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

Atoms and associated spectral properties for positive operators on L^p

Inspired by Schwartz, Jang-Lewis and Victory, who study in particular generalizations of triangularizations of matrices to operators, we shall give for positive operators on Lebesgue spaces equivalent definitions of atoms (maximal irreducible sets). We also characterize positive power compact operators having a unique non-zero atom which appears as a natural generalization of irreducible operators and are also considered in epidemiological models. Using the different characterizations of atoms, we also provide a short proof for the representation of the ascent of a positive power compact operator as the maximal length in the graph of critical atoms

HOW DO GREEK-CYPRIOT PRIMARY SCHOOL TEACHERS PERCEIVE PROFESSIONAL DEVELOPMENT?

The Cypriot Ministry of Education acknowledges the importance of professional development and offers a number of activities in order to continue qualifying teachers. However, the activities offered are limited and are mainly traditional. There is a need to understand the degree to which reforming professional development activities can be successfully introduced into the Cypriot Educational context. Studying teachers‟ perceptions about professional development helps to understand which their needs are, how these needs can be met, and which the potentials to introduce more effective activities are. The study explored teachers‟ perceptions about professional development, brought by a number of Cypriot teachers and headteachers who work in an urban area of Nicosia. Data was gathered through a survey by collecting questionnaires from 123 participants, and through interviews with 6 teachers and 3 headteachers. Combining the quantitative and qualitative approach seemed appropriate for this small-scale research since the quantitative data provided a general image about teachers‟ participation in current professional development, while the qualitative data provided a deeper understanding of participants‟ experiences. The findings showed that teachers tend to participate in in-service training. Though, these activities do not satisfy teachers‟ needs. So, they engage in colleague collaboration and personal study which are more influential for them. Headteachers do no play any role in promoting teachers‟ professional development. The findings show that there are potentials for introducing new activities for professional development but there are some factors which need to be taken into account first

On the apparent permeability of porous media in rarefied gas flows

The apparent gas permeability of the porous medium is an important parameter in the prediction of unconventional gas production, which was first investigated systematically by Klinkenberg in 1941 and found to increase with the reciprocal mean gas pressure (or equivalently, the Knudsen number). Although the underlying rarefaction effects are well-known, the reason that the correction factor in Klinkenberg's famous equation decreases when the Knudsen number increases has not been fully understood. Most of the studies idealize the porous medium as a bundle of straight cylindrical tubes, however, according to the gas kinetic theory, this only results in an increase of the correction factor with the Knudsen number, which clearly contradicts Klinkenberg's experimental observations. Here, by solving the Bhatnagar-Gross-Krook equation in simplified (but not simple) porous media, we identify, for the first time, two key factors that can explain Klinkenberg's experimental results: the tortuous flow path and the non-unitary tangential momentum accommodation coefficient for the gas-surface interaction. Moreover, we find that Klinkenberg's results can only be observed when the ratio between the apparent and intrinsic permeabilities is $\lesssim30$; at large ratios (or Knudsen numbers) the correction factor increases with the Knudsen number. Our numerical results could also serve as benchmarking cases to assess the accuracy of macroscopic models and/or numerical schemes for the modeling/simulation of rarefied gas flows in complex geometries over a wide range of gas rarefaction. Specifically, we point out that the Navier-Stokes equations with the first-order velocity-slip boundary condition are often misused to predict the apparent gas permeability of the porous media; that is, any nonlinear dependence of the apparent gas permeability with the Knudsen number, predicted from the Navier-Stokes equations, is not reliable. Worse still, for some type of gas-surface interactions, even the ``filtered'' linear dependence of the apparent gas permeability with the Knudsen number is of no practical use since, compared to the numerical solution of the Bhatnagar-Gross-Krook equation, it is only accurate when the ratio between the apparent and intrinsic permeabilities is $\lesssim1.5$

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

Electrosprayed mesoporous particles for improved aqueous solubility of a poorly water soluble anticancer agent: in vitro and ex vivo evaluation

open access articleEncapsulation of poorly water-soluble drugs into mesoporous materials (e.g. silica) has evolved as a favorable strategy to improve drug solubility and bioavailability. Several techniques (e.g. spray drying, solvent evaporation, microwave irradiation) have been utilized for the encapsulation of active pharmaceutical ingredients (APIs) into inorganic porous matrices. In the present work, a novel chalcone (KAZ3) with anticancer properties was successfully synthesized by Claisen-Schmidt condensation. KAZ3 was loaded into mesoporous (SBA-15 and MCM-41) and non-porous (fumed silica, FS) materials via two techniques; electrohydrodynamic atomization (EHDA) and solvent impregnation. The effect of both loading methods on the physicochemical properties of the particles (e.g. size, charge, entrapment efficiency, crystallinity, dissolution and permeability) was investigated. Results indicated that EHDA technique can load the active in a complete amorphous form within the pores of the silica particles. In contrast, reduced crystallinity (~79%) was obtained for the solvent impregnated formulations. EHDA engineered formulations significantly improved drug dissolution up to 30-fold, compared to the crystalline drug. Ex vivo studies showed EHDA formulations to exhibit higher permeability across rat intestine than their solvent impregnated counterparts. Cytocompatibility studies on Caco-2 cells demonstrated moderate toxicity at high concentrations of the anticancer agent. The findings of the present study clearly show the immense potential of EHDA as a loading technique for mesoporous materials to produce poorly water-soluble API carriers of high payload at ambient conditions. Furthermore, the scale up potential in EHDA technologies indicate a viable route to enhance drug encapsulation and dissolution rate of loaded porous inorganic materials