Transport of rarefied gas in shales or tight reservoirs: Modeling using macroscopic moment-based equations

The ever-increasing world’s energy demand represents a significant challenge for suppliers.  For the oil and gas sector, responding to this demand has entailed the exploitation of so-called unconventional resources. Extracting hydrocarbons from these reservoirs may require the use of high-end, expensive technology. The Alberta Energy Regulator (AER) in Alberta, Canada, writes that "unconventional oil and natural gas—shale gas in particular—has been called the future of gas supply in North America. (…) It has tremendous economic potential and we know that the interest in these considerable resources will increase".

Natural gas from shales or tight reservoirs is among the unconventional energy sources that are being targeted by the industry. As referred to in this article, according to a report from the U.S. Energy Information Administration published in 2014, natural gas production from shales increased from 4% of the total gas production in 2005 to 40% in 2012.  This report also indicates that shale gas production is expected to rise to 53% in 2040.  Shales commonly consist of mineral (inorganic) regions with micrometer pore size surrounding regions of solid organic porous material with pore size in the nanometer range.  Gas appears either free in the pore space or absorbed in the walls of the organic matrix. Typical conditions in shales are such that the gas molecules mean free path is also in the nanometer range. Therefore, inside the organic porous material, the gas Knudsen number can be in the range of 0.01 to 10, signalling the presence of rarefied effects such as slippage and molecular collisions with the pore walls.

For the purpose of project decision-making and reservoir management, bounding the uncertainty associated with reservoir modelling is crucial, especially in times of relatively low prices, such as the present ones. Besides somewhat fundamental investigations modelling the transport processes at the pore-scale using, for instance, the Lattice -Boltzmann method to solve for the equations of motion or molecular dynamics simulations, employing in many cases a computational domain constructed from detailed images of the actual porous material, from the point of view of reservoir engineering and simulation, the most practical approach is the development of macroscopic models for the apparent permeability of the porous media.

With regard to macro-scale modelling, perhaps the approach that has become the most popular is based on the simple superposition of the Knudsen diffusion model for the rarefied effects and the Poiseuille model for viscous flow in conduits. This renders the so-called “Dusty Gas Model”.  Another approach is to model rarefied effects by adding to the Poiseuille-flow term a term accounting for gas-wall interactions by means of the tangential momentum accommodation coefficient (TMAC).

Even though continuum models for rarefied gas dynamics based on moments such as Grad's or the regularized 13-moment method (R13) have been applied to study gas transport in channels and circular conduits, it has come as a surprised that almost nothing has been done in terms of the application of these results to modelling shale gas transport. We found only the very recent work of Kazemi and Takbiri-Borujeni from 2015 and published in the International Journal of Coal Geology and entitled "An analytical model for shale gas permeability".

In this work, they constructed a model for the apparent permeability of shales by using the expression for the flux of rarefied gas in a channel obtained by Taheri, Torrilhon, and Struchtrup (2009) using the R13 equations. Kazemi and Takbiri-Borujeni compared predictions from their model with published data of apparent permeability versus pressure from core plug experiments performed on a Marcellus shale sample. This comparison shows very good agreement.

Considering that they used the model for flow of rarefied gas in a two-dimensional channel, we certainly believe that there is room for improvement within this framework by adopting analytical results from spatial configurations that better resemble the actual pore-to-pore conduits.