Previous Lesson Complete and Continue  

  An Analytical Model for Shale Gas Permeability

Lesson content locked

Enroll in Course to Unlock
If you're already enrolled, you'll need to login.


- [Ali] Good afternoon everybody, and thank you for attending this presentation today. My name is Ali Takbiri-Borujeni, and today I would like to share some of my research on the analytical model for shale gas permeability. Today I will mainly focus on the core plug scale. This scale consists of inorganic matrix with randomly distributed organic packets within known as kerogen. Some specific problems caused by complexities in this scale motivated me to do this research. So what are these complexities? It's a known fact that shale samples are characterized by ultratight pores and may introduce significant rarefaction effects. By rarefaction effects I mean the transport of fluids in systems so small that the main pure path of the molecules are comparable to the characteristic length of the system. Laboratory measured apparent permeability of this cores, if conducted in low pressure and temperatures need to be extrapolated to the reservoir conditions. And current interpretation techniques may result in incorrect permeability values at reservoir conditions. So before talking about the model that I developed, I would like to give you a brief background on the flow regime characterization. So gas flow regimes are characterized by Knudsen number. Knudsen number is the ratio of gas means free path to the characteristic length of the system. As you can see this figure we can see a schematic depiction of the systems with different Knudsen numbers. And their corresponding modeling techniques. For shale samples and for organic nano pores in shale, Knudsen numbers is known to be larger than point one. Since this value is so high and some of the modeling techniques that we had in the academy like Navier-Stokes equations as you can see them here, like Navier-Stokes equation are shown to give invalid results for this kind of slip flow regimes. But, a Boltzmann equation or Boltzmann Kinetic Equation seems to be valid for all the flow regimes all the Knudsen number ranges and that's what we're gonna use for this study. So in this study I used regularized 13-moment method which is already used for the Boltzmann equation to study flow of gas in macro and nano channels. This figure shows the schematic depiction of the system on their study. As you can see this is basically a straight channel with a know width and pressure girdings and temperature of the walls. By solving the 13 differentials equations that we came up with using R13 method, we are able to solve the equations for velocity, temperature and densities in the channel for different channel sizes. This figure shows the velocity profiles for three different Knudsen numbers. As you can see the velocity in the vicinity of the channel wall, which is know as slip velocity, increases as Knudsen number increase. You can assume that these results are for three different channel widths. Since Knudsen number is inversely proportional to the channel width, so from now on, if I say larger Knudsen number you can imagine that I'm taking about a channel with a smaller width. Therefore, for a smaller channel sizes, we have higher slip velocities. That's one of the consequences of having smaller channels. Now, having the velocity profiles it's possible to measure the permeability of the channel. The original configuration of the permeability term is a bit complicated but is possible to simplify the model into a form which is shown here in the slide is a second order equation in terms of Knudsen numbers. Note that K case of infinity is L is square over 12 which is the permeability of a channel. Or, if you work in oil and gas industry, you're familiar with the fracture permeability which is calculated as a fracture square over 12. This is basically the same thing, so this equation says that for Knudsen numbers that are very small, and when we are close to continuum scale, permeability is basically equal to L is square over 12. But if Knudsen number is bigger, permeabilites that we calculate are higher than the L is square over 12. So this equation can be considered as a general gas permeability equation for channels, and as you can see the permeability approach is through L is square over 12 if Knudsen number reaches zero. We compared our values of A1 and A2 in this equation with the experimental values that are measured in Maurer, et al study in 2003. They measured A1 and A2 for two gases of helium and nitrogen and the values that we got from our model was about 1.14 for A1 and .28 for A2, which are very close to the value that Maurer et al measured in the experiments. So, after getting this model for the permeability, we decided to compare our model against the published model in the literature. Our model seems to agree very well with the experimental results and seem to be superior to most well known models in the oil and gas studies. Namely, the Karniadakis model which is shown with the green dash lines and jaba blue model or NAP model which is shown by dotted blue line. Now that the model is verified, the performance and civility analysis on the results of four different operating parameters such as pressure, temperature, channel size and molecular rate. In this figure we can see that permeability is a function of pressure. This is against the common belief that permeability is constant and is not a functional operating pressure. Therefore, from now on, the values of the permeabilities that we calculate we call apparent permeability to distinguish it with absolute permeability till the first medium. As we decrease pressure, apparent permeability increases. This increase is more significant for smaller channel sizes. And as we can see, if we increase the pressure, or if we decrease the inverse of pressure which is the X-axis, all the values of the apparent permeability reach to absolute permeability or K infinity, and if this ratio in devoy access reaches to one. And that serves as a check to see if our model works correctly. We tried to see the effect of gas molecular weight on the permeability as well and we saw that for the smaller channel width, we saw that the smaller the channel is the higher is the impact of molecular weight on the permeability. This is in contrary to the current model views in oil and gas industry which show a very small impact of molecular weight on the permeability, even for very small channels. We also tried to investigate the impact of temperature on the apparent permeability for different channel sizes, as we can see the effect of temperature on permeability is not negligible for channel sizes smaller than 50 or 60 nano meters. After doing this sensitivity analysis, we tried to include the adsorption effect, or adsorption phenomenon into the model. As you may know, gas sorption capacity is usually defined using volume and pressure isotherms. Isotherms show a non-linear relationship between the volume of adsorbed gas on the surface of the adsorbents and gas pressure at the constant temperature and that's why they are called isotherms. One of the most well known models used in oil and gas industry is Langmuir sorption model. This figure on the right shows a schematic depiction of the Langmuir isotherm. To modify our models, we assume that for an adsorping channel wall, the amount of adsorbed gasses are defined by the Lanmuir isotherm. If we have a isotherm, we know how much of a channel is covered with the gas molecules. And if the channel is covered with the gas molecule, the cross sectional area is smaller by a molecule diameter. Therefore, if we want to calculate the permeability of the whole channel, it's possible to look at it as a series of channels and we can calculate the apparent permeability of the whole model using a harmonic averaging theory. Or, as you can see here, we used the same equation in the same formulation for measuring the apparent permeability of this channel shown in above picture, and as you can see we used harmonic averaging to measure the Q permeability. So, we would like to compare our results for the permeability value that are measured for a morsel of samples. So before that, I would like to give you a brief background on the experimental methods for measuring shale permeability. Conventional steady state models that are usually used for measuring the permeability for conventional samples like sandstones have not been practical because of very low flow rates and extremely long times that are needed to reach a steady state condition for shale. Unsteady state methods based on pressure pulse decay measurement have been extensively used to estimate permeability and diffusion on the shale samples. The unsteady-state methods are faster and can be used to measure permeabilities as low as a few nano-Darcies, however, the interpretations of data obtained from transient techniques introduce a large margin of uncertainty due to the non-uniqueness of the result and reproducibility problems. However, recently a new steady-state technique has been developed in West Virginia University which is capable of measuring shale permeability in order of nano-Darcies. For the remainder of this work, we use the Zamirian et al published results on shale permeability of a Marcellus sample. Zamirian et al tried a method called Klinkenberg correction which is the routing method that we use in oil and gas industry to come up with absolute permeability of the samples. However, the Klinkenberg correction resulted in negative values of the absolute permeability which is very unphysical. Zamirian also tried a new modification to the Klinkenberg model which is known as double-slippage correction model. At the first glance, if you look at this figure on the right of the slide, the results seem to be consistent for all the gases and reaching to the same absolute permeability for all the gases. However, if we zoom into parts of the results, as we can see here, we'll see that the permeability values at higher pressure, which are very important for estimating the absolute permeability, are not considered in the recreation process. So, what we did was to create a bundle of views based on the pore-size distribution that was published in Zamirian's work and we used the same model that we developed in order to calculate the apparent permeability model. I have to note that we used the Langmuir isotherms of a similar study that was done with Dr. Zobaxin at Stanford and we used both result of adsorption for remainder of this work. As you can see we see a good match for the permeability values of nitrogen and CO2, that are measured for a morsel of sample. If we do not consider adsorption in this model, the model should be able to match nitrogen permeability results because nitrogen doesn't show adsorption in this sample. For CO2, the green dash line shows a relatively good match but if we do the correction in our model for the adsorption we see that we will get to an even better match for the permeability in the solid black line. Interesting point of this result is that the absolute permeability of the sample for both gasses are very close, are around 40 nano-Darcies, which confirm that the absolute permeability is not a function of gas depth. They can create an absolute permeability are three times smaller than the one predicted by the double-slippage model. In conclusion, we drive analytical other called, R13 AP model for the apparent gas permeability of nanochannels and the model demonstrates that temperature and gas molecular weight have significant impact on the apparent permeability. The model agrees well with the experimental data of Marcellus shale sample. The extrapolated absolute permeability values for nitrogen and carbon dioxide are the same assuming non adsorbing model and the absolute permeability of carbon dioxide assuming that we have adsorption in the model is determined and is shown to be slightly lower than the case that if we don't have adsorption in the model. So with that, I conclude my presentation, thank you.