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EN
Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the math ematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the frst-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the frst time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave feld in reservoirs with pore/fracture networks, which is of great signifcance for unconventional oil and gas exploration.
2
Content available remote Numerical studies of internal flow in different types of filters
EN
The quality of ambient air attracts considerable, widespread interest. Over the last decades, air purification has become an integral part of HVAC systems, process engineering, automotive and respiratory protection. Efficient separation of micro- and nano- particles is solidly linked with the development of new, sophisticated filtrating materials, as well as generating and validating mathematical models of such porous structures. The paper regards the numerical modeling of various filters. The presented work aims to validate four virtual filtrating materials – the fiberglass HEPA filter, the paper filter used in the automotive industry, knitted wire mesh and polyurethane foam. The pressure drop obtained for the filters under investigation was examined. The CFD results were validated against the data available in the literature. The agreement of the results of numerical and experimental studies proves the suitability of the proposed methods. At the same time, the simplifications employed in the simulations leave room for further improvement in future works.
EN
In this paper, Buongiorno’s mathematical model is adopted to simulate both natural convection and mixed convection of a nanofluid in square porous cavities. The model takes into account the Brownian diffusion and thermophoresis effects. Both constant and variable temperatures are prescribed at the side walls while the remaining walls are maintained adiabatic. Moreover, all boundaries are assumed to be impermeable to the base fluid and the nanoparticles. The governing equations are transformed to a form of dimensionless equations and then solved numerically using the finite-volume method. Thereafter, effects of the Brownian diffusion parameter, the thermophoresis number, and the buoyancy ratio on the flow strength and the average Nusselt number as well as distributions of isocontours of the stream function, temperature, and nanoparticles fraction are presented and discussed.
EN
Thermal conductivity in the boundary layer of heat exchange surface is the crucial parameter of adsorption process efficiency which occurs in the adsorption bed. In order to improve heat transfer conditions in the adsorption chiller, novel constructions of adsorption beds are currently investigated. The porous structure of the sorbent layer causes low thermal conductivity in the adsorption bed. One of the methods to improve heat transfer conditions is a modification of porous media bed structure with glue which is characterized with higher thermal conductivity. The optimum parameters of sorbents and glues to build the novel coated construction, in terms of improving the chiller Coefficient of Performance (COP) were defined in (Grabowska et al. 2018a). The paper implements fuzzy logic approach for predicting thermal conductivity of modified porous media layers. The developed model allows determination of the sorbent layer thermal conductivity based on various input parameters: arithmetic average of particle distribution d, density ρ and thermal diffusivity k. The data from empirical research was used to build up the model by fuzzy logic methods.
EN
In this study we examine the effect of the magnetic field parameter on the growth rate of the Rayleigh-Taylor instability (RTI) in a couple stress fluids. A simple theory based on fully developed flow approximations is used to derive the dispersion relation for the growth rate of the RTI. The general dispersion relation obtained using perturbation equations with appropriate boundary conditions will be reduced for the special cases of propagation and the condition of instability and stability will be obtained. In solving the problem of the R-T instability the appropriate boundary conditions will be applied. The couple-stress parameter is found to be stabilizing and the influence of the various parameters involved in the problem on the interface stability is thoroughly analyzed. The new results will be obtained by plotting the curves between the dimensionless growth rate and the dimensionless wave number for various physical parameters involved in the problem (viz. the magnetic field, couple-stress, porosity, etc.) in the problem. It is found that the magnetic field and couple-stress have a stabilization effect whereas the buoyancy force (surface tension) has a destabilization effect on the RT instability in the presence of porous media.
EN
Elastic waves in fluid-saturated granular media depend on the grain rheology, which can be complicated by the presence of gas bubbles. We investigated the effect of the bubble dynamics and their role in rheological scheme, on the linear Frenkel-Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum and bubbles surrounded by the fluid. We derived the Nikolaevskiy-type equation describing the velocity of the solid matrix in the moving reference system. The equation is linearized to yield the decay rate λ as a function of the wave number k. We compared the λ (k) -dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For both the cases, the λ(k) curve lies entirely below zero, which implies a global decay of the wave. We found that the increase of the radius of the bubbles leads to a faster decay, while the increase in the number of the bubbles leads to slower decay of the wave.
EN
Complexity of the pore geometry and the random nature of flow velocity make it difficult to predict and represent post laminar flow through porous media. Present study experimentally investigates the applicability of Forchheimer and Wilkins equations for post laminar flow where Darcy’s law is invalid due to predominant inertial effect. It is observed that both porosity and media size have significant influence over the coefficients of the Forchheimer coefficients. To incorporate the effect of porosity and media size, behaviour of Forchheimer coefficients are investigated with hydraulic radius as characteristic length. An inversely proportional variation trend is found for all the present and earlier reported data. A new empirical relation between Forchheimer coefficients and hydraulic radius is obtained which can be universally applicable for all media size and porosity. Coefficients of the Wilkins equation are found to be non-deviating for different hydraulic radius in the present study and in the reported literature validating its applicability in predicting the non laminar flow through porous media. Further the Wilkins equation is modified after incorporating the correction factors for better applicability on the field.
EN
In the paper, functions describing different porosity-tortuosity relations were collected, and then the tortuosity values were calculated for a one granular bed consisting of spherical particles with normal distribution of diameters. Information about the bed porosity and particle sizes was obtained from measurements conducted for an artificial granular bed, consisting of glass marbles. The results of calculations were compared with the results of two other methods of tortuosity determination, performed for the same case (details are not described in this paper): the first of them uses the Path Tracking Method, the second one - information about the velocity components in a creeping flow (the Lattice-Boltzmann Method was applied to obtain the velocity field in the flow). The main aim of our article was to test whether the functions linking tortuosity with porosity, which are available in the literature, give similar results as the methods described above. To achieve this aim, the relative errors between results of calculations for the collected formulas and values from the both previous mentioned methods were calculated.
EN
In the case of a two-phase medium – such as the soil, which consists of an elastic skeleton and is filled with pore fluids – stress and strain within the medium are dependent on both phases. Similarly, in the case of heat transfer, heat is conducted through the two phases at different rates, with an additional heat transfer between the phases. In the classical approach to modelling a porous medium, it is assumed that the fluid filling the pore space is water, which is incompressible. In the case of gas, the volume of which is strongly dependent on temperature and pressure, one should take this behavior into account in the constitutive relations for the medium. This work defines the physical relations of a two-phase medium and provides heat transfer equations, constructed for a porous, elastic skeleton with fluid-filled pores, which may be: liquid, gas, or mixture of liquid and a gas in non-isothermal conditions. The paper will present constitutive relations derived from the laws of irreversible thermodynamics, assuming that pores are filled with either a liquid or a gas. These relations, in the opinion of the authors, may be used as the basis for the construction of a model of the medium filled partly with a liquid and partly with a gas. It includes the possibility of independent heat transfer through any given two-phase medium phase, with the transfer of heat between the phases.
10
Content available remote Fenomenologiczne ujęcie hydrodynamiki przepływu gazu przez struktury porowate
PL
Opisano zagadnienia hydrodynamiki przepływu gazu przez struktury porowate oraz dokonano przeglądu wybranych modeli hydrodynamicznych. Przeanalizowano podane w literaturze warunki wykonanych badań dotyczących opisu hydrodynamiki przepływu gazu przez struktury porowate. Zagadnienie procesowe rozpatrzono w kategorii "przegląd badań w zakresie wybranych sposobów opisu hydrodynamiki przepływu gazu przez złoża porowate". Ma ono istotne znaczenie dla poszerzenia wiedzy na temat oceny hydrodynamiki przepływu gazu w mediach porowatych dotąd nierozpoznanych dla rozwoju nowej generacji czystych źródeł energii, zwłaszcza w kontekście produkcji biogazu lub syngazu.
EN
A review, with 57 refs., of selected hydrodynamic models with their characteristics and applicability areas.
EN
In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in R2 or R3. The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.
EN
This paper deals with a numerical solution of two–phase flow problems in porous media. To solve this type of problems, we propose a numerical method based on mixed–hybrid finite element method. The method is briefly introduced for arbitrary dimension but only a 2D case is considered in this work. We implement several variations of this method using different approaches to solve the resulting system of linear algebraic equations. Direct and iterative solvers are used and a parallel implementation of this method based on the domain decomposition method using MPI is described. The accuracy and the computational efficiency of the method is verified using a problem with a known exact solution. Numerical experiments show that the errors are similar for all variations of the method. The method is convergent and the experimental order of convergence is slightly less than one. There are differences in the computational time in favor to the iterative solvers, especially when using finer meshes. For computations on fine meshes it is also advantageous to use the parallelism that significantly speeds-up the computation.
PL
W artykule opisano numeryczne rozwiązanie dla dwufazowego przepływu w porowatych ośrodkach. Dla rozwiązania tego typu problemu zaproponowano numeryczną metodę wykorzystującą mieszany hybrydowy model elementów skończonych. Model opisano w skrócie dla problemu 2D. W pracy zaimplementowano kilka wariantów metody stosując różne podejścia do rozwiązania wynikowego układu równań algebraicznych. Zastosowano bezpośredni i iteracyjny solwer oraz równoległą implementację metodą wykorzystującą dekompozycję domeny z zastosowaniem interfejsu transmisji wiadomości (ang. Message Passing Interfaee MPI). Dokładność i wydajność obliczeniowa metody zostały zweryfikowane dla problemu o znanym rozwiązaniu dokładnym. Przeprowadzone eksperymenty numeryczne wykazały, że błędy dla różnych wariantów metody są zbliżone. Metoda jest zbieżna ze stopniem zbieżności nieco mniejszym od jeden. Różne były natomiast czasy obliczeń na korzyść solwera iteracyjnego, szczególnie dla gęstych siatek. Przy zastosowaniu takich siatek korzystnie jest zrównoleglić obliczenia, co znacznie skraca czas realizacji programu.
EN
Aerosol filtration in fibrous filters is one of the principal methods of accurate removal of particulate matter from a stream of gas. The classical theory of depth filtration of aerosol particles in fibrous structures is based on the assumption of existing single fibre efficiency, which may be used to recalculate the overall efficiency of entire filter. Using “classical theory” of filtration one may introduce some errors, leading finally to a discrepancy between theory and experiment. There are several reasons for inappropriate estimation of the single fibre efficiency: i) neglecting of shortrange interactions, ii) separation of inertial and Brownian effects, ii) perfect adhesion of particles to the fibre, iv) assumption of perfect mixing of aerosol particles in the gas stream, v) assumption of negligible effect of the presence of neighbouring fibres and vi) assumption of perpendicular orientation of homogenous fibres in the filtration structure. Generally speaking, “classical theory” of filtration was used for characterization of the steady - state filtration process (filtration in a clean filter, at the beginning of the process) without deeper investigation of the influence of the nternal structure of the filter on its performance. The aim of this review is to outline and discuss the progress of deep-bed filtration modelling from the use of simple empirical correlations to advanced techniques of Computational Fluid Dynamics and Digital Fluid Dynamics.
EN
The issue of emulsion flow through porous media plays an important role in the development of land purification methods from various oily substances, as well as during the oil extraction process. The concentration of the emulsion dispersed phase exerts a strong influence on the rheological properties. The oil in water emulsions (O/W), that have a concentration of less than 0.5 (or 50%), show Newtonian behavior, and those with higher concentrations non-Newtonian. The flow of such systems through the structure of the porous medium is, therefore, also dependent on the variable rheological properties of the liquid. When the emulsion flows through the porous medium, a reduction of the relative permeability occurs. Due to that the transport may be very effectively limited, as the pores of deposit are being blocked by oil droplets. The paper presents studies on emulsion flow, with various concentrations, through a granular bed. It allowed to track changes in the permeability of the deposit over time. The results of experimental studies of transport and elution of highly concentrated emulsions from granular structures are also presented. This enabled tracking of changes in flow resistance of the emulsion through the porous bed in time. Moreover, the development of the mathematical model, which allows us to define the relationship between the bed blockage degree and the concentration of the emulsion internal phase could be the result of the experimental works. The results of such studies are very widely used in practice, among other things, in the issues of migration of pollutants, such as petroleum substances, in the soil layers. The development of knowledge in this field may contribute to the optimization of existing oil recovery techniques and methods of remediation of soil from organic substances.
EN
Emulsion flow through porous structure is used in many processes in the field of chemical engineering. Good examples of practical applications are Enhanced Oil Recovery (EOR) techniques, soil remediation and treatment of oily wastewater. The emulsion transport in through porous media is not easy to describe, due to rheological behavior of emulsions and porous structure properties. In case of emulsified system flow through porous bed it is possible to observe the retention of oil on porous structure, and reduction of permeability of bed. In the presented study, we tried to investigate the influence of porous bed parameters on oil-in-water emulsion behavior during its flow and elution from porous structure. In experiments we used porous beds with varied particle size range and with different lengths. We measured pressure drop and related permeability changes in time. The turbidimetric and microscopic analysis that we conducted allowed us to check how concentration of emulsion changed during its flow and elution from porous structure.
16
EN
We considered the characteristics of key technological objects involved in gas storage. Mathematical models of groups of hydraulically related objects (system mathematical models) are constructed and described. Problems are set and examples of application of analytical and numerical methods for their solution are provided
PL
Badano imbibicję emulsji w oleofilowych/hydrofobowych strukturach porowatych w celu określenia wpływu różnych niejonowych emulgatorów na zmianę masy absorbowanej emulsji oraz zmianę jej stężenia w czasie trwania procesu w zależności od wysokości penetracji. Stosowano emulsje typu olej-w-wodzie o zawartości fazy rozproszonej 10% i 30% obj., a udział dodanych emulgatorów zmieniał się w zakresie 1–5% obj. Stwierdzono, że stężenie emulgatora, wartość parametru HLB (równowaga hydrofilowo-lipofilowa) i początkowa lepkość wsiąkającego płynu mają duży wpływ na przebieg procesu nasiąkania. Wykazano, że czas krytyczny rośnie wraz ze stężeniem emulgatora, a czas maksymalny procesu maleje wraz ze wzrostem stężenia fazy wewnętrznej. Stwierdzono, że emulgator z etoksylenowego kwasu tłuszczowego zapewniał największą zawartość fazy zdyspergowanej w sorbencie. Dla emulsji o stężeniu 10% największą zaabsorbowaną masę odnotowano w przypadku Rokanolu NL6. Dla emulsji o stężeniu 30% maksymalną masę zaabsorbowanej substancji zaobserwowano w przypadku emulgatora Rokacet O7.
EN
Three com. emulsifiers were used (1–5% by vol.) for dispersing a com. vegetable oil in water to produce resp. emulsions (oil content 10–30% by vol.) used for imbibitions a polypropylene nonwoven (porosity 0.93). The ethoxylated oleic acid-based emulsifier was the most efficient due to providing the highest amts. of the dispersed phase in a sorbent. For 10% emulsions, the largest mass of the imbibed permeant was observed in case of alkoxylated alc.-based emulsifier.
18
Content available remote Transport emulsji typu olej w wodzie przez struktury ziarniste
PL
W pracy przedstawiono badania przepływów emulsji typu olej w wodzie przez złoża ziarniste oraz jej wymywania ze złoża za pomocą wody. Mierzono zmiany oporów przepływu podczas przemieszania się tych płynów przez media ziarniste oraz zmiany struktury emulsji wypływającej ze złoża w czasie trwania procesu i analizowano wpływ stężenia fazy wewnętrznej, stopnia granulacji złoża oraz jego długości. Stwierdzono stopniowy spadek przepuszczalności, czasową redukcję stężenia fazy olejowej oraz zmiany rozkładu rozmiarów kropel wypływającej ze złoża emulsji. Wyniki badań mogą przyczynić się do bardziej precyzyjnej oceny efektywności zarówno wtórnych metod wydobycia ropy jak i remediacji gleb.
EN
The paper presents studies of oil-in-water emulsions flow through granular media and the studies of elution of these liquids from the bed with the use of water. The changes in flow resistance during the movement of such fluids through granular media and changes in the structure of emulsion flowing out from the reservoir during the process were measured and the influence of internal phase concentration, degree of bed granulation and its length were analyzed. A gradual decrease in permeability, temporary reduction of oil phase concentration and changes in the distribution of droplets sizes in emulsion flowing out of the bed was recorded. The research results may contribute to more accurate evaluation of effectiveness of secondary oil recovery methods and soil remediation.
EN
Deep bed filtration is an effective method of submicron and micron particle removal from the fluid stream. There is an extensive body of literature regarding particle deposition in filters, often using the classical continuum approach. However, the approach is not convenient for studying the influence of particle deposition on filter performance (filtration efficiency, pressure drop) when non-steady state boundary conditions have to be introduced. For the purposes of this work the lattice-Boltzmann model describes fluid dynamics, while the solid particle motion is modeled by the Brownian dynamics. For aggregates the effect of their structure on displacement is taken into account. The possibility of particles rebound from the surface of collector or reentrainment of deposits to fluid stream is calculated by energy balanced oscillatory model derived from adhesion theory. The results show the evolution of filtration efficiency and pressure drop of filters with different internal structure described by the size of pores. The size of resuspended aggregates and volume distribution of deposits in filter were also analyzed. The model enables prediction of dynamic filter behavior. It can be a very useful tool for designing filter structures which optimize maximum lifetime with the acceptable values of filtration efficiency and pressure drop.
EN
In the article a sensitivity analysis of linear and nonlinear terms in the Kozeny-Carman and Ergun equations was shown. In the first case the impact of the porosity, tortuosity, specific surface of the porous body and the model constant was investigated. In the second case the porosity, the particle diameter and the sphericity function were taken into account. To express the model sensitivity by numbers, an earlier developed method was used. In this way the order and the importance of the impact of individual parameters was determined. The motivations to create this article were questions, which occurred during developing a novel investigation method, linking the Discrete Element Method and the CFD techniques. The first aim was to predict what will happen, if individual parameters will be set with an error: which data should be set as accurately as possible and which data are not very important for the result value. The second intention was to state which of parameters used in porous media investigations should be expressed by functions and which by constant values. The article may be treated as set of pointers helping in using of Kozeny-Carman and Ergun laws or as an example of research methodology based on the sensitivity analysis.
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