A Novel Method for Gas–Water Relative Permeability Measurement of Coal Using NMR Relaxation

Using the conventional volumetric method in unsteady-state relative permeability measurements for unconventional gas reservoirs, such as coal and gas shale, is a significant challenge because the movable water volume in coal or shale is too small to be detected. Moreover, the dead volume in the measurement system adds extra inaccuracy to the displaced water determination. In this study, a low-field nuclear magnetic resonance (NMR) spectrometer was introduced into a custom-built relative permeability measurement apparatus, and a new method was developed to accurately quantify the displaced water, avoiding the drawback of the dead volume. The changes of water in the coal matrix and cleats were monitored during the unsteady-state displacement experiments. Relative permeability curves for two Chinese anthracite and bituminous coals were obtained, matching the existing research results from the Chinese coalbed methane area. Moreover, the influences of confining pressure on the shape of the relative permeability curve were evaluated. Although uncertainties and limits exist, the NMR-based method is a practical and applicable method to evaluate the gas/water relative permeability of ultra-low permeability rocks.     

A Fractal Model for Gas–Water Relative Permeability in Inorganic Shale with Nanoscale Pores

A reliable gas–water relative permeability model in shale is extremely important for the accurate numerical simulation of gas–water two-phase flow (e.g., fracturing fluid flowback) in gas-shale reservoirs, which has important implication for the economic development of gas-shale reservoir. A gas–water relative permeability model in inorganic shale with nanoscale pores at laboratory condition and reservoir condition was proposed based on the fractal scaling theory and modified non-slip boundary of continuity equation in the nanotube. The model not only considers the gas slippage in the entire Knudsen regime, multilayer sticking (near-wall high-viscosity water) and the quantified thickness of water film, but also combines the real gas effect and stress dependence effect. The presented model has been validated by various experiments data of sandstone with microscale pores and bulk shale with nanoscale pores. The results show that: (1) The Knudsen diffusion and slippage effects enhance the gas relative permeability dramatically; however, it is not obviously affected at high pressure. (2) The multilayer sticking effect and water film should not be neglected: the multilayer sticking would reduce the water relative permeability as well as slightly decrease gas relative permeability, and the film flow has a negative impact on both of the gas and water relative permeability. (3) The increased fractal dimension for pore size distribution or tortuosity would increase gas relative permeability but decrease the water relative permeability for a given saturation; however, the effect on relative permeability is not that notable. (4) The real gas effect is beneficial for the gas relative permeability, and the influence is considerable when the pressure is high enough and when the nanopores of bulk shale are mostly with smaller size. For the stress dependence, not like the intrinsic permeability, none of the gas or water relative permeability is sensitive to the net pressure and it can be ignored completely.     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

A New Stable Bubble Element for Incompressible Fluid Flow Based on a Mixed Petrov–Galerkin Finite Element Formulation

A new mixed Petrov–Galerkin formulation employing the MINI element with a non-confirming bubble function for an incompressible media governed by the Stokes equations, which is equivalent to the stabilized finite element by P ¹-P ¹ approximation, is proposed. The new formulation possesses better stability properties than the conventional Bubnov–Galerkin formulation employing the MINI element. In this aspect, the stabilizing effect of this formulation is evaluated by a stabilizing parameter determined by both shapes of the trial and the weighting bubble functions.     

A Permeability–Throat Diameter Correlation for a Medium Generated with Delaunay Tessellation and Voronoi Algorithm

Transport in Porous Media - Describing different transport phenomena through porous media at micro- and mesoscale represents a convenient pre-step in experimental characterization for materials. A...     

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

A Local Thermal Non-Equilibrium Solution Based on the Brinkman–Forchheimer-Extended Darcy Model for Thermally and Hydrodynamically Fully Developed Flow in a Channel Filled with a Porous Medium

Transport in Porous Media - A general analytical solution procedure for the Brinkman–Forchheimer-extended Darcy model has been proposed to obtain local thermal non-equilibrium solutions for...     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

A Highly Conductive Porous Medium for Solid–Gas Reactions: Effect of the Dispersed Phase on the Thermal Tortuosity

The heat transfer in a highly conductive material constituted by a graphite matrix in which a granular phase is dispersed is studied. The effective thermal conductivity of this anisotropic porous composite medium used in solid–gas reactors can vary largely with the component fractions. The effect of the dispersed grains on the deformable structure of the matrix is considered. A model developed on the basis of thermal tortuosity by analogy with mass transfer is adequately correlated with experimental results.     

Porosity–Permeability Relations for Evolving Pore Space: A Review with a Focus on (Bio-)geochemically Altered Porous Media

Reactive transport processes in a porous medium will often both cause changes to the pore structure, via precipitation and dissolution of biomass or minerals, and be affected by these changes, via changes to the material’s porosity and permeability. An understanding of the pore structure morphology and the changes to flow parameters during these processes is critical when modeling reactive transport. Commonly applied porosity–permeability relations in simulation models on the REV scale use a power-law relation, often with slight modifications, to describe such features; they are often used for modeling the effects of mineral precipitation and/or dissolution on permeability. To predict the reduction in permeability due to biomass growth, many different and often rather complex relations have been developed and published by a variety of authors. Some authors use exponential or simplified Kozeny–Carman relations. However, many of these relations do not lead to fundamentally different predictions of permeability alteration when compared to a simple power-law relation with a suitable exponent. Exceptions to this general trend are only few of the porosity–permeability relations developed for biomass clogging; these consider a residual permeability even when the pore space is completely filled with biomass. Other exceptions are relations that consider a critical porosity at which the porous medium becomes impermeable; this is often used when modeling the effect of mineral precipitation. This review first defines the scale on which porosity–permeability relations are typically used and aims at explaining why these relations are not unique. It shows the variety of existing approaches and concludes with their essential features.     

A novel method for designing S-box based on chaotic map and Teaching–Learning-Based Optimization

A new method for obtaining strong S-boxes based on chaotic map and Teaching–Learning-Based Optimization (TLBO) is presented in this paper. Our method presents eight rounds; each round contains two transformations: row left shifting and columnwise rotation. The vectors for the transformations are different from one round to another, and they are controlled by two keys to the logistic map. These two keys are optimized by using TLBO which aims to construct a strong S-box that satisfies to the criteria set in advance. Test for the following criteria such as bijectivity, nonlinearity, strict avalanche criteria, equiprobable inputs/outputs XOR distribution is analyzed. Additionally, we will provide many comparisons with other S-boxes and test of the sensitivity to keys. The results of performance test show that the proposed design S-boxes presents good cryptography proprieties and can resist to several attacks.     

A droplet transport model for channel and pipe flow based on particle kinetic theory and a stress–ω turbulence model

A model for droplet transport in turbulent gas–liquid flow in horizontal channels or pipes is developed. We invoke the kinetic theory of Reeks and co-workers for inertial particles, and the Reynolds stress–ω model of Wilcox for the gas turbulence. We introduce a suitable turbulence boundary condition at the gas–liquid interface. The full kinetic model is compared to experimental data using high density SF₆ gas and Exxsol oil, and the model gives a good prediction of the droplet distribution.     

A study on the optimization of the angle of curvature for a Ranque–Hilsch vortex tube,using both experimental and full Reynolds stress turbulence numerical modelling

The working tube is a main part of vortex tube which the compressed fluid is injected into this part tangentially. An appropriate design of working tube geometry leads to better efficiency and performance of vortex tube. In the experimental investigation, the parameters are focused on the working tube angle, inlet pressure and number of nozzles. The effect of the working tube angle is investigated in the range of θ = 0–120°. The experimental tests show that we have an optimum model between θ = 0 and θ = 20°. The most objective of this investigation is the demonstration of the successful use of CFD in order to develop a design tool that can be utilized with confidence over a range of operating conditions and geometries, thereby providing a powerful tool that can be used to optimize vortex tube design as well as assess its utility in the field of new applications and industries. A computational fluid dynamics model was employed to predict the performances of the air flow inside the vortex tube. The numerical investigation was done by full 3D steady state CFD-simulation using FLUENT6.3.26. This model utilizes the Reynolds stress model to solve the flow equations. Experiments were also conducted to validate results obtained for the numerical simulation. First purpose of numerical study in this case was validation with experimental data to confirm these results and the second was the optimization of experimental model to achieve the highest efficiency.     

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.     

A Phase-Partitioning Model for CO<Subscript>2</Subscript>–Brine Mixtures at Elevated Temperatures and Pressures: Application to CO<Subscript>2</Subscript>-Enhanced Geothermal Systems

Correlations are presented to compute the mutual solubilities of CO₂ and chloride brines at temperatures 12–300°C, pressures 1–600 bar (0.1–60 MPa), and salinities 0–6 m NaCl. The formulation is computationally efficient and primarily intended for numerical simulations of CO₂-water flow in carbon sequestration and geothermal studies. The phase-partitioning model relies on experimental data from literature for phase partitioning between CO₂ and NaCl brines, and extends the previously published correlations to higher temperatures. The model relies on activity coefficients for the H₂O-rich (aqueous) phase and fugacity coefficients for the CO₂-rich phase. Activity coefficients are treated using a Margules expression for CO₂ in pure water, and a Pitzer expression for salting-out effects. Fugacity coefficients are computed using a modified Redlich–Kwong equation of state and mixing rules that incorporate asymmetric binary interaction parameters. Parameters for the calculation of activity and fugacity coefficients were fitted to published solubility data over the P–T range of interest. In doing so, mutual solubilities and gas-phase volumetric data are typically reproduced within the scatter of the available data. An example of multiphase flow simulation implementing the mutual solubility model is presented for the case of a hypothetical, enhanced geothermal system where CO₂ is used as the heat extraction fluid. In this simulation, dry supercritical CO₂ at 20°C is injected into a 200°C hot-water reservoir. Results show that the injected CO₂ displaces the formation water relatively quickly, but that the produced CO₂ contains significant water for long periods of time. The amount of water in the CO₂ could have implications for reactivity with reservoir rocks and engineered materials.