Austin Journal of Hydrology is an open access, peer reviewed, scholarly journal dedicated to publish articles in all areas of Hydrology.
The aim of the journal is to provide a platform for engineers, scientists and academicians all over the world to endorse, discuss and share various new issues and developments in diverse areas of hydrology.
Austin Journal of Hydrology accepts original research articles, review articles, case reports, commentaries, clinical images and rapid communication on all the aspects of Hydrology.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The current study examines the generation and propagation of a Third order solitary water wave along
the channel. Surface displacement and wave profi le prediction challenges are interesting subjects in the
fi eld of marine engineering and many researchers have tried to investigate these parameters. To study the
wave propagation problem, here, fi rstly the meshless Incompressible Smoothed Particle Hydrodynamics
(ISPH) numerical method is described. Secondly,
This document summarizes research on oblique shock waves that appear in supersonic carbon dioxide two-phase flow, as occurs in ejector refrigeration cycles. It presents:
1) Theoretical analyses showing that two types of oblique shock waves can occur - weak shocks where flow remains supersonic, and strong shocks with large pressure recovery and subsonic flow.
2) An experiment using a carbon dioxide two-phase flow channel to observe these shock waves.
3) Equations governing compressible two-phase flow and the conditions under which strong and weak oblique shock waves form, to compare with experimental results.
This document provides an introduction to fluid mechanics fundamentals, including:
- The four primary dimensions in fluid mechanics are mass, length, time, and force. All other variables can be expressed in terms of these.
- Liquids can be treated as incompressible for most fluid mechanics problems since pressure changes are typically not large enough to cause changes in density.
- Viscosity describes a fluid's resistance to shear forces or layer sliding, and can result in either laminar (smooth) flow or turbulent (chaotic) flow.
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...zoya rizvi
This document discusses computational fluid dynamics (CFD) and its application to aerodynamics. It begins by introducing CFD and the governing equations of fluid dynamics - the continuity, momentum, and energy equations. These partial differential equations can be used to model fluid flow. The document then examines the finite control volume approach and substantial derivative used to develop the Navier-Stokes equations from fundamental principles. An example application of CFD to aerodynamics is provided. The document aims to explain the methodology of CFD, including establishing the governing equations and interpreting results.
The document discusses transport phenomena and provides definitions and examples of key concepts in vector and tensor analysis used to describe transport phenomena. It defines transport phenomena as dealing with the movement of physical quantities in chemical or mechanical processes. There are three main types of transport: momentum, energy, and mass transport. Vector and tensor quantities like velocity, stress, and strain gradient are used to describe transport phenomena. Tensors have a magnitude and direction(s) and transform under coordinate system rotations. The document provides examples of scalar, vector, and tensor notation and the Kronecker delta, alternating unit tensor, and mathematical operations on vectors like addition, dot product, and cross product.
Dimensional analysis Similarity laws Model laws R A Shah
Rayleigh's method- Theory and Examples
Buckingham Pi Theorem- Theory and Examples
Model and Similitude
Forces on Fluid
Dimensionless Numbers
Model laws
Distorted models
This document provides an overview of unsteady-state flow and the diffusivity equation, which is used to model pressure changes over time in reservoirs. It discusses the assumptions and solutions of the diffusivity equation, including the Ei-function and dimensionless pressure drop solutions. The constant-terminal pressure and constant-terminal rate solutions are examined. Graphs demonstrate how pressure profiles change over different times based on these solutions. The document also explores using dimensionless variables to simplify analyses of unsteady-state flow regimes.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The current study examines the generation and propagation of a Third order solitary water wave along
the channel. Surface displacement and wave profi le prediction challenges are interesting subjects in the
fi eld of marine engineering and many researchers have tried to investigate these parameters. To study the
wave propagation problem, here, fi rstly the meshless Incompressible Smoothed Particle Hydrodynamics
(ISPH) numerical method is described. Secondly,
This document summarizes research on oblique shock waves that appear in supersonic carbon dioxide two-phase flow, as occurs in ejector refrigeration cycles. It presents:
1) Theoretical analyses showing that two types of oblique shock waves can occur - weak shocks where flow remains supersonic, and strong shocks with large pressure recovery and subsonic flow.
2) An experiment using a carbon dioxide two-phase flow channel to observe these shock waves.
3) Equations governing compressible two-phase flow and the conditions under which strong and weak oblique shock waves form, to compare with experimental results.
This document provides an introduction to fluid mechanics fundamentals, including:
- The four primary dimensions in fluid mechanics are mass, length, time, and force. All other variables can be expressed in terms of these.
- Liquids can be treated as incompressible for most fluid mechanics problems since pressure changes are typically not large enough to cause changes in density.
- Viscosity describes a fluid's resistance to shear forces or layer sliding, and can result in either laminar (smooth) flow or turbulent (chaotic) flow.
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...zoya rizvi
This document discusses computational fluid dynamics (CFD) and its application to aerodynamics. It begins by introducing CFD and the governing equations of fluid dynamics - the continuity, momentum, and energy equations. These partial differential equations can be used to model fluid flow. The document then examines the finite control volume approach and substantial derivative used to develop the Navier-Stokes equations from fundamental principles. An example application of CFD to aerodynamics is provided. The document aims to explain the methodology of CFD, including establishing the governing equations and interpreting results.
The document discusses transport phenomena and provides definitions and examples of key concepts in vector and tensor analysis used to describe transport phenomena. It defines transport phenomena as dealing with the movement of physical quantities in chemical or mechanical processes. There are three main types of transport: momentum, energy, and mass transport. Vector and tensor quantities like velocity, stress, and strain gradient are used to describe transport phenomena. Tensors have a magnitude and direction(s) and transform under coordinate system rotations. The document provides examples of scalar, vector, and tensor notation and the Kronecker delta, alternating unit tensor, and mathematical operations on vectors like addition, dot product, and cross product.
Dimensional analysis Similarity laws Model laws R A Shah
Rayleigh's method- Theory and Examples
Buckingham Pi Theorem- Theory and Examples
Model and Similitude
Forces on Fluid
Dimensionless Numbers
Model laws
Distorted models
This document provides an overview of unsteady-state flow and the diffusivity equation, which is used to model pressure changes over time in reservoirs. It discusses the assumptions and solutions of the diffusivity equation, including the Ei-function and dimensionless pressure drop solutions. The constant-terminal pressure and constant-terminal rate solutions are examined. Graphs demonstrate how pressure profiles change over different times based on these solutions. The document also explores using dimensionless variables to simplify analyses of unsteady-state flow regimes.
The document covers reservoir engineering concepts including solutions to the diffusivity equation for radial flow of single-phase and compressible fluids. It discusses the pD and Ei-function solutions, and presents the unified steady-state flow regime equations for radial flow of single-phase and compressible fluids using the pD-function, m(p)-function, and pressure-squared approximations. It also covers the pseudo-steady state flow regime and relationships between pressure functions.
This document provides an overview of Darcy's law and permeability concepts covered in a reservoir engineering course. It begins by defining permeability as a property that controls fluid flow in porous media. Darcy's law is then introduced, which relates fluid flow rate to pressure drop, permeability, fluid properties, and geometry. The document goes on to discuss permeability measurement techniques, effective and relative permeability, reservoir properties, heterogeneity, and applications of Darcy's law to linear and radial flow models.
This document provides an overview of a reservoir engineering course focused on Darcy's Law and permeability. It covers key topics like laboratory analysis of rock properties including porosity, saturation and permeability. It also discusses linear and radial flow models based on Darcy's Law and techniques for determining permeability in the laboratory and averaging permeabilities for heterogeneous reservoirs. The document emphasizes that permeability is an important property that controls fluid flow in reservoirs and was first mathematically defined by Henry Darcy. It provides the equations for linear and radial flow based on Darcy's Law.
This document provides an overview of steady state radial flow in reservoirs. It discusses steady state flow of incompressible, slightly compressible, and compressible fluids. For incompressible fluids, Darcy's law is used to calculate flow rates. For compressible fluids, the real gas potential and pseudopressure are introduced to account for compressibility. Flow rates can be expressed in terms of average reservoir pressure or approximated using the p-squared method. The document also covers multiphase flow, flow ratios of water-oil and gas-oil, and pressure disturbance for a shut-in well.
This document contains 51 short answer questions related to aerodynamics and compressible flow. The questions cover topics like gas dynamics, compressible versus incompressible flow, compressibility, types of compressibility, properties of perfect gases, adiabatic and isentropic processes, Mach number, flow regimes, continuity, momentum, and energy equations. Many questions also focus specifically on nozzle flow, including definitions of different types of nozzles, choking, expansion, under-expanded versus over-expanded nozzles, and nozzle efficiency.
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
This document summarizes key concepts from a reservoir engineering course, including pseudosteady-state (PSS) flow regimes for radial flow of slightly compressible (SC) and compressible (C) fluids. It discusses how the PSS flow condition is reached after transient flow, and how average reservoir pressure changes at a constant rate in PSS. Equations are provided for calculating flow rates of SC and C fluids in PSS, along with approximations that account for skin effect and non-ideal assumptions.
This document outlines the contents of the course ME 6604 Gas Dynamics and Jet Propulsion. It contains 5 units:
1. Basic concepts and isentropic flows, including concepts of compressible flow, stagnation properties, and flow through nozzles and diffusers.
2. Flow through ducts, including Fanno flow with friction and Rayleigh flow with heat transfer.
3. Normal and oblique shocks, including governing equations and properties across shock waves.
4. Jet propulsion, including theories of jet propulsion and performance of ramjets, turbojets, turbofans and turboprops.
5. Space propulsion, including rocket propulsion principles, types of rocket
This document outlines the key concepts in reservoir engineering. It discusses reservoir characteristics including fluid types, flow regimes, and geometries. It then covers steady-state and unsteady-state flow, defining transient flow as the period where the reservoir boundary has no effect. The document derives the diffusivity equation from continuity, transport, and compressibility equations. It discusses the assumptions and solutions of the diffusivity equation, including constant-terminal pressure and rate solutions.
Characteristics of shock reflection in the dual solution domainSaif al-din ali
This document summarizes a numerical study that investigates the use of laser energy deposition to induce transitions between regular and Mach shock reflections in supersonic flow over dual wedge configurations. The study validated its numerical approach by comparing results to an experiment involving laser deposition in front of a sphere. Simulations then examined how varying the position and amount of laser energy deposition could influence transition characteristics in the dual solution domain over wedge configurations. Key findings included how transition time and occurrence depended on deposition parameters and position relative to the shock waves and wedges.
This document outlines key concepts in gas dynamics and compressible flow. It defines gas dynamics as the branch of fluid dynamics concerned with compressible flow. Some main topics covered include the fundamental laws of thermodynamics, definitions of basic terms like system and state, and equations for the conservation of mass, momentum and energy. It also discusses different types of flow and processes like steady/unsteady, laminar/turbulent and adiabatic. Stagnation properties of gases are defined using equations relating stagnation temperature, pressure and density to static properties.
This document provides an overview of a reservoir engineering course covering topics like:
- PSS and skin concepts for radial flow of single- and multi-phase fluids
- Turbulent versus laminar flow and models for turbulent/non-Darcy flow
- The concept of superposition and its applications, including effects of multiple wells, rate changes, boundaries, and pressure changes
- Transient well testing methods and the information they provide about a reservoir's properties
Effect of Longitudinal Velocity of the Particle of the Dusty Fluid with Volum...inventionjournals
The effect of finite volume fraction of suspended particulate matter on axially symmetrical jet mixing of incompressible dusty fluid has been considered. Here we are assuming the velocity and temperature in the jet to differ only slightly from that of surrounding stream, a perturbation method has been employed to linearize the equation those have been solved by using Laplace Transformation technique. Numerical computations have been made to find the solutions of the longitudinal perturbed fluid velocity and longitudinal perturbed particle velocity.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
This document summarizes a numerical study on free-surface flow conducted using a computational fluid dynamics (CFD) solver. The study examines the wave profile generated by a submerged hydrofoil through several test cases varying parameters like the turbulence model, grid resolution, and hydrofoil depth. The document provides background on the governing equations solved by the CFD solver and the interface capturing technique used to model the free surface. Five test cases are described that investigate grid convergence, the impact of laminar vs turbulent models, the relationship between hydrofoil depth and wave height, and the effect of discretization schemes.
The document discusses control volume analysis and the conservation laws applied to control volumes, including:
- Conservation of mass for fixed, moving, and deforming control volumes
- Linear momentum (Newton's second law) for fixed control volumes
- The energy equation for fixed control volumes, including work, heat transfer, and applications to steady, incompressible flow
- An example problem calculating the work output of steam passing through a turbine using the energy equation
This presentation is made to provide the overall conceptual knowledge on Chilton Colburn Analogy. It includes basis, importance, assumption, advantages, limitations and applications in addition to the derivation. Make It Useful!
This document provides a detailed table of contents for a course on transport phenomena. It outlines topics related to momentum transport and fluid dynamics including the equation of continuity, viscous stress, and momentum balances. It also covers mass transfer in reactive and non-reactive systems, including diffusion, mass transfer coefficients, and applications to reactors. Finally, it provides an overview of the first lecture which defines vectors, fluxes, and derives the equation of continuity for incompressible fluids.
This chapter discusses differential analysis of fluid flow. It introduces the concepts of stream function and vorticity. The key equations derived are:
1) The differential equations of continuity, linear momentum, and mass conservation which relate the time rate of change of fluid properties like density and velocity within an infinitesimal control volume.
2) The Navier-Stokes equations which model viscous flow using Newton's laws and relate stresses to strain rates via viscosity.
3) Equations for inviscid, irrotational flow where viscosity and vorticity are neglected.
4) The stream function, a potential function whose contour lines represent streamlines, allowing 2D problems to be solved using a
Analysis of the Interaction between A Fluid and A Circular Pile Using the Fra...IJERA Editor
The purpose of this research is to study the interaction between a fluid and a circular pile, located downstream
from a fan-shaped dam, through the fractional Navier-Stokes equations, and in particular, its approximation to
the boundary layer. The flow region is divided into zones according to the vorticity transport theory of
turbulence. First, we consider the limit of the spatial occupancy index close to 1. Then, a stream function is
introduced, and for the potential zone, we consider a complex potential, using the inverse distances on a circle.
In the other limit, when the spatial occupation index approaches 0, we consider the equations of the boundary
layer in the limit of fully developed turbulence. Next, for the last approaches, a new stream function and
velocities in their radial and polar components are obtained. We also find the asymmetry of the pressure
distribution around the pile, based on the viscosity and considering that the pressure drag force and the friction
coefficients are proportional to the inverse of the Reynolds number. We conclude that D'Alembert's paradox and
Thomson's theorem has been resolved. For applications, in the case of the turbulent wake, we are interested both
in the orientation given by the pile symmetry axis and its extension. The criterion that should be satisfied is: the
diameter of the pile, on the border of inequality, must be located as proportional average between the length of
the turbulent wake and twice the characteristic length associated with the dam, whose aspect ratio, in turn, to the
pile diameter, determines the contraction factor.
This document discusses the application of vector integration in various domains. It begins by defining vector calculus concepts like del, gradient, curl, and divergence. It then presents several theorems of vector integration. Next, it explains how vector integration can be used to find the rate of change of fluid mass and analyze fluid circulation, vorticity, and the Bjerknes Circulation Theorem regarding sea breezes. It also discusses using vector calculus concepts in electricity and magnetism.
Dipolar interaction and the Manning formulaIJERA Editor
In this work we want to show that the mathematical model of quantum mechanics, led to its classical approach, is able to reproduce as close macroscopic experimental results captured by the Manning formula, sufficiently verified through their diverse applications in hydraulics. Molecular interaction between the fluid and the wall of the vessel is studied decomposing the Hamiltonian in two parts: free, and interacting. Scaling process is considered from molecular to hydraulic. Participation of the symmetries of Saint-Venant equation in the hydraulic gradient is taken into account. Correlations between different variables are set. The magnitude of scale change is estimated. We conclude that the Compton wavelength induces to the Boussinesq viscosity concept and the characteristic length of the viscous sublayer.
The document covers reservoir engineering concepts including solutions to the diffusivity equation for radial flow of single-phase and compressible fluids. It discusses the pD and Ei-function solutions, and presents the unified steady-state flow regime equations for radial flow of single-phase and compressible fluids using the pD-function, m(p)-function, and pressure-squared approximations. It also covers the pseudo-steady state flow regime and relationships between pressure functions.
This document provides an overview of Darcy's law and permeability concepts covered in a reservoir engineering course. It begins by defining permeability as a property that controls fluid flow in porous media. Darcy's law is then introduced, which relates fluid flow rate to pressure drop, permeability, fluid properties, and geometry. The document goes on to discuss permeability measurement techniques, effective and relative permeability, reservoir properties, heterogeneity, and applications of Darcy's law to linear and radial flow models.
This document provides an overview of a reservoir engineering course focused on Darcy's Law and permeability. It covers key topics like laboratory analysis of rock properties including porosity, saturation and permeability. It also discusses linear and radial flow models based on Darcy's Law and techniques for determining permeability in the laboratory and averaging permeabilities for heterogeneous reservoirs. The document emphasizes that permeability is an important property that controls fluid flow in reservoirs and was first mathematically defined by Henry Darcy. It provides the equations for linear and radial flow based on Darcy's Law.
This document provides an overview of steady state radial flow in reservoirs. It discusses steady state flow of incompressible, slightly compressible, and compressible fluids. For incompressible fluids, Darcy's law is used to calculate flow rates. For compressible fluids, the real gas potential and pseudopressure are introduced to account for compressibility. Flow rates can be expressed in terms of average reservoir pressure or approximated using the p-squared method. The document also covers multiphase flow, flow ratios of water-oil and gas-oil, and pressure disturbance for a shut-in well.
This document contains 51 short answer questions related to aerodynamics and compressible flow. The questions cover topics like gas dynamics, compressible versus incompressible flow, compressibility, types of compressibility, properties of perfect gases, adiabatic and isentropic processes, Mach number, flow regimes, continuity, momentum, and energy equations. Many questions also focus specifically on nozzle flow, including definitions of different types of nozzles, choking, expansion, under-expanded versus over-expanded nozzles, and nozzle efficiency.
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
This document summarizes key concepts from a reservoir engineering course, including pseudosteady-state (PSS) flow regimes for radial flow of slightly compressible (SC) and compressible (C) fluids. It discusses how the PSS flow condition is reached after transient flow, and how average reservoir pressure changes at a constant rate in PSS. Equations are provided for calculating flow rates of SC and C fluids in PSS, along with approximations that account for skin effect and non-ideal assumptions.
This document outlines the contents of the course ME 6604 Gas Dynamics and Jet Propulsion. It contains 5 units:
1. Basic concepts and isentropic flows, including concepts of compressible flow, stagnation properties, and flow through nozzles and diffusers.
2. Flow through ducts, including Fanno flow with friction and Rayleigh flow with heat transfer.
3. Normal and oblique shocks, including governing equations and properties across shock waves.
4. Jet propulsion, including theories of jet propulsion and performance of ramjets, turbojets, turbofans and turboprops.
5. Space propulsion, including rocket propulsion principles, types of rocket
This document outlines the key concepts in reservoir engineering. It discusses reservoir characteristics including fluid types, flow regimes, and geometries. It then covers steady-state and unsteady-state flow, defining transient flow as the period where the reservoir boundary has no effect. The document derives the diffusivity equation from continuity, transport, and compressibility equations. It discusses the assumptions and solutions of the diffusivity equation, including constant-terminal pressure and rate solutions.
Characteristics of shock reflection in the dual solution domainSaif al-din ali
This document summarizes a numerical study that investigates the use of laser energy deposition to induce transitions between regular and Mach shock reflections in supersonic flow over dual wedge configurations. The study validated its numerical approach by comparing results to an experiment involving laser deposition in front of a sphere. Simulations then examined how varying the position and amount of laser energy deposition could influence transition characteristics in the dual solution domain over wedge configurations. Key findings included how transition time and occurrence depended on deposition parameters and position relative to the shock waves and wedges.
This document outlines key concepts in gas dynamics and compressible flow. It defines gas dynamics as the branch of fluid dynamics concerned with compressible flow. Some main topics covered include the fundamental laws of thermodynamics, definitions of basic terms like system and state, and equations for the conservation of mass, momentum and energy. It also discusses different types of flow and processes like steady/unsteady, laminar/turbulent and adiabatic. Stagnation properties of gases are defined using equations relating stagnation temperature, pressure and density to static properties.
This document provides an overview of a reservoir engineering course covering topics like:
- PSS and skin concepts for radial flow of single- and multi-phase fluids
- Turbulent versus laminar flow and models for turbulent/non-Darcy flow
- The concept of superposition and its applications, including effects of multiple wells, rate changes, boundaries, and pressure changes
- Transient well testing methods and the information they provide about a reservoir's properties
Effect of Longitudinal Velocity of the Particle of the Dusty Fluid with Volum...inventionjournals
The effect of finite volume fraction of suspended particulate matter on axially symmetrical jet mixing of incompressible dusty fluid has been considered. Here we are assuming the velocity and temperature in the jet to differ only slightly from that of surrounding stream, a perturbation method has been employed to linearize the equation those have been solved by using Laplace Transformation technique. Numerical computations have been made to find the solutions of the longitudinal perturbed fluid velocity and longitudinal perturbed particle velocity.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
This document summarizes a numerical study on free-surface flow conducted using a computational fluid dynamics (CFD) solver. The study examines the wave profile generated by a submerged hydrofoil through several test cases varying parameters like the turbulence model, grid resolution, and hydrofoil depth. The document provides background on the governing equations solved by the CFD solver and the interface capturing technique used to model the free surface. Five test cases are described that investigate grid convergence, the impact of laminar vs turbulent models, the relationship between hydrofoil depth and wave height, and the effect of discretization schemes.
The document discusses control volume analysis and the conservation laws applied to control volumes, including:
- Conservation of mass for fixed, moving, and deforming control volumes
- Linear momentum (Newton's second law) for fixed control volumes
- The energy equation for fixed control volumes, including work, heat transfer, and applications to steady, incompressible flow
- An example problem calculating the work output of steam passing through a turbine using the energy equation
This presentation is made to provide the overall conceptual knowledge on Chilton Colburn Analogy. It includes basis, importance, assumption, advantages, limitations and applications in addition to the derivation. Make It Useful!
This document provides a detailed table of contents for a course on transport phenomena. It outlines topics related to momentum transport and fluid dynamics including the equation of continuity, viscous stress, and momentum balances. It also covers mass transfer in reactive and non-reactive systems, including diffusion, mass transfer coefficients, and applications to reactors. Finally, it provides an overview of the first lecture which defines vectors, fluxes, and derives the equation of continuity for incompressible fluids.
This chapter discusses differential analysis of fluid flow. It introduces the concepts of stream function and vorticity. The key equations derived are:
1) The differential equations of continuity, linear momentum, and mass conservation which relate the time rate of change of fluid properties like density and velocity within an infinitesimal control volume.
2) The Navier-Stokes equations which model viscous flow using Newton's laws and relate stresses to strain rates via viscosity.
3) Equations for inviscid, irrotational flow where viscosity and vorticity are neglected.
4) The stream function, a potential function whose contour lines represent streamlines, allowing 2D problems to be solved using a
Analysis of the Interaction between A Fluid and A Circular Pile Using the Fra...IJERA Editor
The purpose of this research is to study the interaction between a fluid and a circular pile, located downstream
from a fan-shaped dam, through the fractional Navier-Stokes equations, and in particular, its approximation to
the boundary layer. The flow region is divided into zones according to the vorticity transport theory of
turbulence. First, we consider the limit of the spatial occupancy index close to 1. Then, a stream function is
introduced, and for the potential zone, we consider a complex potential, using the inverse distances on a circle.
In the other limit, when the spatial occupation index approaches 0, we consider the equations of the boundary
layer in the limit of fully developed turbulence. Next, for the last approaches, a new stream function and
velocities in their radial and polar components are obtained. We also find the asymmetry of the pressure
distribution around the pile, based on the viscosity and considering that the pressure drag force and the friction
coefficients are proportional to the inverse of the Reynolds number. We conclude that D'Alembert's paradox and
Thomson's theorem has been resolved. For applications, in the case of the turbulent wake, we are interested both
in the orientation given by the pile symmetry axis and its extension. The criterion that should be satisfied is: the
diameter of the pile, on the border of inequality, must be located as proportional average between the length of
the turbulent wake and twice the characteristic length associated with the dam, whose aspect ratio, in turn, to the
pile diameter, determines the contraction factor.
This document discusses the application of vector integration in various domains. It begins by defining vector calculus concepts like del, gradient, curl, and divergence. It then presents several theorems of vector integration. Next, it explains how vector integration can be used to find the rate of change of fluid mass and analyze fluid circulation, vorticity, and the Bjerknes Circulation Theorem regarding sea breezes. It also discusses using vector calculus concepts in electricity and magnetism.
Dipolar interaction and the Manning formulaIJERA Editor
In this work we want to show that the mathematical model of quantum mechanics, led to its classical approach, is able to reproduce as close macroscopic experimental results captured by the Manning formula, sufficiently verified through their diverse applications in hydraulics. Molecular interaction between the fluid and the wall of the vessel is studied decomposing the Hamiltonian in two parts: free, and interacting. Scaling process is considered from molecular to hydraulic. Participation of the symmetries of Saint-Venant equation in the hydraulic gradient is taken into account. Correlations between different variables are set. The magnitude of scale change is estimated. We conclude that the Compton wavelength induces to the Boussinesq viscosity concept and the characteristic length of the viscous sublayer.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
This document discusses rotating Boussinesq equations, which model the motion of a viscous incompressible stratified fluid in a rotating system. It derives the rotating Boussinesq equations, considers them with and without Reynolds stress terms, and discusses existence and uniqueness of solutions. It focuses on Faedo-Galerkin approximations and LaSalle invariance principle to assess stability and attraction of solutions.
This document discusses rotating Boussinesq equations which model the motion of viscous stratified fluids under rotation. It derives the rotating Boussinesq equations, considers them with and without Reynolds stress terms, and introduces the initial-boundary value problem. The focus is on existence and uniqueness of solutions, Lyapunov functions for stability analysis, and Faedo-Galerkin approximations.
This document summarizes a journal article that models a dam break over a rubble mound using the shallow water equations and the Lax-Friedrichs numerical scheme. The model is validated against experimental data. A grid refinement study is performed using three grid sizes to calculate the grid convergence index, which indicates the smallest grid size needed for a grid-independent solution. The model accurately simulates stationary water and matches experimental data for a dam break over a triangular obstacle.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
1) The document discusses modeling gravitational fields within general relativity by mapping geodesic motion in curved spacetime to motion along worldlines in Minkowski spacetime.
2) It presents equations showing the equivalence between motion in a gravitational field described by a metric and motion in flat spacetime under the influence of a force field.
3) As an example, it models the Schwarzschild metric by considering dust moving radially in Minkowski spacetime, recovering the Schwarzschild solution from Einstein's field equations in vacuum.
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations
with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if
quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave
packet around the classical path.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations
with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if
quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave
packet around the classical path.
1) The paper investigates whether quantum variations around geodesics could circumvent caustics that occur in certain space-times.
2) An action is developed that yields both the field equations and geodesic condition. Quantizing this action provides a way to determine the extent of the wave packet around the classical path.
3) It is shown that replacing plane wave solutions with wave packets in the path integral still yields acceptable results. Determining if the distribution matches expectation values and variances is key to establishing geodesic completeness with quantum variations.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations
with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if
quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave
packet around the classical path.
This document describes the method of multiple scales for extracting the slow time dependence of patterns near a bifurcation. The key aspects are:
1) Introducing scaled space and time coordinates (multiple scales) to capture slow modulations to the pattern, treating these as separate variables.
2) Expanding the solution as a power series in a small parameter near threshold.
3) Equating terms at each order in the expansion to derive amplitude equations for the slowly varying amplitudes, using solvability conditions to constrain the equations.
4) The solvability conditions arise because the linear operator has a null space, requiring the removal of components in this null space for finite solutions.
FDM is an older method than FEM that requires less computational power but is also less accurate in some cases where higher-order accuracy is required. FEM permit to get a higher order of accuracy, but requires more computational power and is also more exigent on the quality of the mesh.29-Jun-2017
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What's the difference between FEM and FDM? - FEA for All
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Kinetic studies on malachite green dye adsorption from aqueous solutions by A...Open Access Research Paper
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In 2022, the first commercial application of MetaArray™ was performed at the site. MetaArray™ utilizes statistical analysis, such as principal component analysis and multivariate analysis to provide evidence that reductive dechlorination is active or even that it is slowing. This creates actionable data allowing users to save money by making important site management decisions earlier.
The results of the MetaArray™ analysis’ support vector machine (SVM) identified groundwater monitoring wells with a 80% confidence that were characterized as either Limited for Reductive Decholorination or had a High Reductive Reduction Dechlorination potential. The results of MetaArray™ will be used to further optimize the site’s post remediation monitoring program for monitored natural attenuation.
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2. Austin J Hydrol 2(1): id1011 (2015) - Page - 02
Shaul Sorek Austin Publishing Group
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macroscopic Navier-Stokes (NS) equation. These at the REV scale
can vary from inertia fluxes in the form of a nonlinear wave equation,
Forchheimer’s law expressing the transmittance of fluid inertia to
the solid matrix through their microscopic solid-fluid interface, or
conform to Darcy’s law when friction at that interface is dominant.
Spatial Scaling of the General Macroscopic
Balance Equations
Addressing Bear and Bachmat (1990) the spatial averaging over
the volume, U0α
, of the α phase within the, U0
, REV volume dictates
the relations,
1 2 1 2 1 2 1 21 2
; ,e e e e e e e e ee e
α
α α α α α α α
−
= = + + = +
(1)
Where
00
1
( )i i
U
e e dU
U αα
α
≡ ∫ denotes the i intensive macroscopic
quantity of the α phase, spatially averaged over U0α
and ( )i i ie e e
α
= −
denotes its deviation from ie
α
. The Left Hand Side (LHS) of can,
e.g., refer to the microscopic phase extensive quantity (E) being the
momentum (E≡M) vector (i.e. eM
=ρV; e1
≡ ρ phase density; e2
≡V phase
velocity vector).
Modified rule (Bear and Bachmat, 1990) for averaging a spatial
derivative reads
0
* *
0
1 1
, ,
j
aj S i j aj S j
ji
e e e
T x ds T x ds
x x U x U
α α
αβ αα
α α
ξ ξ
θ θ
∂ ∂ ∂
≅ + ∫ ≡ ∫
∂ ∂ ∂
(2)
where
x denotes a distance vector from the REV centroid to
the microscopic interface surface (Sαβ
) between the α and β phases,
θα
≡U0α
/U0
denotes the volume ratio associated with the α phase,
e eα
α
θ= with
00
1
( )
U
e edU
U α
≡ ∫ denoting average of the α phase over a REV,
ξ denotes a unit vector outward to an interface and T*
α
denotes the α
phase tortuosity tensor associated with ( Sαα
) the α- α part of the REV
enclosing interface.
Note that for E≡U0α
we obtain e (≡dE/dU0α
) = 1 for which 1e
α
=
leading to e αθ= . While (2) is authenticated, the non-modified rule
0
1
i
i
i S
e e
x x
dS
U αβ
ξ
∂ ∂
+
∂ ∂
= ∫ for averaging a spatial derivative, when E≡U0α
, results
with the relation
0
1
.i
i S
ds
x U αβ
αθ
ξ
∂
= −
∂ ∫
In [2-4] it was assumed that:
Assumption 1 [A.1] The microscopic interface surface between
two adjacent phases is material to the extensive quantity flux of the
phases.
Assumption 2 [A.2] The phase intensiv,e,e, qsantity accounts for
a high Strouhal number Ste
»1
/
,
e e
e c c
c
L t
St
V
≡ with characteristic quantities
Le
c,
te
c,
Vc
of length, time and phase velocity, respectively).
We note that by virtue of [A.2], the hydrodynamic derivative of
a macroscopic quantity can be approximated by its corresponding
temporal derivative at a fixed frame of reference, and together with
[A.1] it is replaced by the macroscopic temporal derivative (i.e.,
D
Dt t t
e e e∂ ∂≅ =
∂ ∂
, respectively).
In view of (1), [A.1], [A.2] and elaborating on Bear and Bachmat
(1990), the general macroscopic balance equation of a phase extensive
(E) quantity reads,
(3)
in which
α
V denotes its macroscopic velocity vector, [ ( ) ]
E E
e
α α
−≡J V V
denotes the macroscopic diffusive flux vector obtained by empirical
relation, habitually, following a potential flux pattern ΓE
denotes the
rate of generating the phase extensive quantity per its unit mass. Note
that (3) accounts for 1 2 1 2[ ( ) ]e e e ee e e e
α α
α α α α
α
α α
= + = ++V V V V V
the macroscopic
flux vector of the phase extensive quantity which is decomposed into
the advective flux ( )e
α α
V , the dispersive flux ( )e
α
V
e and 1 2e e
α
α
V
that
accounts for the deviation from the advective flux. Supplement to [1]
and following [4], we write,
Assumption 3 [A.3] Phase motion is not of a Brownian type,
as the vanishing of its macroscopic intensive quantity will cause the
disappearance of the deviation from that intensive quantity (i.e.
0 0, 0e e iff e
α
= ⇒ = =
).
Hereinafter, for the sake of reducing the overload of notations,
we will refer to the α phase macroscopic quantities without the
( ) ( )
00
1
[ ]
U
dU
U α
α
α
⋅⋅ ≡ ⋅⋅∫ designator.
By the premise of [A.3], in terms of the α phase macroscopic
quantities for any spatial scale, we write
(4)
in which ΛE
denotes a factor (e.g. obtained empirically) that,
depending on the e quantity, can be a scalar, an element of a vector or
of a tensor. By virtue of (4) we can thus relate the dispersive flux in (3)
to the advective flux, reading
(5)
where Λv
is a tensor quantity. Addressing (3), the phase
macroscopic quantity of the product of deviations from their
corresponding intensive quantities (in what follows denoted by dE
)
can by virtue of (4) be related to the product of their macroscopic
Figure 1: Examples of porous materials: (a) Beach sand; (b) Sandstone; (c)
Limestone; (d) Rye bread; (e) Wood; (f) Human lung; (g) Spheres packing for
granular structures; (h) Crushed limestone for construction. (After D. A. Nield
and A. Bejan, “Convection in Porous Media”, Springer, 2006).
1 2 1 2 1 2 1 2[ ( )] [ ( ) ( )] EE
e e e e e e e e e
t
α α α
αα α α α α
α
α
α α αθ θ θ θ ρ
∂
= − ⋅ + +
∂
Γ+ + +V V J
∇
Ee e= Λ
V V; ,Ee e= ⋅ Λ = ⋅V A V, A = V V
Λ Λ
3. Austin J Hydrol 2(1): id1011 (2015) - Page - 03
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quantities,
1 2 12 12 11 2 2,E
e e e ed e e e e≡ = Λ Λ ≡ Λ Λ ⋅
(6)
Referring to (3) we can make an assumption that addresses the
deviation from the advective flux.
Assumption 4[A.4] The product of intensive quantities is
dominant over that of their corresponding deviations 1 2 1 2
.e e e e
≫
As by virtue of (6) we have 121 2 1 2 ,ee e e e= Λ
the [A.4] assumption
leads to 12
1eαε ≡ Λ from which we obtain 12
, 1e e eB BαεΛ = = ⋅
On the premise of (6) and [A.4], we rewrite (3) as a balance
equation addressing the advective, dispersive and diffusive fluxes, the
assumed smaller term associated with deviation from the intensive
quantity (εα
θBe
e1
e2
) and its corresponding assumed smaller term of
deviation from the advective flux (εα
θBe
e1
e2
V),
(7)
Referring to (3), different forms of balance equations will
emanate subject to assumptions made when comparing between
decompositions of the phase extensive quantity flux [ ]E E
e e e= + +V V V J
(e.g. neglecting the momentum dispersive flux, when addressing
medium heterogeneity). Let us thus consider the assumption that,
Assumption 5[A.5] The advective flux is dominant over the
dispersive flux .e eV V
≫ As by virtue of (5) we have e e⋅=V A V
, the
[A.5] assumption leads to ε≡ A <<1 so that , 1.ε =A= B B
On the premise of (5) , [A.4] and [A.5] we rewrite (3) as a balance
equation addressing the advective and diffusive fluxes, the assumed
smaller term associated with deviation from the intensive quantity
and the assumed smaller terms of deviation from the advective flux
and the dispersive flux (θ(εα
Be
I+εB).e1
e2
V),
(8)
where I denotes the unit tensor.
As both solutions of (7) and (8) when ε≈0(εα
) are functions of ε
<< 1, we can expand their dependent variables using terms associated
with a polynomial of ε powers. Approximate solutions to (7) and (8)
can thus be obtained by solving an infinite set of the corresponding
balance equations of descending order magnitudes. While the first
(i.e. when ε =0 ) balance equation may be nonlinear and of the highest
order of magnitude, the others are linear balance equations as they
are combined with variables obtained from the solution of equations
that are of higher order magnitude. In view of (7) we hence obtain the
general primary (dominant) macroscopic balance equation for any α
phase corresponding to the REV (larger) length scale in the form of
(9)
for which the combined, respectively, dispersion and diffusion fluxes
( )E
e +V J
habitually follow a macroscopic extension suggested for the
empirical microscopic law associated with JE
. This commonly has
the pattern of a potential flow, and thus (9) conforms to a PDE with
hyperbolic or parabolic characteristics.
By virtue of (6) we obtain the equation of a lesser order of
magnitude than (9), i.e. the secondary macroscopic balance equation
emerging from (7) and assumed to correspond to a length span much
smaller than that of (9), reads
(10)
We note that (9) and (10) are based on a different premise and
thus can represent different characteristics. At the REV (larger)
spatial scale and subject to different assumptions, the governing
balance equation (9) can accommodate pure advection, advection
– dispersion or pure dispersion fluxes. The secondary governing
balance equation (10) describes the concurrent pure advection
(i.e. hyperbolic PDE characterized by a sharp front migration of
the state variable) of the deviation (dE
) from the intensive quantity
and its corresponding deviation from the advective flux, at a spatial
scale much smaller than that of the REV. The solution for dE
of (10)
depends also on V, obtained from the solution of (9). Moreover, one
can seek further resolution modes of dE
and V, via an iterative process
between (10) and the combined form of (9) and (10) [i.e. the original
form of (7)].
By virtue of [A.5], another possible form emanating from (8) is a
general primary (dominant) macroscopic balance equation for any α
phase valid at the REV (larger) length scale that reads
(11)
By virtue of , for ( )0 α
ε ε≈ and on the basis of (8), concurrent
to (11) yet with a smaller order of magnitude and at a length span
much smaller than the typical length scale of the REV, the secondary
macroscopic balance equation for any α phase reads
(12)
We note that unlike (10) and by virtue of [A.5], deviation from
the advective flux 1 2( )e e V
and the dispersive flux( )e V
, both assumed
smaller than the advective flux (e1
e2
V), are accounted for in (12). The
discussion and procedure referring to (7), (9) and (10) corresponds,
respectively, also to (8), (11) and (12). Sorek et al. [4] demonstrate the
form and validity of the implication of (9) and (10) its corresponding
secondary balance equation, as well as (11) and (12) its corresponding
secondary balance equation. These two fundamental group forms (9),
(10) and (11), (12), address, respectively, in [4] the balance equations
of a component mass transport and phase energy (accounting for the
advective and dispersive fluxes, when addressing their larger scale)
and phase momentum (neglecting its dispersive flux for its larger
scale).
Spatial scaled momentum balance equations of a phase
Following[2,4],herethephaseextensivequantityisitsmomentum
(i.e. E≡M), for which its macroscopic quantities are eM
≡ρV (i.e. e1
M
≡ρ
e2
M
≡V), the momentum diffusive flux JM
= -σ (σ-phase stress tensor)
and the momentum source flux ΓM
=F (F- specific body force acting
on the phase). Following (6) and [A.4] we consider that ρ ρ = M
V V
d .
Thus, by virtue of (11), the primary macroscopic momentum balance
equation of the α phase at the larger spatial scale of the typical REV
length reads
(13)
By virtue of (12) and in view of (5) and (6), the concurrent
( ) ( ) ( ) ( )1 2 1 2 1 2 1 2 ,E E
e ee e B e e e e B e e
t t
α αθ ε θ θ θρ θ ε ε
∂ ∂ + = − ⋅ + + Γ − ⋅ + ⋅ ∂ ∂
V J I B V∇ ∇
( )1 2 1 2
E E
e e e e e
t
θ θ θρ
∂
= − ⋅ + + + Γ
∂
V V J
∇
( ) ( ).E E
d d
t
θ θ
∂
= − ⋅
∂
V∇
( ) ( )1 2 1 2
E E
e e e e
t
θ θ θρ
∂ = − ⋅ + + Γ ∂
V J∇
( ) ( ){ }12
1
E E
ed d
t
θ θ
−∂ = − ⋅ + Λ ⋅
∂
I A V∇
( ) ( ) ( )1 2 1 2 1 2 1 2[ ( )] .EE
e ee e B e e e e e B e e
t t
α αθ ε θ θ θ ε θρ
∂ ∂
+ = − ⋅ + + + − ⋅
∂ ∂
ΓV V J V
∇ ∇
( ) ( )
t
θρ θ ρ θρ⋅
∂
=−
∂
− + V VV F∇ σ
4. Austin J Hydrol 2(1): id1011 (2015) - Page - 04
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secondary momentum balance equation of the α phase at a scale much
smaller than that of the typical REV length, reads,
(14)
Where, as in (12), the momentum dispersive flux ([ ]ρV) V
is
accounted for in (14). Letting the α phase represent a Newtonian
fluid (i.e. its diffusive momentum flux reads σ=τ-PI, where P denotes
pressure and the constitutive potential flow pattern for the shear
stress is, ( )T
µ λτ= + + ⋅ V V VI∇ ∇ ∇ , for which µ and λ denote, respectively,
the first and second viscosities), then the macroscopic Navier-Stokes
equation resulting from (13) can govern the propagation of shock
waves through porous media [7], conform to Forchheimer’s law
accounting for the transfer of fluid inertia through the microscopic
solid-fluid interface or to Darcy’s law when friction at that interface
is dominant [2]. At the much smaller scale, the hyperbolic PDE of
(14) describes an inertia momentum balance equation in the form of
a wave equation that governs the concurrent propagation of the dM
quantity.
Spatial scaled energy balance equations of a fluid
Following Sorek et al. [4], consider a porous medium saturated
by a Newtonian fluid with heat (H) as its extensive quantity (E≡H) for
which its intensive quantity is eH
≡Cv
ρT (i.e. eH
1
≡ cv
ρ, eH
2
≡T) where Cv
denotes the fluid constant specific heat at fixed volume, T denotes its
temperature and let us also account for ΓH
denoting the specific heat
source. Following Bear and Bachmat [1] and in view of (1) and (3)
we obtain the fluid macroscopic heat balance equation
(15)
where φ denotes porosity, the fluid dispersive heat flux )(c Tυ ρ V
and
its diffusive heat flux JH
are assumed of the same order of magnitude
and (following Fourier’s law, using a potential flow pattern) are
assembled into ( )[ ]H
c cT Tυ υρ∗
⋅ = +V J
− ∇Λ with ∗
Λ denoting the combined
constant thermal dispersion and diffusion (conductance) tensor.
Note that (15) accounts for the fluid viscous dissipation heat flux
(τ: ∇ V), although habitually considered negligible. The fluid heat
flux associated with volumetric deformation ( )[ ]T P T υ
∂ ∂ ⋅V∇ at fixed
volume (v) is expressed in terms of the constant fluid compressibility
coefficients (βT
/βP
)v
with (1 )( ][ )P T
Pβ ρ ρ≡ ∂ ∂ due to its pressure
(P) change at fixed temperature and (1 )( ][ )T P
Tβ ρ ρ≡ ∂ ∂ due to
thermal change at fixed pressure. In view of (9) and (15), we obtain
the primary macroscopic heat balance equation in the form
(16)
In view of (10) and (15), at the smaller scale, we obtain the
secondary macroscopic heat balance equation in the form
(17)
for which H
d c Tυ ρ≡
denotes the deviations product from the
intensive heat quantity. We note that while (16) is characterised by
advective and dispersive heat transfer, concurrent at the smaller scale
(17) addresses pure advective flux of heat transfer.
Spatial scaled component mass balance equations in a
fluid
In Sorek and Ronen [3] we consider a γ component for
which mγ
denotes its mass so its extensive quantity is E ≡ mγ
with
the intensive quantity ( )m
e C
γ
ρω≡ = given by its concentration
( )1 2 1 2; ,m m m m
C e e e e
γ γ γ γ
ω ρ= ≡ ≡ related to its mass fraction (ω), and let us
account for the specific component source ( )mγ
Γ . Let us consider
a saturated domain, following Fick’s law based on a potential flow
pattern we combine the component diffusive flux vector ( )mγ
J and its
dispersive flux vector ( ])[C ρω=V V
into ( )m
h C C
γ
− ⋅ = +D V J
∇ for which Dh
denotes a hydrodynamic dispersion tensor. In view of (9), at the REV
spatial scale, we obtain the primary component mass balance equation
in the form
(18)
We note that (18) describes the transport of a component
subject to advective and dispersive flux mechanisms, interpreted by
hyperbolic and parabolic PDE characteristics, respectively.
In view of (10), at the smaller scale, the secondary component
mass balance equation becomes
(19)
where ( )m
d
γ
ωρ≡
denotes the deviations product from the component
concentration. We note that concurrent to the advective dispersive
transport described by (18), the transport at the adjacent scale,
significantly smaller than that of the REV typical length, is governed
by (19) which describes a pure advection mechanism.
Supporting observations of particles migration at the
scale of several pores
We note that the form of macroscopic balance equations at the
smaller (e.g. several pores) scale is proven to be hyperbolic PDE’s. This
suggests a “bulk flow” phenomenon, at that scale, of moving particles
aggregated by a shock wave drive. We suggest that this “bulk flow”
mechanism is governed by a wave equation for the fluid momentum
balance equation and pure advection transport for the particle’s mass
balance equation. Numerical predictions or simulations addressing
a sensitivity analysis cannot alter the basic characteristics of the
balance hyperbolic PDEs at the smaller scale. We thus maintain
that validation of the theoretical development should at first rely on
verifying that observations at the several pores scale comply with
features of the theoretical model at that scale. Field observations
under natural gradient flow conditions (specific discharge of 4 to
16/myr and 35% porosity) in a contaminated sandy aquifer show,
micro-scale variations in the flux, mineralogical composition and size
of suspended particles [9,10]. Along a 16 m saturated section below
the water table the average concentration of particles in groundwater
varied between 1 and 40 mg/L, but high concentrations of up to 5000
mg/L were also detected. The particles were composed of CaCo3
(11-
57%), quartz (7-39%) and clays (8-43%). Most of the particles were
within the 140-3000 nm size range with size modes varying from310-
660 nm. The transport of large amounts of particles under the above
mentioned natural flow conditions within a porous medium (average
grain size of 125 nm) is striking. Motion of different particle parcels
observed along the depth of a well and sampled at different time
periods, under natural flow conditions and due to a sudden single
pumping excitation (100 L/mm), are depicted in Figure 2. We note in
Figure 2 that the single pumping caused the rise of colloids turbidity
( ) ( ){ } ( )
1
V V V,M M
d d
t
ρ ρθ θ
−∂
= − ⋅ + ⋅ ≡ Λ Λ Λ ∂
I V∇ Ω Ω Λ
( ) ( )*
: HT
P
c T c T T T
t
υ υ
υ
β
φ ρ φ ρ φ φ φρ
β
∂ = − ⋅ − ⋅ + − ⋅ + Γ ∂
V V V∇ Λ ∇ ∇ ∇τ
( ) ( )V m
hC C C
t
γ
φ φ φρ
∂
= − ⋅ − ⋅ + Γ ∂
D∇ ∇
( ) ( ),m m
d d
t
γ γ
φ φ
∂
= − ⋅
∂
V∇
T T
t
c T T c Tυ υφ ρ ρ φ ρ ρ φ ∗
+ +
∂
− ⋅
= − ⋅ ∂
V
∇Λ∇
( ) ( ): : HT
P
T T
υ
β
φ φ φρ
β
+ ⋅ ⋅
+ − + + Γ
V V V V
∇ ∇ ∇ ∇τ τ
( ) ( ) ( ) ( ): ,H H T
P
d d
t
T
υ
β
φ φ φ φ
β
∂
= − ⋅ + − ⋅
∂
V V V
∇ ∇ ∇τ
τ
5. Austin J Hydrol 2(1): id1011 (2015) - Page - 05
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(a measure of the degree to which a fluid loses its transparency due to
the presence of suspended particles). This can be explained in view of
(13) concerning temporal scaling (explained in the following section
3) for the large spatial scale of a fluid momentum saturating a porous
matrix [11], confirmed by simulation of its simplified form [12] and
validated through different experiments [13]. In view of these [11-13],
we maintain that at the start of pumping, inertia temporally governs
the flow regime and a wave driven transport at that instant caused the
rise of colloids turbidity (Figure 2). At the several pores scale, (14)
and (19) suggest that displacement be governed by the combination
of wave influenced motion and pure advective transport mechanisms,
respectively. The several pores scale motion of condensed parcels of
particles that complies with the hyperbolic forms of (14) and (19),
is also demonstrated in Figure 2. Vertical difference of magnitude
in particles concentration (Figure 2) corresponds to two different
natural flow fields reflecting a higher specific discharge at the deeper
depth and thus transporting significantly more particles at that depth
[9].
At the several pores scale, (14) and (19) suggest that displacement
be governed by the combination of wave influenced motion and
pure advective transport mechanisms, respectively. Motion of
fronts (observed as continuous bulk displacement or “piston
flow”) remaining sharp, is a characteristic solution of hyperbolic
PDEs. The motion of condensed parcels of particles exhibited in
Figure 2, noticeably complies with the typical solution featured
by the hyperbolic forms of (14) and (19). Hence, we maintain that
observations depicted in Figure 2 validate that the PDEs of (14)
and (19) govern the hydrodynamics at the scale of several pores.
Vertical difference of magnitude in parcels of particles concentration
(Figure 2) corresponds to two different natural flow fields reflecting
a higher specific discharge at the deeper depth and thus displacing
significantly more particles at that depth [9]. At the pores scale and
for two time snap shots, Figure 3 depicts bulk motion of dominant
mean size (determined by their weight percentage) particles along
depth. Observations depicted in Figure 3 thus attest by the mass bulk
displacements, the hydrodynamics of wave influenced motion and
pure advective transport resulting from the hyperbolic PDEs of (14)
and (19), respectively.
Temporal Scaling of the Transport
Phenomena through Porous Media
In what follows we neglect the secondary balance equations of
the fluid momentum (14) and (19) the component mass. Actually,
such a decision leans towards a qualitative rather than a quantitative
premise. The resolution of the relative influence between the smaller
and bigger spatial scales should rely on quantitative (e.g. numerical
exercises) measures and be investigated thoroughly. We thus
consider as dominant the macroscopic balance equations of the fluid
momentum (13) in the form of the NS equation and refer to (18) for
the mass balance equation of a γ component.
In Sorek et al. [11] we further develop the time evolving
approximate forms of these balance equations, after an abrupt rise
of the fluid pressure. We consider linear equilibrium adsorption
isotherm, deformable saturated porous matrix and a varying fluid
density ρ = ρ (P,C). On the basis of dimensional analysis we find [11]
that the balance equations of the fluid momentum and mass and
the component mass, conform to different dominant forms as time
evolves after the onset of the abrupt pressure rise. We will specifically
draw our attention to the first two evolution periods as well as
summarize aspects presented in [11].
Figure 2: Particles concentration at different sampling time (4, 15, 23, 49
and 69 days) intervals along a well depth (vertical difference of magnitude
in particles concentration corresponds to two different natural flow fields with
higher specific discharge at the deeper depth), under natural flow conditions
and due to a pumping (100 L/m) excitation, after Ronen et al. [9]. Figure 3: Particle size distribution for sampling intervals of two profiles, after
Ronen et al. [2].
6. Austin J Hydrol 2(1): id1011 (2015) - Page - 06
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Considering a fluid without mass source, without diffusion and
dispersion fluxes of its mass (i.e. , , 0, 0, 0m m m
E m e ρ ρ≡ = Γ = = =V J
), then in
view of (11) we write the fluid mass balance equation in the form
(20)
Following (13) we obtain the fluid momentum balance equation
in the form [14]
(21)
where g denotes gravity acceleration, Z denotes altitude, qr
[≡φ (V-Vs
)]
denotes the specific fluid flux vector relative to (Vs
) the solid phase
velocity vector, cf
denotes a shape factor, ∆ denotes the hydraulic
radius of the pore space and [ ](1 ) )(
fS
ij i jfS
S
ij S dSα δ ξ ξ′ ≡ −∫ denotes the ij
components of a cosine direction tensor at the microscopic Sfs
fluid-
solid interface, and δ denotes the Kronecker delta tensor. Sorek et
al. [2] extend the macroscopic NS equation of (21) when, by virtue
of (21), P∇ through Sfs
is replaced with all of the microscopic NS
equation flux terms ( ( ) [ ( ) ] ).
fS fS
i ij j j k j k j j k j
S S
P x V t V V x gd Z x SxS dx xξ ρ ρ τ ξ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ −∂= ∂∫ ∫
Subject
to some assumptions, Sorek et al. [2] obtain an extended macroscopic
NS equation that reads
(22)
In which 1[ ( ) ( ) ]
fS
ikji kj k j
fs
S
F x ds
S
δ ξ ξ≡ −∫
denotes the kji components of
the Forchheimer 3rd
rank tensor. For a fully developed flow regime
[i.e. 2
( ) ( ) ·ft cφρ µφ ′∂ ∂ + ⋅ ∆V V V V∇ α ] (22) reduces to Forchheimer Law
which together with the assumption of a Reynolds number less than
a unit (22) reduces to Darcy’s Law [2]. Note that it is based on rigors
macroscopic continuum mechanic approach such that at a point (i.e.
within the REV) there is an account for the interaction of at least two
phases also through their common microscopic interface. Thus, e.g.,
Darcy’s Law was proven [2] to be a derivative of the macroscopic
NS momentum balance equation which validated the originally
suggested empirical relation for flow through a porous medium bulk.
One can, however, find microscopic derivation of the equivalent to
Darcy’s Law when it is considered as a phenomenological filtration
relation for, say, membrane technology [15].
Following an abrupt up-rise of pressure, the investigation for the
approximate forms of the balance equations will be sought in terms
of pressure rate, which can be obtained from the differentiations of
ρ = ρ (P,C). Hence, (21) will be substituted by a combined form of
(20) and (21). Considering no mass source for the component and by
virtue of (9), we write the component mass balance equation in the
fluid accounting also for its adsorption on the solid matrix, i.e. the
component transport equation, in the form
(23)
in which Cs
(= ρs
kd
C; where S
ρ denotes the solid density and kd
denotes the partitioning coefficient) denotes the component mass
per unit volume of solid matrix. Analyzing (23) in orders of the
pressure rate magnitude during different time scales, we assume a low
solubility (i.e. concentration is practically sought in terms of the fluid
volume) and that changes in the total stress of the porous matrix are
of the same order as pressure (i.e. ϕ = ϕ (P). To address the transport
phenomena during different time scales [11] we will analyze the
order of magnitude of the terms appearing in the balance equations
set (20), (21) and (23). This set is thus rewritten in nondimensional
forms obtained by subdividing all terms by the one associated with
the pressure rate.
At the first time t1
(=0+
) being the onset instant of the abrupt
pressure rise, starting with fluid at rest with no loss of generality,
we have P C P C
1t t t t ; L L Lc c c c c c= = = ≅ ≅V V
. Where ( )c
denotes a characteristic
quantity, and the characteristic velocity Ve
c
≅ Le
c
/te
c
thus becomes
VP
c
= Vc
c
= VV
c
= Vc
resulting in a Strouhal number being
Following Sorek [11] at t1
the first time scale addressing the up-rise of
the pressure impulse, we obtain:
The fluid mass balance equation conforming to
0,
DP
Dt
= (24)
and the fluid momentum balance equation reads
(25)
Assuming that adsorption is a slow process in comparison to
change in pressure rate such that it does not affect the component
mass, the component transport equation becomes
( )1 0,
DP
C
Dt
φφ β− = (26)
where ( ) ( ){ }1 1 1 Pφβ φ φ ≡ − − ∂ − ∂ denotes the matrix compressibility
and ( )D Dt t≡ ∂ ∂ + ⋅V ∇ denotes the hydrodynamic time derivative
operator. By virtue of (24) and (25) the magnitude of the pressure rise
will propagate without attenuation
1 0
( .)t
P Const+=
= as if through an
incompressible fluid, and (26) is automatically fulfilled. On the basis
of the dimensional analysis the order of magnitude for the pressure
propagation distance will be ( )( )1 1 0 ,c cL 0 t Sφ ρ=
for which ( )0 1PS φφβ φ β ≡ + − denotes the specific storativity of the
porous medium.
At the second consecutive evolution time t2
period, we have
t2
=tP
c
= tV
c
= tC
c
; LP
c
>LV
c
≅ LC
c
leading to, VP
c
> VC
c
= VV
c
= Vc
, however
the velocity Strouhal number (Stv
=1) remains as was at the first time
scale. It is proven Sorek [11] that at this second time scale the fluid
mass balance equation is described by (20) while the fluid momentum
balance equation conforms to a non-linear wave equation that reads,
(27)
The dimensional analysis asserts that the non-linear wave in
the form of (27) prevails during an order of time magnitude being
( )2 ( )c c ct 0 P V gρ= , and the order of magnitudes of the characteristic
pressure ( 2
c c cP Vρ> ) with L2
=0(Pc
/ρc
g) as its propagation distance.
We note from the dimensional analysis [11] that during the time
period starting at 2t , the solution for P and V of (27) can be obtained
separately with no reference to the solution of the component
concentration, as during time increments resolving for the evolution
of P and V of (27) we have ρ ≅ρ(P.) The update, by the end of
each time increment, of fluid density associated with component
concentration (ρ=ρ(P,C)) will be after solving for the component
transport equation.
( ) ( )
t
φρ φρ
∂
= − ⋅
∂
V∇
( ) ( )1 ,S hC C C C
t
φ φ φ
∂
+ − = − ⋅ ⋅ ∂
V - D∇ ∇
L
1.
V
e e
e c c
c
t
St ≡ =
2
2
( ) ( ) ( · )f
r r r
c
P g Z
t
φρ φ ρ µ λ µ∗∂
′+ ⋅ − + ⋅ + + ⋅ + ∇ −
∂ ∆
V
V V)= T q q q∇ (∇ ∇ ∇∇ α
2 2
( ) ( P Z) ·
2
f fc c
g
t
φρ φ ρ φρ µφ∗∂
′+ ⋅ ≅ − + ⋅ − ⋅ −
∂ ∆ ∆
V
V V T V V V∇ ∇ ∇ αF
0Pφ ∗
⋅T∇ =
( ) 0
D P
g
Dt ρ
∗
+ + Ζ ⋅
V
T
∇
∇ =
7. Austin J Hydrol 2(1): id1011 (2015) - Page - 07
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The component transport equation becomes,
(28)
Hence, the pre-evaluation of P and V can be interpreted in (28) as
known source terms. The dimensional analysis suggests that the order
of magnitude of the component concentration spatial distribution
will be to a distance ( )2C
2
cL 0 V g= .
Levi-Hevroni et al. [16] apply a dimensional analysis accounting for
the fluid momentum in the form of (22) to investigate the wave period
starting at t2
(following an abrupt change in fluid’s pressure and
temperature) for the fluid heat transfer and flow problem, and solid
deformationconcerningathermoelastichighlyflexibleporousmatrix.
We note in Figure 4 that the 1-D numerical simulation conducted by
Levi-Hevroni et al. [16] yield excellent agreement with 1-D numerical
simulation for almost non-deformable porous materials Levy et al.
[17]and with shock-tube experimental observations Levy et al.[18] for
the pressure histories of air at various locations along the shock wave
propagation pass. The contact surface of the gas which originally filled
the pores and was pushed out, was found experimentally by Skews
et al. [19] and reported in Levi-Hevroni et al. [16] to be reproduced
numerically. The calculated gas density field and the trajectory of the
foam front edge are depicted in Figure 5.
Burde and Sorek [12] investigated the component transport
during the wave period starting at the second time scale t2
. A
simplified analytical solution is applied for the 1-D case of the set (20),
(27) and (28) where the flow problem is addressed by the traveling
wave (kinematic wave) case (Landau and Lifshitz, 1987) for which
V=V(ρ); P=P(ρ) and thus the fluid mass and momentum balance
equations are rewritten in a similar wave equation form in terms of V
or P. The solid matrix is considered as slightly deformable following
the assumption that V ,St tφ φ∂ ∂ ∂ ∂ in which ( )s
denotes a quantity
associated with the solid porous matrix. Component migration Burde
and Sorek [12] due to the action of a compaction wave (e.g. change in
injection rate) or an expansion wave (e.g. change in pumping rate),
both when being emitted at the surface (x=0), are depicted in Figure
6. Solutions Burde and Sorek [12] are presented as time evolves from
t2
, and the component displacement is continuously governed by the
wave form of (27).
Simulations findings of Burde and Sorek [12] for migration of
a component governed by continuous propagation of a shock wave,
was verified to be consistent with solutes displacements observed
through various experimental setups under the action of emitting a
succession of compaction waves [13].
Further for a deformable matrix, during the compaction wave
period at the second time increment, on top of the fluid mass (20) and
momentum (22) balance equations the solid mass and momentum
balance equations read respectively [21].
(29)
(30)
where σα
(α≡ f, S) denotes the α phase stress tensor with σ’s
as
the matrix effective stress, and the phases tortuosity tensors are
related by *
(1 ) S
φ φ∗
+ − = δT T . Considering an elastic matrix undergoing
deformations we refer to a constitutive relation for, σ’s equivalent to
the microscopic Hooks law, and a strain tensor eS
that read,
(31)
where Sµ′ denotes the macroscopic Lamé constant related to the
matrix shear strain, I denotes the unit tensor and ws
denotes the matrix
displacement vector. Referring to the set of fluid , (20), (22) and matrix
(29) - (31) mass and momentum balance equations together with the
solute mass balance (23) equation, the 1D component displacement
due to compaction waves propagation are solved numerically
using the Total Variation Diminishing (TVD) scheme [21]. Solute
extraction is simulated in view of applying at the medium surface a
sequence of pressure pulses each generating an expansion wave. The
inwards expansion wave generates a pressure gradient contradicting
the direction of the wave propagation. The fluid flow governed by
the pressure gradient, will thus displace the component towards
Figure 4: Experimental results and numerical predictions [dotted line - Levy
et al. [17] and solid line - Levi-Hevroni et al. [16] of the gas pressure histories
(after Levi-Hevroni et al. [16] using a 40 mm long silicon carbide sample with
10 pores/Inch and average porosity of 0.728 ± 0.016. Incident-shock wave
Mach number in this experiment was MS
=1.378. (a) Upstream 43 mm ahead
the porous material front edge. (b) Along the shock tube side-wall inside the
porous material, 23 mm from end-wall. (c) At the shock tube end-wall.
0
0
1 1 P s d
s d
SDC
P
C Dt S
φ
φβ ρ κφ
β ρ κ
φ
+−
= − ⋅ − ⋅
V V∇ ∇
( ) ( )1 1 0,s s s
t
φ ρ φ ρ
∂
− ⋅ − = ∂
+ V∇
( )( )1 ,s s fφ= − −'σ σ σ
( ) ( ) ( ) * '
2
1 1 1 ,
2
0f
s s s s s s sP
t
c
φ ρ φ ρ φρφ
∂
− + ⋅ − + − ⋅ − ⋅ − = ⋅
∆ ∂
V V VV V∇ ∇ ∇ FT ó
( )' 1
2 , ,
2
T
S S S S S S S Sλ µ ′⋅ + = +
∇ ∇ ∇σ = w I w we e
8. Austin J Hydrol 2(1): id1011 (2015) - Page - 08
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the medium surface. Using such a procedure [21] the component’s
mass extraction is compared between pumping and application
of expansion waves emitted from the surface. Pumping refers to
Darcy’s law for which drag is assumed dominant at the solid-fluid
interface, neglecting gravitational body force, the matrix momentum
balance equation (30) is assumed static and (31) accounts for its stress
constitutive law. (Sorek and Ohana, 2009) develop an approximate
analytical solution for the specific component’s extraction mass
flux Qsolute-p
for constant pumping intensity at the surface. Their
comparison (Table 1) with Qsolute-w
for specific solute extraction mass
flux due to emitting expansion waves proves the overwhelming
advantage of the latter.
Bear and Sorek [5] discuss two additional forms of dominant
governing fluid mass and momentum balance equations due
to the onset of an abrupt pressure rise. Following the first two
aforementioned evolution periods, at the start of the third ascending
time scale, fluid’s inertial and drag momentum fluxes are of the
same order expressed by the full extent of NS equation that can also
conform to Forchheimer Law [2]. During the fourth evolution time
period, drag associated with viscosity dominates the flow regime
and inertial momentum flux becomes negligible in comparison. The
resulting viscous creep flow is expressed by the Brinkman momentum
equation, when friction between two adjacent fluid layers dominates,
or by Darcy’s linear momentum, when friction at the solid-fluid
interface dominates [2].
Conclusion
The macroscopic balance equations describing the transport
phenomena through porous media are spatial and temporal
dependent. Space wise each balance equation is decoupled into
a primary equation addressing the REV length span scale and a
secondary equation, concurrent to the primary one, valid at a length
scale smaller than that of the REV. The latter is associated with
intensive quantities deviating from their corresponding averaged
Figure 5: Calculated gas density field and front edge trajectory of the flexible
elasto-plastic porous sample (after Levi-Hevroni et al. [12]. We note the
contact surface due to gas emerging from the sample porous matrix, after
it had reached its utmost deformation. This very large deformation depleted
the sample pores and forced the gas to flow across the sample/gas interface.
Simulation results are similar to the experimental observations of Skews et
al. [19].
terms. The primary balance equation can conform to a pure
hyperbolic PDE addressing advective fluxes, PDE with hyperbolic-
parabolic characteristics expressing advective-diffusive (with/without
dispersive) fluxes or pure parabolic PDE governing diffusive (with/
without dispersive) flux mechanisms. The secondary balance equation
remains a pure hyperbolic PDE for any extensive quantity of a phase
or component. The secondary macroscopic balance equations are
habitually neglected, yet these can be significant in cases of deviation
from the average fluid velocities and/or for heterogeneity in fluid
and matrix properties. Validation of the theoretical development
relies on verifying that field observations under natural gradient flow
conditions indicate that particles are displaced in parcels at several
pores scale and thus comply with features of the mass and momentum
balances at that scale.
Approximate macroscopic mass and Navier-Stokes fluid balance
equations and the component mass balance equation through
different time scales were developed following the onset of an abrupt
pressure change. Numerical simulations were consistent and in
excellent agreement with experimental observations.
Figure 6: 1-D simulations, for a general unit system, of a component
migration through a saturated porous medium, as time evolves starting at the
second time scale following an abrupt up-rise of pressure Burde and Sorek
[12]. With ( )0
that denotes a value at the reference x=0, t=0, solutions are also
depicted in terms of 0
(component's mass in the fluid) (unit volume of porous m.)
(same values at the reference x=0, t=0)
( )f CC φ φ
≡
,
of χ≡ρS
kd
that addresses a measure of adsorption, of 0 0
[(1 ) ]P TSr φ λ ∗
≡ − that
provides a measure of the matrix stiffness ( λS
denotes the Lame’ coefficient,
analogous to Young’s elastic modulus of Hooks law), and for values of
0 0
(component's mass in the p. m.) (unit volume of p. m.)
(same values at the reference x=0, t=0)
(1 )
(1 )
Pm CC
φ χ φ
φ χ φ
+ −
≡ + −
. Waves are being emitted
at the surface (x=0), with C0
(x,0) denoting the initial concentration distribution
and for χ = 0.25, r =0.25 . (A) Temporal concentration along depth under the
action of continuous propagation of a compaction wave. (B) Temporal mass
extraction subject to the action of continuous expansion wave propagation.
Depth[m]
Pervious Semi-Pervious Impervious
Sand & Grevel;
Frectured Rocks
Silt; Loess;
Layered clay; Oil
Reservoir Rocks
Unweathered
clay;
Limestone;
Dolomite
Water 30 0.06
5.0*E-05 to
5.0*E-07
5.0*E-09
Air 10 0.002
2.0*E-06 to
2.0*E-08
2.0*E-010
Table 1: Values of Qsolute-p
/Qsolute-w
for typical matrix properties.