Principal Axes and Values of the Dispersion Coefficient in the 2D Axially Symmetric Porous Medium

2010 ◽  
pp. 351-355 ◽  
Author(s):  
Leonid G. Fel ◽  
Jacob Bear
Keyword(s):  
Porous Medium ◽  
Dispersion Coefficient ◽  
Axially Symmetric ◽  
Principal Axes

Hydrodynamic Dispersion Coefficient of Urea in Silty Loam Soil

2019 ◽  
Vol 6 (04) ◽  
Author(s):  
RAM PAL ◽  
H C SHARMA ◽  
M IMTIYAZ
Keyword(s):  
Transport Phenomenon ◽  
Porous Medium ◽  
Contaminant Transport ◽  
Dispersion Coefficient ◽  
Hydrodynamic Dispersion ◽  
Soil Columns ◽  
Soil Medium ◽  
Adverse Health Effects ◽  
Silty Loam ◽  
Loam Soil

The modern theme of agriculture is not only to increase production but also to minimize undesirable environmental effects. Leaching of surface-applied fertilizer is the major source of groundwater pollution. Nitrogenous fertilizers are the most popular among the Indian farmers, which on leaching reach the groundwater in different forms (NH4-N, NO3-N, etc). NO3-N leaches faster than other types, remains in-reactive in groundwater, moves with the velocity of groundwater and contaminates it. Contamination arises when NO3-N accumulates in groundwater and consumed in high amount by humans and animals, may result in adverse health effects. For the study of contaminant transport phenomenon in porous medium, a general convection dispersion equation is used, in which dispersion coefficient is one of the primary parameters necessary to be determined for a particular soil. Keeping it in view a study was conducted to assess different available techniques to determine the dispersion coefficient with the help of soil columns having silty loam soil as soil medium. The value of the dispersion coefficient obtained for silty loam soil, by this method was equal to 0.00576 m2.

scholarly journals Temperature effect in potassium and nitrate ions in soil transport

2008 ◽  
Vol 28 (3) ◽  
pp. 438-447 ◽  
Author(s):  
Adriano D. M. A. Gonçalves ◽  
Jarbas H. Miranda ◽  
Paulo Rossi ◽  
José F. G. Sabadin ◽  
Marcos Y. Kamogawa
Keyword(s):  
Porous Medium ◽  
Dispersion Coefficient ◽  
Potassium Nitrate ◽  
Positive Influence ◽  
Retardation Factor ◽  
Nitrate Transport ◽  
Transport Parameters ◽  
Medium Temperature ◽  
Soil Transport ◽  
Solute Dynamics

When doing researches on solute dynamics in porous medium, the knowledge of medium characteristics and percolating liquids, as well as of external factors is very important. An important external factor is temperature and, in this sense, our purpose was determining potassium and nitrate transport parameters for different values of temperature, in miscible displacement experiments. Evaluated parameters were retardation factor (R), diffusion/dispersion coefficient (D) and dispersivity, at ambient temperature (25 up to 28 ºC), 40 ºC and 50 ºC. Salts used were potassium nitrate and potassium chlorate, prepared in a solution made up of 5 ppm nitrate and 2.000 ppm potassium, with Red-Yellow Latosol porous medium. Temperature exhibited a positive influence upon porous medium solution and upon dispersion coefficient.

The Growth of Mixing Zone in Heterogeneous Porous Media

2012 ◽  
Author(s):  
M. R. Othman ◽  
R. Badlishah Ahmad ◽  
Z. May
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Short Duration ◽  
Dispersion Coefficient ◽  
Fluid Velocity ◽  
Heterogeneous Porous Media ◽  
Mixing Zone ◽  
Zone Size ◽  
Model Mixing ◽  
Dispersive Mixing

Dengan menggunakan penyelesaian analitikal yang merangkumi fraktal eksponen, pembesaran jarak pencampuran telah dapat ditentukan bagi model satu dimensi. Size zon pencampuran didapati meningkat apabila media berliang menjadi semakin heterogen. Dalam media berliang yang heterogen, saiz zon pencampuran meningkat apabila pemalar penyerakan meningkat terutama sekali pada aliran jangkamasa singkat relatif. Terdapat tiga faktor penting mempengaruhi saiz zon pencampuran penyerakan, ΔxD. Perkara terpenting dalam kajian ini ialah keheterogenan takungan, yang dipersembahkan oleh fraktal eksponen, β. Hasil kajian mendapati bahawa apabila β menjadi kecil (media berliang menjadi semakin heterogen), saiz zon pencampuran meningkat. Satu lagi faktor mempengaruhi ΔxD ialah pemalar penyerakan bersandar masa, Κ(tD). Di dalam takungan heterogen, zon pencampuran meningkat dengan peningkatan nilai pemalar penyerakan pada aliran jangkamasa singkat relatif. Bagi aliran jangkamasa panjang relatif, bagaimanapun, ΔxD terus meningkat walaupun Κ(tD) menjadi tetap. Faktor ketiga ialah purata kelajuan bendalir, ν. Zon pencampuran mempunyai perkaitan songsang dengan kelajuan bendalir dengan cara ΔxD meningkat apabila ν berkurangan. Kata kunci: Kehomogenan; keheterogenan; pekali penyerakan; eksponen fraktal; zon pencampuran; media berliang Utilizing currently available analytical solutions that incorporate fractal exponent, the growth of mixing length of injected solvent was determined for a one-dimensional model. Mixing zone size was found to increase as porous medium becomes increasingly heterogeneous. In a heterogeneous porous media, mixing zone size increases as dispersion coefficient increases particularly at relatively short duration of flow. There are three important factors influencing the size of the dispersive mixing zone, ΔxD. Of particular importance in this study is reservoir heterogeneity, which is represented by a fractal exponent, β. It was discovered that as β becomes smaller (porous medium becomes increasingly heterogeneous), the size of the mixing zone increases. Another factor affecting ΔxD is time dependent dispersion coefficient, Κ(tD). In a heterogeneous reservoir, mixing zone increases with increasing value of dispersion coefficient at relatively short duration of flow. For relatively long period of flow, however? ΔxD continues to increase even though Κ(tD) remains constant. The third factor is average fluid velocity, ν. Mixing zones have inverse relationship with fluid velocity in that ΔxD increases as ν decreases. Key words: Homogeneity; heterogeneity; dispersion coefficient; fractal exponent; mixing zone; dimensionless concentration; porous media

Analysis of the Influence of Fluid Flow on the Plasticity of Porous Rock Under an Axially Symmetric Punch

1974 ◽  
Vol 14 (03) ◽  
pp. 271-278 ◽  
Author(s):  
Milos Kojic ◽  
J.B. Cheatham
Keyword(s):  
Plastic Deformation ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Pressure Gradient ◽  
Stress Tensor ◽  
Unit Volume ◽  
Fluid Pressure ◽  
The Body ◽  
Axially Symmetric

Introduction A number of problems occur in the fields of drilling and rock mechanics for which consideration must be given to the interaction of fluid flow and rock deformation. Such problems include those of borehole stability, chip removal from under a drill bit, drilling in the presence of a fluid pressure gradient between the drilling fluid and formation fluid, and drilling by use of hydraulic jets. We have recently developed a general theory of the influence of fluid pressure gradients and gravity on the plasticity of porous media. The solution of the problem considered here serves as an example of the application of that theory. The illustrative problem is to determine the load required on a flat problem is to determine the load required on a flat axially symmetric punch for incipient plasticity of the porous medium under the punch when fluid flows through the bottom face of the punch. The rock is assumed to behave as a Coulomb plastic material under the influence of body forces plastic material under the influence of body forces due to fluid pressure gradients and gravity. Numerical methods that have been used by Cox et al. for analyzing axially symmetric plastic deformation in soils with gravity force are applied to the problem considered here. Involved is an iterative process for determining the slip lines. The fluid flow field ‘used for calculating the fluid pressure gradient is based upon the work by Ham pressure gradient is based upon the work by Ham in his study of the potential distribution ahead of the bit in rotary drilling. The effective stresses in the porous rock and the punch force for incipient plasticity are computed in terms of the fluid plasticity are computed in terms of the fluid pressure and the cohesive strength and internal pressure and the cohesive strength and internal friction of the rock. PLASTICITY OF POROUS MEDIA PLASTICITY OF POROUS MEDIA A recently developed general theory of plasticity of porous media under the influence of fluid flow is summarized in this section. The equation of motion for the porous solid for the case of incipient plastic deformation reduces to the following equilibrium equation:(1) where Ts is the partial stress tensor of the solid; Fs is the body force acting on the solid per unit volume of the solid material; P is the interaction force between the solid and the fluid; and is the porosity, which is defined as the ratio of the pore porosity, which is defined as the ratio of the pore volume to the total volume of the solid-fluid mixture. The partial stress tensor Ts can be considered as the effective stress tensor that is used in sod mechanics. With the acceptance of the effective stress principle defined in Ref. 5, the yield function, f, in the following form is satisfied for plastic deformation of the porous medium. plastic deformation of the porous medium.(2) where EP is the plastic strain tensor and K and the work-hardening parameter. From the equation of motion for the fluid, the interaction force P can be expressed in the form(3) where is the inertial force of the fluid per unit volume of the mixture and F is the body force acting on the fluid per unit volume of fluid. For the case of incipient plastic deformation the solid can be considered static (velocities of the solid particles are zero), and the problem of determining particles are zero), and the problem of determining the fluid flow field is the one usually analyzed in petroleum engineering. petroleum engineering. Consider a flow of be fluid such that the inertial forces of the fluid can be neglected and assume that Darcy's law is applicable. SPEJ P. 271

SECOND ORDER NUCLEAR QUADRUPOLE EFFECTS IN SINGLE CRYSTALS: PART I. THEORETICAL

1953 ◽  
Vol 31 (5) ◽  
pp. 820-836 ◽  
Author(s):  
G. M. Volkoff
Keyword(s):  
Magnetic Field ◽  
Electric Field ◽  
Single Crystals ◽  
Electric Quadrupole ◽  
Second Order ◽  
Order Effects ◽  
Axially Symmetric ◽  
Crystalline Electric Field ◽  
Resonance Frequencies ◽  
Principal Axes

The dependence of electric quadrupole splitting of nuclear magnetic resonance absorption lines in single crystals on crystal orientation in an external magnetic field is investigated theoretically following earlier work of Pound, of Volkoff, Petch, and Smellie, and of Bersohn. Explicit formulae are given, applicable to non axially symmetric crystalline electric field gradients (η ≠ 0), and valid up to terms of the second order in the quadrupole coupling constant [Formula: see text], for the dependence of the absorption frequencies on the angle of rotation of the crystal about any arbitrary axis perpendicular to the magnetic field. Some formulae including third order effects in Cz are also given. It is shown that an experimental study of the dependence of this splitting on the angles of rotation about any two arbitrary mutually perpendicular axes is sufficient, when second order effects are measurable, to yield the values of | Cz |, η, and the orientation of the principal axes of the electric field gradient tensor at the nuclear sites. In the case that the direction of one of the principal axes is known from crystal symmetry, a single rotation about this axis gives the complete information.A new method of determining nuclear spin I is proposed which depends on comparing first and second order shifts of the resonance frequencies of the strong inner line components. The method will be of interest in those cases where the total number 2I of line components can not be unambiguously ascertained owing to the outer line components being excessively broadened and weakened by crystal imperfections.

A Mathematical and Experimental Examination of Transverse Dispersion Coefficients

1968 ◽  
Vol 8 (02) ◽  
pp. 195-204 ◽  
Author(s):  
Robert C. Hassinger ◽  
Dale U. Von Rosenberg
Keyword(s):  
Porous Medium ◽  
Dispersion Coefficient ◽  
Packed Column ◽  
Packing Material ◽  
Longitudinal Dispersion ◽  
Physical Parameters ◽  
Dispersion Coefficients ◽  
Molecular Diffusivity ◽  
Transverse Dispersion ◽  
Wide Range

Abstract Transverse dispersion has received considerably less treatment in the literature than has longitudinal dispersion. Different methods for determining transverse dispersion coefficients have been used in different investigations, and the results obtained have not been consistent enough to permit accurate generalizations as to the effect of various physical parameters on the magnitude of these coefficients. A numerical solution to the differential equation describing transverse dispersion in the absence of longitudinal dispersion was obtained to enable one to calculate the dispersion coefficient from experimental results. The more general dispersion equation including longitudinal dispersion also was solved numerically to give quantitative limits of a dimensionless group within which the assumption of negligible longitudinal dispersion is justified. Possible experimental procedures were examined, and one utilizing a cylindrical packed column was chosen for the determination of transverse dispersion coefficients. Values of these coefficients were determined for a system of two miscible organic fluids of equal density and viscosity, for two sizes of packing material over a wide range of flow rates in the laminar regime. The dispersion coefficient was found to decrease, for a constant value of the product of packing size and interstitial velocity, as the size of the packing material particles increased. Introduction Longitudinal dispersion has received extensive treatment in the literature, and consequently is better understood than its orthogonal counterpart, transverse dispersion. Many mathematical models of dispersion processes assume that transverse dispersion is rapid enough to damp out any radial concentration gradients and therefore may be neglected. Laboratory and production results, however, indicate that this is a poor assumption. Various experimental procedures for determining transverse dispersion coefficients have been used in previous investigations, but the results have generally been expressed by similar correlations. The transverse dispersion coefficients obtained, however, have often varied considerably for given values of the correlation parameters. We feel that further experimental determinations of transverse dispersion coefficients will help alleviate some of the inconsistencies in these empirical correlations. One assumption implicit in all previous investigations is that of negligible longitudinal dispersion in the experimental system. An attempt to justify this assumption often is made using intuitive reasoning, but it is apparent that this reasoning must break down as the condition of zero flow rate is approached. A mathematical examination of the equations describing the system yields physical limits outside of which the assumption of negligible longitudinal dispersion is invalid. Background In a porous medium, the "effective molecular diffusivity" De is less than the molecular diffusivity D measured in the absence of a porous medium, due to the tortuous path which a diffusing molecule must travel. Various authors have reported values of the ratio De/D in the range of 0.6 to 0.7. When there is fluid flow within the porous medium, mass transfer occurs by convective dispersion as well as by molecular diffusion. These are separate phenomena and can be treated as such on a microscopic scale. However, the mathematical complexity is such that only extremely simple geometries could be considered, and the results hardly would be applicable to the complex geometries existent in actual porous media. SPEJ P. 195ˆ

Numerical investigation of simultaneous effects of nanofluid flow and porous baffle on thermal energy transfer and flow features in a circular channel

2021 ◽  
pp. 1-21
Author(s):  
Hossein Namadchian ◽  
Javad Sodagar Abardeh ◽  
Ahmad Arabkoohsar ◽  
K.A.R. Ismail
Keyword(s):  
Porous Medium ◽  
Energy Transfer ◽  
Thermal Energy ◽  
Simple Algorithm ◽  
Darcy Number ◽  
Circular Channel ◽  
Axially Symmetric ◽  
Nanofluid Flow ◽  
Two Phase ◽  
Porous Region

Abstract In the present work, the forced-convection heat transfer features of different nanofluids in a circular channel with porous baffles are numerically investigated. Nanofluid flow in the porous area is simulated by the simultaneous use of Darcy-Brinkman-Forchheimer and two-phase mixture models. The flow is considered to be laminar, two-dimensionall, steady, axially symmetric, and incompressible. The simulations are conducted in Fluent software and by using the finite volume method and SIMPLE algorithm. The influences of various parameters, including Reynolds number, volume fractions of nanoparticles, Darcy number, porous region height, and various nanofluid types on the nanofluid flows and their thermal energy transfer features, are investigated. Results show that porous blocks significantly change the flow characteristics and thermal energy transfer features. For instance, at low Darcy numbers, the permeability of the porous region decreases, and the porous baffles have greater resistance against the nanofluid flow. As a result, the vortex area becomes stronger and taller, and streamlines near obstacles are tighter. However, in high Darcy numbers, due to the high permeability of the porous medium, the flow will be the same as the flow in the channel without barriers, and the porous baffles will not have much influence on the flow. For example, at Darcy number Da = 10-4 the vortex area almost disappears. The growth of conductivity ratio increases the local Nu in the vicinity of the barriers. Properties of the porous medium and nanofluid flow affect the thermal energy transfer rate, and it can be improved by making appropriate changes to these features.

The Research of Thermal Dispersion in Porous Medium with Different Skeleton Structures

2014 ◽  
Vol 1049-1050 ◽  
pp. 148-153
Author(s):  
Hong Ren Zhan ◽  
Hong Feng Xu ◽  
Shi Fang Li ◽  
Xian Zhen Zhang ◽  
Ya Xia Li
Keyword(s):  
Numerical Simulation ◽  
Porous Medium ◽  
Dispersion Coefficient ◽  
Influence Factors ◽  
Simulation Method ◽  
Particle Model ◽  
Thermal Dispersion ◽  
Darcy Velocity ◽  
Axial Temperature ◽  
Circular Particle

A numerical simulation method is hereby adopted in this paper to calculate and analyze the thermal dispersion influence factors of three particle-shaped porous mediums of two-dimensional porous medium with different skeleton structures. Result shows that the thermal dispersion influence factors include fluid thermal properties, Darcy velocity, porosity and skeleton structure, etc. Under the same condition, the thermal dispersion coefficient of a circular particle model is maximum, and it increases with the rise of Ped number; with the increase of the porosity or the reduction of Darcy velocity, the thermal dispersion coefficient reduces increasingly. When considering the thermal dispersion effect, the axial temperature variation of fluid is more applicable for numerical simulation, which proves that it shall consider the thermal dispersion during researches on porous medium models.

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