham, E. A. ** Matrix permeability measurements of gas shales: gas slippage and adsorption as sources of systematic error**. Geological Science, University of British Columbia, Vancouver, March, 2011

O’Reilly, D. J., Foshee, D. L., Nichols, M. W. ** Prudent Development: Realizing the Potential of North America’s Abundant Natural Gas and Oil Resources**. National Petroleum Council. <http://www.npc.org/NARD-ExecSummVol.pdf> (2011). Accessed 26 July 2017

Pan, Z., Ma, Y., Connell, Luke D., Down, David I., Camilleri, M. ** Measuring anisotropic permeability using a cubic shale sample in a triaxial cell**. Elsevier, J. Nat. Gas. Sci. Eng. 26 336-344 (2015)

Pathi, V. S. M. ** Factors affecting the permeability of gas shales**. Geological Science, University of British Columbia, Vancouver, October, 2008

Yang, B., Kang, Y., You, L., Li, X., Chen, Q. ** Measurement of the surface diffusion coefficient for adsorbed gas in the fine mesopores and micropores of shale organic matter**. Elsevier, Fuel J. 181 793-804 (2016)

Zhang, L., Li, D., Lu, D., Zhang, T. ** A new formulation of apparent permeability for gas transport in shale**. Elsevier, J. Nat. Gas. Sci. Eng. 23 221-226 (2015)

Zhang, J., Scherer, G. W. ** Permeability of shale by the beam-bending method**. Elsevier, Int. J. Rock Mech. Min. Sci. 53 179-191 (2012)

- Appendix

Table 1 – Collision, kinetic and effective diameters of hydrogen, nitrogen and methane (Cao et al, 2016)

Table 2 – Final results of permeability in the PoroPDP-200 and the improved apparatus (Cao et al, 2016)

Method |
Sample |
Parameter considered |
Parameter calculated with conventional approach |
Parameter calculated with effect approach |
Error (%) |

Adsorption effect | Pure shale | Permeability (
k) |
1.79*10-6 mD | 3.10*10-6 mD | 73. 2 % |

Adsorption effect | Pure shale | Permeability (
k) |
6.20*10-5 mD | 1.22*10-5 mD | 97.3 % |

Adsorption effect | Sand shale | Permeability (
k) |
4.90*10-4 mD | 7.65*10-4 mD | 56.1 % |

Surface diffusion effect | γ-alumina | Permeability
k.ρμ.L |
3.15*10-6molm2.s.Pa | 4.20*10-6molm2.s.Pa | 25 % |

Surface diffusion effect | γ-alumina | Permeability
k.ρμ.L |
3.20*10-6molm2.s.Pa | 4.40*10-6molm2.s.Pa | 27.2 % |

Surface diffusion effect | γ-alumina | Permeability
k.ρμ.L |
3.20*10-6molm2.s.Pa | 4.00*10-6molm2.s.Pa | 20 % |

Anisotropic effect | Shale | Gas production rate | 31250 m3/day | 20000 m3/day | 36 % |

Anisotropic effect | Shale | Gas production rate | 20000 m3/day | 8750 m3/day | 56.3 % |

Anisotropic effect | Shale | Gas production rate | 11250 m3/day | 4750 m3/day | 57.8% |

Table 3 – Relative error comparison between the proposed methods for permeability

Figure 1 – Improved apparatus for determination of permeability (Cao et al, 2016)

Figure 2 – Comparison of permeability for (A) adsorptive and (B) low adsorption shale samples (Cao et al, 2016)

Figure 3 – Permeability test deviations – (A) adsorptive and (B) low adsorption sample (Cao et al, 2016)

Figure 4 – Experimental and model data for permeability with nitrogen (Zhang et al, 2015)

Figure 5 – Experimental and model data for permeability with CO_{2 }(Zhang et al, 2015)

Figure 6 – Effect of pressure and temperature on the surface permeability (Zhang et al, 2015)

Figure 7 – Effect of pressure and temperature on the apparent permeability (Zhang et al, 2015)

Figure 8 – Effect of porosity on the apparent permeability (Zhang et al, 2015)

Figure 9 – Effect of pore radius on the apparent permeability (Zhang et al, 2015)

Figure 10 – Effect of surface fractal dimension on the apparent permeability (Zhang et al, 2015)

Figure 11 – Cubic shale sample in rubber sleeve (Pan et al, 2015)

Figure 12 – Apparatus for anisotropic effect on permeability (Pan et al, 2015)

Figure 13 – Pressure profile of reservoirs vs. time (Pan et al, 2015)

Figure 14 – Pressure difference profile for horizontal direction 1 (Pan et al, 2015)

Figure 15 – Pressure difference profile for horizontal direction 2 (Pan et al, 2015)

Figure 16 – Pressure difference profile for vertical direction (Pan et al, 2015)

Figure 17 – Estimation of gas production rate at different values of permeability (Pan et al, 2015)