Computational study of gas transport in shale at pore-scale and beyond

Unconventional gas resources like shale are poised to enter a golden age thanks to the worldwide increased exploitation potential. Yet, the future of these resources is far from assured since it is still subject to many uncertainties, mainly with regard to gas recoverability. The macroscopic flow properties that allow the prediction of production are directly linked to the microscopic flow where underlying rarefaction effects play a major role. This is due to the pore size which is as small as a few nanometers and comparable to the mean free path. Thus, an in-depth understanding of gas transport in ultra-tight porous media is crucial for the accurate determination of flow properties in shale rocks. This thesis is a fundamental research aiming to aid and benefit shale gas exploration and development. The objective of this work is to provide useful insights for such non-equilibrium porous media flows, where the conventional fluid mechanics theory fails. Even though there are multiple heuristic permeability models in the literature, I find them unsuitable to provide reliable apparent permeability estimates since they often include simplifications of the flow mechanisms and matrix complexity. Notably, I hereby establish/prove the limitations of the accuracy of the Navier-Stokes equations to the first order of Knudsen number. We also thoroughly analyse Klinkenberg's slip factor behaviour for a wide range of gas rarefaction, utilising gas kinetic theory, for both simple and complex porous media. Moreover, using controllably random porous media, I systematically quantify the impact of numerous structural characteristics, i.e. porosity, tortuosity, specific surface area, heterogeneity and degree of anisotropy, on both intrinsic and apparent permeability.;One of the key contributions of this work is a new semi-analytical permeability formulation derived using the produced simulation results. This expression, suitable for both isotropic and anisotropic two-dimensional porous media, accounts for the aforementioned properties as well as for continuum and slip flow. The main advantage of the proposed formulation is the fact that it does not entail any experimental or numerical data as input, unlike other established models. Shale is intrinsically multiscale, thus the direct simulation of transport in all scales is not feasible. Upscaling from the pore-scale is indispensable in order to eventually obtain the essential macroscopic properties in the field-scale. For this reason, I examine well-known analytical and numerical upscaling techniques, verifying the sensitivity and accuracy of the latter ones. Studying microscale sample images, we need to consider the appearance of microfractures. The difference of the characteristic length scales between the nanopores and the microfractures requires a hybrid upscaling method such as the Brinkman approach.;The suitability of this model is extensively validated on fractured porous media of interest, especially on the grounds that the exact form of the effective viscosity is still a matter of discussion. We perform this validation comparing numerous direct simulation results with the corresponding ones from the Brinkman solution. Different values of the effective viscosity are investigated, along with a variable permeability model applied at the vicinity of the fluid-porous interface. Due to lack of an appropriate universal treatment of the transition zone of random porous media, we consider effective viscosity equal to fluid viscosity. The accuracy of the Brinkman approach is further examined using several two and three-dimensional random porous media containing fractures, as well as considering rarefied conditions. Although I find that heterogeneity and anisotropy increase the error of the effective permeability derived from the Brinkman approach, generally, the effective permeability extracted from this coarse-scale model compares favourably to its fine-scale counterpart obtained from the Stokes and Boltzmann model equations for porous media flows. Finally, I conclude that neglecting the rarefaction effects leads to a significant underestimation of the effective permeability of fractured ultra-tight porous media.Unconventional gas resources like shale are poised to enter a golden age thanks to the worldwide increased exploitation potential. Yet, the future of these resources is far from assured since it is still subject to many uncertainties, mainly with regard to gas recoverability. The macroscopic flow properties that allow the prediction of production are directly linked to the microscopic flow where underlying rarefaction effects play a major role. This is due to the pore size which is as small as a few nanometers and comparable to the mean free path. Thus, an in-depth understanding of gas transport in ultra-tight porous media is crucial for the accurate determination of flow properties in shale rocks. This thesis is a fundamental research aiming to aid and benefit shale gas exploration and development. The objective of this work is to provide useful insights for such non-equilibrium porous media flows, where the conventional fluid mechanics theory fails. Even though there are multiple heuristic permeability models in the literature, I find them unsuitable to provide reliable apparent permeability estimates since they often include simplifications of the flow mechanisms and matrix complexity. Notably, I hereby establish/prove the limitations of the accuracy of the Navier-Stokes equations to the first order of Knudsen number. We also thoroughly analyse Klinkenberg's slip factor behaviour for a wide range of gas rarefaction, utilising gas kinetic theory, for both simple and complex porous media. Moreover, using controllably random porous media, I systematically quantify the impact of numerous structural characteristics, i.e. porosity, tortuosity, specific surface area, heterogeneity and degree of anisotropy, on both intrinsic and apparent permeability.;One of the key contributions of this work is a new semi-analytical permeability formulation derived using the produced simulation results. This expression, suitable for both isotropic and anisotropic two-dimensional porous media, accounts for the aforementioned properties as well as for continuum and slip flow. The main advantage of the proposed formulation is the fact that it does not entail any experimental or numerical data as input, unlike other established models. Shale is intrinsically multiscale, thus the direct simulation of transport in all scales is not feasible. Upscaling from the pore-scale is indispensable in order to eventually obtain the essential macroscopic properties in the field-scale. For this reason, I examine well-known analytical and numerical upscaling techniques, verifying the sensitivity and accuracy of the latter ones. Studying microscale sample images, we need to consider the appearance of microfractures. The difference of the characteristic length scales between the nanopores and the microfractures requires a hybrid upscaling method such as the Brinkman approach.;The suitability of this model is extensively validated on fractured porous media of interest, especially on the grounds that the exact form of the effective viscosity is still a matter of discussion. We perform this validation comparing numerous direct simulation results with the corresponding ones from the Brinkman solution. Different values of the effective viscosity are investigated, along with a variable permeability model applied at the vicinity of the fluid-porous interface. Due to lack of an appropriate universal treatment of the transition zone of random porous media, we consider effective viscosity equal to fluid viscosity. The accuracy of the Brinkman approach is further examined using several two and three-dimensional random porous media containing fractures, as well as considering rarefied conditions. Although I find that heterogeneity and anisotropy increase the error of the effective permeability derived from the Brinkman approach, generally, the effective permeability extracted from this coarse-scale model compares favourably to its fine-scale counterpart obtained from the Stokes and Boltzmann model equations for porous media flows. Finally, I conclude that neglecting the rarefaction effects leads to a significant underestimation of the effective permeability of fractured ultra-tight porous media

Intrinsic and apparent gas permeability of heterogeneous and anisotropic ultra-tight porous media

Accurate prediction of unconventional gas production requires deep understanding of the permeability of complex rock samples. Several predictive expressions of permeability, which include either simplifications of the porous media structure or the flow mechanisms, have been proposed recently. The main objective of this research is to quantify the impact of solid matrix complexity on both intrinsic and apparent permeability. To this end, numerous two-dimensional random porous media structures are constructed using the quartet structure generation set algorithm. Parametric and statistical analysis reveals the importance of the specific surface area of pores, tortuosity, heterogeneity and degree of anisotropy. Special focus is given to the directional dependency of the permeability on isotropic and anisotropic geometries, considering the great impact of anisotropy on the laboratory evaluation of permeability data and the anisotropic nature of shale rocks. Simulation results, for the same value of porosity, clearly indicate the drastic improvement of permeability due to the reduction of specific surface area of pores and their height to width ratio. This suggests that rock matrix complexity has significant impact on permeability and should not be neglected while forming permeability formulations for porous media. Finally, the results of the apparent permeability, obtained by solving the gas kinetic equation, are taken into consideration to demonstrate the enhancement ratio, slip factor and their correlation with the aforementioned parameters. Semi-analytical expressions for intrinsic and apparent permeability, considering continuum and slip flow respectively, are derived. The proposed formulations, suitable for both isotropic and anisotropic structures, have the advantage of not entailing any numerical or experimental data as input

Shale gas permeability upscaling from the pore-scale

The effective permeability of large shale samples provides useful insights into shale gas production. However, its determination can only be achieved through upscaling, since the direct pore-scale simulation of gas flows in large rock samples is not feasible due to the high computational cost and absence of pore connectivity in sample images. Although the Brinkman formulation is widely used in the permeability upscaling of conventional rocks, how to choose the effective viscosity in this coarse-scale model is not clear. Moreover, its application in shale rocks, where the rarefaction effects are important so that the conventional Navier-Stokes equations are inadequate, is rare, and its accuracy has not been assessed. This study aims to address the above two problems, by comparing the Brinkman solutions of several two-dimensional and three-dimensional random porous media containing fractures with the fine-scale solutions of the Stokes and Boltzmann equations (for continuum and rarefied gas flows, respectively). It is found that the use of the fluid viscosity in the Brinkman model, instead of the controversial effective viscosity, leads to accurate results for the cases considered. Additionally, the macroscopic quantity in rarefied gas flows in shale rocks is well predicted by the Brinkman model for a wide range of gas rarefaction, since the error is found to be less than 7%, while neglecting the rarefaction effects leads to significant underestimation of effective permeability (up to 90% in the cases studied). Although heterogeneity and anisotropy of the porous medium increase the error of the effective permeability derived from the Brinkman model, generally speaking, the effective permeability extracted from this coarse-scale model compares favorably to its fine-scale counterpart